New theory of odd numbers 8th problem of Hilbert

Transcription

1 New theory of odd numbers 8th problem of Hilbert F r a n c e 2 3 a v e n u e d e W a i l l y C r o i s s y WOLF François;WOLF Marc A new theory on odd composites and odd primes is presented. This theory is built on a new repository of odd numbers named space W. This space allows to distinguish the composite numbers (UNNP) from prime uneven numbers (PUN). The organization of these odd numbers is determined from structural elements of the space W. These elements allow us to understand the distribution of prime numbers, and also the oscillatory and the fractal properties of odd primes. Many conjectures are explained: the Riemann hypothesis, the Goldbach conjecture, the twin prime conjecture, conjecture Cramer and Legendre. F r a n c o i s. w o l t s o f t e m a i l. c o m m a r c. w o l t s o f t e m a i l. c o m 1 6 / 1 2 / Auteurs : François et Marc WOLF Page 1

2 Table of Contents Theory W : study of odd numbers: prime and composite numbers Chapter I Characterization of odd primes... 6 Introduction Context Construction of space W Structure of the space W Organisation of odd numbers Features of the remarkable numbers remarkable points in space N Common points between the sequences Primality test in space W Structure of a «base unit» Definition of a base unit Feature of a base unit Singular points inside each base unit These points are divisible neither by «3» nor by «5» These points do not belong to a couple of twin prime numbers Natural Period of primes Determination of prime numbers PUN and composite numbers UNNP Determination of the natural period of primes Density of primes Basic pattern Combined patterns and Goldbach s conjecture Distribution of primes: Brocard's conjecture and Legendre's conjecture Mathematical characterization of odd primes - Riemann Hypothesis Riemann Hypothesis How to characterize an odd prime number using trigonometric functions? Method for determining the twin primes (Twin Prime Conjecture) Origin of twin primes Derivative Space W Study of the first term of the basic schema Determination of the equation of twin primes Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 2

3 5.3 Resolution of the equation Definition of the condition 1: Resolution of the condition 1: Definition of the Condition 2 : Mathematical form of twin primes Mathematical form of a pair of twin primes Mathematical form of the set of prime numbers Mathematical form of the composite numbers Formula to count the pairs of twin primes Enumeration of twin primes Formula of the twin primes Determination of the evolution of the number of twin primes in base units Determination of the evolution of the number of twin primes in the natural periods Conclusion Representation of prime and composite numbers by binary values Study of the sequence of points j= Study of the sequence of points j= Study of the sequence of points j Decomposition of an odd number into prime factors Odd Primes and twin primes Description of the distribution of binary values "11" Formulas connecting prime numbers Formulas connecting the pairs of twin primes Twin prime conjecture Détermination of the parameters Determination of the formulas for the parameters (A) and (B) Formula f(j) for (A) Formula f(j) for (B) Enumeration of prime factors Determination of the coefficient a B a Alternating series: a B b Convergence of the alternating series: a B c Alternating series: estimation of the value of a B Infinity of prime numbers Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 3

4 5.9.4 Determination of the formula f(j) for the parameter (C) Enumeration of the pairs of prime factors Determination of the coefficient a C a Alternating series: a C b Convergence of the alternating series a C c Alternating series: estimation of the value of a C Infinity of the pairs of twin primes Conclusion Method for determining the couples of prime numbers that compose an even number (Goldbach's Conjecture) Determination of Goldbach's equation Definition and context Definition of the equations Resolution of Goldbach's equation Constraints Solution The foundations of a mathematical proof Determination of the mathematical condition Numerical computations Conclusion Examples The even numbers that are studied: 10, 14, 36, Decomposition of the even numbers according to the mathematical form of the even number mathematical forms of primes that correspond to solutions of Goldbach's equation 226 Conclusion References Indexes of graphics APPENDIX 1 : Distribution of points PUNv within the natural period of the basic pattern Mt(1) APPENDIX 2: representation of the points connecting to the 15 formulas giving birth to the first prime number of a pair of twin primes APPENDIX 3: results of the study of < APPENDIX 4: result of the calulation of the elements Un of the aleternating series ab according to the parameter Nmax APPENDIX 5: number of main combinations of pairs of prime factors according to the number of the pairs of primes NC for Nmax= Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 4

5 APPENDIX 6: Algorithm to enumerate the combinations of pairs of prime factors APPENDIX 7: result of the calculation of the elements Unc of the alternating series ac according to the parameter Nmax APPENDIX 8: table with the pair of values (1, Ni) APPENDIX 9: Evolution of the value (B) according to the parameter j Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 5

6 Chapter I Characterization of odd primes The purpose of this study is to answer the following questions: - What are the structural elements for understanding the organization of primes? - What is the distribution of prime numbers? - How to determine exact formulas? - How to demonstrate conjectures? The object of the study is mathematical, but the process is scientific. We defined a workspace W devoted to the odd numbers. In order to study the properties of the odd numbers, a measurement interval and a measurement unit are required. Indeed, the current study of prime numbers has only one measuring interval whose boundaries are 0 and infinity. The results obtained correspond to the behavior of the properties of prime numbers such as, when N tends to infinity. The study of the evolution of the properties of the odd numbers within this interval has to highlight the structure of these numbers. Introduction The prime uneven numbers represent a subset of odd numbers. The knowledge of the organization and structure of odd numbers is essential to better understand the properties of odd primes. The study of the organization and structure of odd numbers requires the definition of a specific new workspace that is named space W. This space with twodimensional structure shows the regular structure of the odd numbers. It allows to study the properties of odd numbers and therefore those of odd primes. Odd numbers are decomposed into two categories: the uneven numbers non-prime called as composites (UNNP) and the prime uneven numbers (PUN). In the paragraphs that follow, we will show that these odd numbers have a structure consisting of three structural elements : - The points UNNP are characterized by an affine parametric function which we call basic schema (first structural element). This basic schema will allow the introduction of trigonometric functions that characterize the points PUN. - The study of the properties of the odd numbers is realized within a structure called "base unit" (second structural element). This structure serves as a measurement interval. - The evolution of these properties is related to the basic pattern (third structural element). Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 6

7 This structure and these elements are used to validate conjectures such as the Riemann hypothesis, the twin prime numbers, Legendre, Brocard. But it also explains other conjectures like Goldbach's conjecture and Cramér's conjecture. Note: The graphics, as well as verification of theoretical and numerical results are made with computers MAPLE v A specific program in C / C ++ was carried out to evaluate the performance of new formulas and new algorithms for counting prime numbers. The following definitions are used in this study. Formula/abréviations Representation/Explanations Comments [ ] brackets in a formula The brackets in a formula corresponding to the integer part of the value. Brackets are also represented as a function : Set N, or N This is the set of natural integers. When we use the reference to a space or a set, the letter designating that Space N The set of odd positive natural integers greater space / set is indicated in bold. than "1" is defined as the space N. In this study, only the odd numbers are studied. Space W This space is composed by the index k of the odd numbers. An index k is also named one point of the space W. The value of k represents the rank of an odd natural integer greater than zero «in average» This term is used to describe the evolution of the properties of the odd numbers. The study shows and explains why the values of these properties vary. UNNP Uneven number non-prime PUN Prime uneven number UNNP is the abbreviation of an odd number which is non-prime. This is a composite number or composite. PUN is the abbreviation of an odd number which is prime. This is a prime number. Table 1: Definition of some terms used in this study (Lexicon) 1- Context The numbers generated in the space W correspond only to odd numbers. One value of «k» corresponds thus to an odd number in the space N. A point UNNP is the index of this odd number non-prime in the space W. A point PUN is the index of this odd number prime in the space W. A prime number is a natural number that has exactly two distinct positive integers as divisors (1 and itself). Thus, neither the number 1 is prime because it has only one positive integer divider nor the number 0 because it is divisible by all integers. The first odd prime is then the number 3. The even and odd numbers are divided into three groups. The composite numbers, the prime numbers and the neutral elements : zero and one. The primes are constituted by a single even number equal to «2», and an infinite number of odd numbers. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 7

8 Even numbers are excluded. The issues raised in this chapter is to determine the odd primes. The function representing an odd number in the set of natural numbers N is as follows: with An odd number «Ni» is indexed by the parameter «number.» that represents the rank of this Note : The indexes of two consecutive odd numbers are distant of one unit. Hence the consecutive index of an odd number with the index is. Two operations are performed on the function : - The number «1» is excluded because it corresponds neither to a composite number nor at a prime number. The first odd-number of the function is then the number «3». Thus the function Ni(k) becomes : with Équation 1 : relationship between the index k of an odd number and its value in the space N The relationship between the indexes and is : - The parameter «k» corresponds to a parametric function affine. This is an arithmetic progression. This parameter is obtained by the following formula: with and the parameter «j»,. Équation 2 : function corresponding to the basic schema of the space W Hence The function is the base of the construction of the workspace which will be called space W. This function represents the set of indexes of the odd numbers of space N, with a lag of one unit because the first odd number "1" is excluded. The first index corresponds to the first odd number. The numbers generated by this function only correspond to the odd numbers of space N. This workspace W offers two types of numbers: - the uneven number non-prime (UNNP) or odd composites, - the prime uneven number (NIP) or odd primes. 2- Construction of space W The space W represents the values of the index of an odd number of the space N. This index corresponds to the index given by the equation 1. This equation makes the link between the space W and the space N. One expresses the relationship between the index and the odd number, using the term "correspondence" : Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 8

9 «The number corresponds to an odd number PUN or UNNP.» The values of the index correspond to series of points k obtained through the parameter «n». The values of the parameters belong to the set of natural numbers. For each value of the parameter «j», a series of points is generated by the linear function given by Equation 2. This serie of points is written with the following notation:. Each series of points corresponds to a sequence «j». The space W consists of two axes: - an x axis named k. One notes on this axis the values provided by the series. - a y axis named j. A linear function is defined for each value of the parameter «j». This function generates a series of points with the parameter «n». Definition of basic schema: This is the first structural element of odd numbers. Only the odd numbers «Ni» greater or equal to «3» are taken into account. These numbers are shown on the graph below in blue. Figure 1 : Graphical representation of odd integers in the space N To build the space W, the axis of abscissas x named «k», is taken as the axis describing the indexes of the odd numbers, with the exception of the first odd number «1». The index of numbers belonging to the space N is represented by the parameter k in the space W, as shown schematically in the figure below. Figure 2 : Graphical representation of the odd integers with their index in the space W The odd multiples of odd numbers are connected in the space W by following affine functions : The series, corresponding to odd multiples of the odd number «3» is given by the following linear function: with «n» belonging to the set of natural numbers N and j=0. This is a linear function, a particular case of an affine function because the value of the y-intercept is zero. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 9

10 Value of «n» Value of Odd numbers that are multiples of «3» The series, corresponding to odd multiples of the odd number «5» is given by the following linear function: with «n» belonging to the set of natural numbers N and j=1. Value of «n» Value of Odd numbers that are multiples of «5» The series, corresponding to odd multiples of the odd number «7» is given by the following linear function: with «n» belonging to the set of natural numbers N and j=2. Value of «n» Value of Odd numbers that are multiples of «7» And so on... For each value «j», an affine function determines the set of indexes which correspond to the odd numbers that are multiples of the coefficient of this function named. This coefficient is an odd number whose index is obtained by this affine function with the parameter's value n = 0. An additional axis, named «j» is created to represent these affine functions in the space W. The axis «j» allows the representation of these affine functions independently each others in space W. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 10

11 sequence «j» Constant Coefficient Odd numbers that are multiples of Proof : Hence Hence Hence If «n» is strictly greater than zero, the affine functions only describe the indexes corresponding to odd multiples of odd numbers. The set of indexes of odd numbers strictly greater than «1» is represented in the space W by an affine parametric function called "Basic Schema". This function corresponds to the first structural element of the odd numbers. This set is given by the following formula: with Équation 3 : Basic schema: affine function corresponding to the odd numbers greater than «1». This function is used to generate the set of numbers that make up the space W. Definitions : - the constant is appointed «lag» of a sequence «j» - the coefficient is appointed period of the sequence «j» written Note : the values of the parameters, and are positive odd numbers or positive even numbers. Hence. These numbers are represented on the graph given Figure 3. Definition 2.1: The rank or index of any odd number Ni greater than or equal to 3 can be represented by the following parametric affine function with the parameters «j» and «n» belonging to natural numbers N. This function corresponds to the basic schema of odd numbers. The relationship between an odd number and the affine parametric function is as follows : with. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 11

12 Any odd number greater or equal to 3 can be described using two parameters. The index of the odd numbers can thus be represented in a two-dimensional space with the formula. The workspace W describes the set of values depending on the parameters «j» et «n». Definition 2.2: The rank or index of any odd number Ni greater than or equal to 3 multiple of a prime odd number, can be represented by the following affine function parameter with the parameters «j» belonging to the natural integers N and «n belonging to the nonzero integers N*. The relationship between an odd multiple number of an odd prime number and the affine parametric function is as follows: with. One obtains thus all the odd multiples of each odd prime with the following formula. If, then one gets all composites (UNNP). Note: the odd number "1" does not belong to the space W. Définition 2.3: The space W described the indexes of odd numbers whose values belong to the set of odd natural numbers minus the number 1. This space is named space N. The odd numbers are described in space N thanks to their index : Space N The indexes of the odd numbers define the space W. They are described by the following formula: Space W We get two sets of numbers related to the parameter : - } : set containing all odd numbers greater than or equal to 3 with. - set containing only the odd multiples of odd primes with. If we replace the parameter «n» by «n+1», then the set E2 is described by : The set of odd primes (PUN), named Eip, is obtained by subtracting the set E2 to the set E1. We get thus. The set Eip is the complementary set of E2 in the set E1 Reminder: we do not process the indexes of the even numbers in the space W. Indeed an even number corresponds in the space W to an index with a value belonging to the set of rational numbers. Only integers are processed in the space W. The following formula is used to connect the odd numbers defined in the space N and the indexes of odd numbers defined in the space W. What represent the axis «k» and the axis «j»? - The axis «k» represents the index «k» of the odd numbers. An index k is also called a point in space W. For each odd number Ni, a value is defined. The relationship between and «Ni» is : hence. Note : The study of the primality of the number Ni in the space W is therefore performed in the interval [0 ; ]. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 12

13 - The axis «j» is used to represent for each value «j», a set of points related by the formula. This set of points represents the odd multiples of the odd number in space N, defined by the period whose the index is «j». A sequence «j» is defined by this set of points. The primality of an odd number Ni is studied with all the odd numbers that lie between zero and the value corresponding to the index of the integer value of the square root of the studied odd number. The relationship between and «Ni» is as follows: [ ] hence Note : The study of the primality of the number «Ni» is performed in the interval [0 ; ]. The primality of an odd number is studied with the value of the number, and the integer value of the square root of the number. This value of the square root defines the upper limit of the study of primality. In the space W, corresponds to the value of the number, and jmax corresponds to the odd integer value of the square root of the number. The space W allows to link these two dimensions: the axis and the axis. The space W is represented Figure 3. The parameters of the Figure 3 are the followings: On the axis «j», the interval represented is included between 0 and =5 On the axis «k», the points of each suite arithmetical are represented with «n» included between 0 and =10. On the graphic, the values corresponding to a value identical to the parameter, are linked with dotted line. The formula used to determine the point sequences on the axis "j" corresponds to the equation 3. Description of the graphic Figure 3 : The graphic corresponds to the space W and represents a part of the first five sequences «j». Some numbers PUN are also represented This graphic represents the space W and allow us to describe it. The axis of abscissa represents the values of indexes of odd numbers. The values of some prime odd numbers are represented below the values of. The axis of ordinates represents the values of the parameter. The sequence of points in red on each horizontal line, i.e. for each value of, corresponds to the multiples of the sequence «j» having as period. Thus for, the series of points corresponds to the multiples of 3. For, the series of points corresponds to the multiples of 5. And so on.. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 13

14 The set of points in red obtained for the set of odd multiples of prime odd numbers (UNNP). They are the composite numbers. The projection of these points on the axis is represented by some arrows in red color. All the values of does not correspond to the points in red. The values of, with no correspondence with the points in red, correspond to the prime odd numbers (PUN). The indexes of numbers PUN correspond to the values which are not generated by the basic schema, as showed below. Figure 3 : The graphic corresponds to the space W and represents a part of the first five sequences «j». Some numbers PUN are also represented The points generated by the sequences with are not taken into account in order to obtain only the points UNNP. This allows to distinguish the points UNNP from the points PUN. If the period of a sequence corresponds to a composite number, the indexes generated by this sequence correspond to a subset of a set of points generated by one of the previous sequences. This means that a sequence, whose the period corresponds to a composite number, do not generate any new points UNNP. This sequence does not increase the number of points UNNP. We then indicate than the sequence does not densify the axis «k» with some new points UNNP. Only the sequences «j» whose the period corresponds to a prime number generate some points corresponding to of new values UNNP. Only these sequences densify the axis «k». The density of UNNP corresponds to the ratio number of points UNNP by number of points PUN, as. This ratio increases when a new sequence with a period equals to a prime number appears [5]. This property is linked to the rarefaction of prime numbers. The abscissa of a point corresponds to the index of an odd number represented on the axis k on the graphic. The ordinate of a point corresponds to the index of the square root of the odd Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 14

15 number. The parameter «j» allows to make the link between the index of the value N of the number as * +, and the index of the value of the integer part of the square root of the number as * Structure of the space W 3.1- Organisation of odd numbers Definition 3.1 : This space W has remarkable points that explain the densification of the axis «k» by the points generated by sequences «j». This densification corresponds to an increase in the number of composite numbers, and therefore the scarcity of prime numbers. Note : the existence of these remarkable Points has led to define a base unit which can measure this densification. In space W, it exists remarkable points related to sequences «j». These points have the particularity to be the starting point of an increase in the density of the axis «k». There is a remarkable point for each value of «j». However, the increase of the density is conditioned by the fact that the period of the sequence «j» must be a prime number. The densification is mathematically obtained using the basic schema: with and with the period equal to which must correspond to a prime number. Reminder: we do not take into account the points obtained with the value, because they match all odd points, primes and composites. If we wish to study both types of odd numbers, the primes (PUN) and the composites (UNNP), we will study on the axis «j» only points generated by defined sequences with the following condition :. For, new points appear giving additional k values to those obtained for. For example, for, additional points exist from the value. This point is a remarkable point. There is one for each value of «j». From these remarkable points, it is observed in Figure 6 on page 17 that new points appear to densify the axis «k». These remarkable points are named GW 1. They are connected by the following formula : with n = j + 1 with j 0 Équation 4 : formula of a remarkable point with j 0 The relationship between «n» and «j» is proved below. Proof : 1 GW is the name of Genevieve Wolf in tribute to our Mother. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 15

16 For any sequence «j» with a period equal to «exists.», a remarkable point The distance between two remarkable points GW(j) is named base unit, noted is the second structural element. We will calculate the distance between these remarkable points. The figure 6 page 17 shows that the distance between 2 points GW(j) is connected to the period of the sequence «j». This distance is equal to. The first point is obtained with the sequence j=0, With the sequence j = 1, the obtained point is Hence with. Table 2 : demonstration of the formula by induction j n Demonstration Comments 0 1 The values of period follows an arithmetic sequence: With = + 2 with 1 2 =3 And it corresponds to the set of odd numbers except for With the number «1» This n-1 n With With and i0=3 Hence With and Résultat We get with the following formula : Hence avec j 0 or Hence ( * The formula is valid only for values of. The formula, also noted, is valid for. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 16

17 Another demonstration is presented below: The remarkable points of each sequences «j» are connected with the following formula Hence Definition : The base unit points minus one with j 0. is defined as equal to the distance between two consecutive Hence Each base unit consists of a number of points equal to the distance of a base unit more «1», because the number of points which constitutes the number of a set of intervals is always greater of one unit relative to number of intervals. Hence the following formula :. The graph below shows remarkable points and the base unit. The value of a base unit is not a constant. It is connected to the sequence. «j». Its distance increases linearly with «j». Note : As the basic schema is the same regardless of the scale, we observe the same patterns within the base unit when the scale increases. We get a pattern that repeats itself whatever the scale of the measure. This behavior is similar to a fractal dimension. (See paragraph 3.6- Density of primes page 32) Space W Figure 4 : Representation of the basic schema and the base unit Ugw(j), and the equivalence of the points k(j) and k(m) connected by arrows with a green color. The remarkable points GW(j) does not correspond to this form :. Proof : Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 17

18 ( + We know that Hence is not divisible by 3. (See paragraph 3.4 for a proof) Features of the remarkable numbers remarkable points in space N The remarkable points are the indexes of odd perfect square. These remarkable points correspond to the square of odd numbers such as,,, etc. j GW(j) Ni = 2 * GW(j) The formula in space N, leads to the following formula: Proof : we have and, hence : After factorization, the following result is obtained: hence These remarkable points GW(j) correspond to odd numbers squared. At each sequence «j» is associated a remarkable point. When the period of a sequence is equal to a prime number, the density of points UNNP increases from this remarkable point Common points between the sequences The sequences generate sets of points. These sequences have common points. The identification of these points helps to understand the search condition of the primality of an odd number. This condition corresponds to investigate whether an odd number N is a multiple of an odd prime number included in the following interval [ ]. Why [ ]? The answer is given in paragraph Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 18

19 At each point «k» corresponds a value which depends on the parameter «n» and the parameter «j» provided by the basic schema:. For each value of «n», k is connected to «j» by an affine function. Each point k(j) is located on a straight line : with, et Hence : For each value of, is an affine function with that is a subset of the values of, hence. Each point k(m) is located on a straight line : with, and The intersection of these lines gives the points with. All points and are equal from the points for a given integer greater than zero. The points on the straight lines and are interconnected by the following equation: with, for a given value. The points are shown in Figure 4 page 17. The points having the same value are connected together by a green arrow. From the remarkable point and for a given value, the axis «k» is no more densified with points for values of greater than. Indeed, all points are the same as the points computed with a value of equal to à Primality test in space W The study of the primality of a number Ni involves dividing this number by all prime numbers between 3 and the square root of that number by taking only the integer part of the result of the square root. In the space W, the index of the square root of this number corresponds to the sequence of the remarkable point. The study then focuses on the divisibility of this number with the period of the sequences between 0 and. The first step is to position on the axis «k» the index of the number «Ni» defined by. The following formula provides as a function of Ni :. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 19

20 Hence the index. The second step is to calculate the maximum value «made on the axis «j» with the following formula:» for which the calculations are The resolution of this equation leads to the following single positive solution : * + : the function retrieves the integer part of a number. The value «the formula : * +» is the index of the square root of an odd number «Ni» in space W. Hence The points taken into account in the calculations of the primality of an odd number Ni that has the index belong to the interval [0 ; ] on the axis «k», and to the interval [0 ; ] on the axis «j». In summary, this result demonstrates that an odd number "Ni" is prime if it is divisible by none of prime numbers in the range [3 ; ]. For a given sequence «j», certain points generated are different of the points generated by the previous sequences if and only if two conditions are met: The value of the period of the sequence «j» corresponds to a primes. The sequence «j» starts to densify the axis «k» only from the remarkable point. This means that for values of «j» greater than the value * +, the sequences «j» do not generate new points that allows to densify the axis «k» in the interval defined by the base unit with. The study of the primality of a number Ni is reduced to study the divisibility of the number Ni in the interval [0 ; ] Structure of a «base unit» Definition of a base unit Definition 3.3.1: A base unit corresponds to the distance between 2 consecutive points GW(j) minus one. This distance corresponds to. This is a measuring interval to determine the evolution of the properties of the odd numbers. Hence the interval defined by :.. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 20

21 This interval consists of a number of points equal to the value provided by the formula :. This unit is used to measure the properties of prime numbers. The base unit is thus a measurement interval. Most of the properties of prime numbers can not be generalized when the value of the odd number «Ni» tends to infinity. These examples are from the book of Jean-Paul Delahaye «Merveilleux nombres premiers» [1] : It was noted that the numbers have no common factors before. We could have thought that this formula could generate infinitely many primes whatever the value n. Similarly, Pierre de Fermat noting that the formula gives prime numbers for the following values of «n» : 0,1,2,3,4 ; he conjectured that the function gives always a prime number, which is not true for n = 5. Why doesn't the generalization of formulas to higher scales work? Why appears a limit from which the formulas do not provide the expected result? The structure and organization of primes are characterized by a basic schema and a base unit. These elements show that any measure of the properties of prime numbers must be done using discrete mathematics. Indeed, the base unit is linked to the basic schema by the parameter «j». The study of the properties of odd numbers requires to study the evolution of these properties within the base units. We have «Ni» the value of an odd number. When the value of that number is growing, the number of calculations required to study the properties of this number increases with the parameter «j». But the value of this parameter does not grow linearly. This parameter is related to the square root of «Ni». Only discrete mathematics allows to study the properties of odd numbers in order to obtain accurate results within base units. The knowledge of this structure validates the changes in properties of prime numbers regardless of the scale at which we stand. We can extrapolate the results even to infinity. For this it is necessary to carry out the measurements of these properties within basic units. The measures that are made within an interval that is greater than base units do not allows to predict the evolution of these properties at any scale. This is related to the density of UNNP that evolves when a new sequence «j» occurs with a period whose value is a prime number. The change of the density means a change in the distribution of prime numbers and therefore a change of the properties studied. These base units thus allow us to analyze the properties of prime numbers including the evolution of these properties such that the density of primes when «j» tends to infinity However, the paragraph 3.5- allows to introduce the concept of natural frequency and natural period which explains why it is necessary to go toward very large numbers to find counterexamples of the conjectures. Indeed the natural period corresponds to the multiplication of prime numbers in the range [0 ; ]. This period thus increases very rapidly as a factorial function. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 21

22 The graph below shows the index of an odd number in space W. Figure 5 : Representation of an odd number in the space W For instance, with N=37, we get and. Question : how to study the primality of this number Ni? The square root of 37 has an integer part equal to 6. But the nearest and lowest odd number is 5. This number corresponds to 1 in the space W. So to find out if the number 37 is an odd prime number, we must study the sequences "j "inferior and equal to 1. None of the points generated by the sequence j = 0 and j = 1 corresponds to k = 17. The number 37 is thus a prime number Feature of a base unit The density of the axis «k», for a given interval, corresponds to the number of UNNP points compared to the total number of existing points UNNP+PUN. If the base unit has no longer primes from a point, this means that the density of the axis «k» is equal to "1" from this point. Indeed, a sequence «j» densifies the axis k on condition that the period of this sequence corresponds to a prime number. If the density of composites (UNNP) for a given sequence is equal to "1", then the following values of «j» can not increase this density. There will therefore be no more prime numbers beyond this sequence. However, it has been proven by Euclid [2] that there is an infinity of prime numbers. There are obligatorily one or more prime numbers in each of the base units. In addition, the number of primes increases when the base unit increases with the value of «j». Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 22

23 This study demonstrates that result Singular points inside each base unit In the space W, the singular points are defined by the following formula: In the space N, these points correspond to the odd numbers obtained by the following formula:. These points have two properties which are demonstrated in the paragraph hereafter: These points are divisible neither by «3» nor by «5». They do not belong to a couple of twin prime numbers. They are between two points non-prime which the values, in the space W, are and. Hence < < < < In the theory W, the first singular point are located just before the remarkable point obtained for i.e.. The value of the first singular point is then, which corresponds to an odd number equal to. This prime singular point belongs to the base unit. It corresponds to the last point of this base unit. For, we note than the we obtain a point "singular" for (i.e. ) which belongs to a couple of twin prime numbers: and give respectively the twin prime odd numbers. This is an exception relative to the nonexistence of the remarkable point These points are divisible neither by «3» nor by «5» The singular point for the value, in the space W, is given by the formula: In the space N, the formula become: Note: The first sequence «j» corresponds to and therefore to Fgwm1(0)=23. However positive odd number «7». In order to take into account this point, we have expanded the definition of the parameter «j» to -1. We have replaced by. Hence the obtaining of the polynomial formula of the second degree: with An other mathematical formulation is possible with and. This allows to obtain the following formula: Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 23

24 with The odd numbers generated with this formula are divisible neither by «3» nor by «5». Proof 1- They are not divisible by 3 due to the fact of their position in the space W (Definition) Mathematically, we are going to demonstrate than the formula is not divisible by 3. As j an natural integer. As Hence Hence, with However is never divisible by 3. is therefore not divisible by 3. Proof: As a natural integer. We are going to demonstrate that is never divisible by 3. The Euclidean division of by 3 can only take one of the three following forms: n = 3q, n = 3q + 1 or n = 3q + 2. Let s examine each of these cases: If n = 3q, with 0 1 < 3, therefore is not divisible by 3 because the rest 1 is not null. If n = 3q +1, is not divisible by 3 because the rest 2 is not null. If n = 3q+2, is not divisible by 3 because the rest 2 is not null. with 0 2 < 3, therefore with 0 2 < 3, therefore 2- They are no more divisible by «5». The explication is the following: Let s examine the formula ( ) - A number to the square such as can only be terminated by the following values:. - The term generates a number which is terminated by the values 0 or 5 Therefore the sum of the 2 previous numbers can only be terminated by the following values:. At least, if we multiply by «2» the sum generated, we obtain a number which can only be terminated by the following values:. In the space N, we must again multiply by 2 and add 3 because. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 24

25 We will have thus only some values which are terminated by the following figures:. Only the numbers which are terminated by 0 and 5 are divisible by «5». The value thus generated is therefore never divisible by «5» These points do not belong to a couple of twin prime numbers It is to note, as confirmed by the graphic Figure 3, than these points do not belong to the twin prime numbers, because they are located between two points UNNP. Proof: therefore 1- Point corresponds to an odd number to the square as showed in the paragraph 2.1. This point is therefore never a prime number. 2- Point In the space W, we have the following formulas: Hence However Hence To obtain the odd numbers, in the space of natural integers N, the following formula is applied: Hence The value always, in the space N, to a point divisible by with. The point can therefore not correspond to an odd number prime. Conclusion: the numbers generated by the function never belong to a couple of twin prime numbers, except for the first point «j = 0». This point corresponds to in the space W, i.e. the odd number «7» in the space N. This exception can be attributed to the fact that it does not exist a point. The graphic below represents the particular position of these singular points in the space W. These points are represented in the form of a brown/gray square on the axis. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 25

26 Figure 6 : Representation of singular points Theorem 3.4: As the set of odd numbers Ni, corresponding to the singular points in the space W, described by the following polynomial formula: with And the following equivalent formula: with and The odd numbers generated by these formulas have the following properties: - They do not belong to the twin prime numbers. - They are divisible neither by «3» nor by «5» Summary: The space W shows a regular stucture of odd numbers owing the basic schema. This structure has led to define a base unit. This base unit is the interval in which the properties of prime numbers are studied. Inside each interval is present a singular point. However, to get measurements without oscillatory variations, it is necessary to extend the interval of the measurement to natural period of the system Natural Period of primes Definition : The natural period is defined as the measurement interval for the study of the properties of prime numbers. The natural frequency of prime numbers is described as the inverse of the natural period of primes. The natural period of prime numbers is defined by the function. The natural frequency of prime numbers is defined by the function. Postulate : The values of the density of odd composites (UNNP) does not oscillate around a mean value if the measurements are performed within the natural period. The study of the density within this natural period provides stable values. The studied property relates to the density of odd composite numbers (UNNP). UNNP density is defined as the ratio between the number of UNNP points and the number of PUN points. Auteurs : François et Marc WOLF mathscience.tsoft .com Page 26

27 Determination of prime numbers PUN and composite numbers UNNP The measure of density of UNNP points within the base units shows a variation of the value obtained around a mean value. The measured density value oscillates around an average value as shown Figure 17 : Curve representing the evolution of the ratio UNNP / PUN based on the base unit j page Erreur! Signet non défini.. Why have we got values of the density that oscillate around a mean value according to the units of measurement? Each base unit is linked to a «j» sequence. A sequence «j» generates a set of points corresponding to odd multiples of odd number defined by the period of this sequence that is equal to. Two scenarios are presented : - If the period of the sequence corresponds to a composite number, all the numbers generated by this sequence correspond to the points generated by one or more of the preceding sequences. Hypothesis: The density UNNP/PUN within the base unit of the sequence «j» should therefore not be modified with respect to the density of the previous sequence. Indeed, no new point UNNP is created with this new sequence «j». However, this density fluctuates as shown in Figure 17 : Curve representing the evolution of the ratio UNNP / PUN based on the base unit j page 84. This fluctuation is due to the measurement interval. The interval of the based unit of a sequence is greater than that of a sequence. - If the period of the sequence corresponds to a prime number, a part of the generated numbers by this sequence correspond to new UNNP points. The density of UNNP points is modified and increases because the number of points UNNP increases more than the number of points PUN. The classical approach to the distribution of prime numbers is to consider that the composite numbers correspond to multiple of prime numbers. The prime numbers generate the composites. The approach to the distribution of primes, in this study, is to consider that the prime numbers are the rule. The numbers are all of prime numbers. Compound numbers correspond to the numbers generated by the sequences. These composites remove the primes. The composites generate the prime numbers. The results show that the measurement of the density of the points UNNP varies from a sequence to a sequence. This variation is due to the interval taken for measuring the density of UNNP points. Density is measured in the range bounded by the base unit. The following table provides the number of points UNNP within the base units UGW (j) for the sequences "j" in the range [0 ; 3]. The sequences "j" generate identical UNNP points. These points are only counted once. (Voir Figure 4 : Representation of the basic schema Auteurs : François et Marc WOLF mathscience.tsoft .com Page 27

28 and the base unit Ugw(j), and the equivalence of the points k(j) and k(m) connected by arrows with a green color. page 17 to determine the points UNNP). Sequence «j» Interval of measure on axis «k» Number of points in the interval Number of UNNP points 0 [3 ; 10] [11 ; 22] [23 ; 38] [39 ; 58] Table 3: number of points UNNP in the base units Determination of the natural period of primes If we extend the measurement of UNNP points in the intervals which are consecutive to the base unit of the sequence «j» with a number of points equal to the base unit, the results show values that oscillate around a mean value. The UNNP points taken into account correspond to the points generated by the sequences «j» in the range. Sequence «jmax» Interval of measure on axis «k» Number of UNNP points 0 [11 ; 18] 3 [19 ; 26] 2 [27 ; 34] 3 [35 ; 42] 3 [43 ; 50] 2 1 [23 ; 34] 6 [35 ; 46] 6 [47 ; 58] 5 2 [39 ; 54] 9 [55 ; 70] 9 [71 ; 86] 9 [87 ; 102] 9 3 [59 ; 78] 11 [79 ; 98] 9 [99 ; 118] 11 Table 4: number of points UNNP in intervals which extend the base units Comments Values alternated periodically The stabilization of the measurement of the density requires defining a suitable interval to the measurements of density, depending on the sequence. Definition: the natural period of sequence is defined as being equal to the multiplication of periods of the sequences «j» whose value corresponds to a prime number, and only for the sequences in the range [0 ; ]. Hence the following formula for obtaining the natural period of a sequence is : Auteurs : François et Marc WOLF mathscience.tsoft .com Page 28

29 Sequence «jmax» If the value of the formula corresponds to a prime number then the function returns the value of the prime number, so. If the value of the formula corresponds to a composite number then the function returns the value «1». The following table provides natural periods for the following sequences in the range [0 ; 4] : Natural period Number of points in the interval of measure on axis «k» Multiplication of prime numbers Comments The period for the sequence «j=0» corresponds to the value «3» which is a prime number The period for the sequence «j=1» corresponds to the value «5» which is a prime number The period for the sequence «j=2» corresponds to the value «7» which is a prime number The period for the sequence «j=3» corresponds to the value «9» which is a composite number. This value is not taken into account in the calculation of the natural period The period for the sequence «j=4» corresponds to the value «11» which is a prime number. so on Table 5: Determination of natural periods for sequences «j» These natural periods depends on the «j» sequence. These natural periods correspond to an interval larger than the base units. The chart Figure 7 : below represents the base units and the natural periods for sequences «j» in the range. The first point of each natural period of a «j» sequence begins at the remarkable point of the sequence «j» so. The base unit and the natural period of a sequence «j» starts at the same point. Auteurs : François et Marc WOLF mathscience.tsoft .com Page 29

30 Figure 7 : Chart representing the base unit ) and the natural period of the system per sequence «j» The natural periods correspond to intervals whose value increases rapidly relative to the base units. Auteurs : François et Marc WOLF mathscience.tsoft .com Page 30

31 In base units, we have defined the UNNP numbers and the PUN numbers. Within natural periods we will define the virtual points and virtual points. Let us consider the sequences and. The UNNP points correspond to the points generated by these two sequences with within the base unit " ". The complementary points in the base unit are the PUN points. The interval of a natural period is larger than a base unit, with the exception of the first base unit with. The number of points in the natural period for is equal to 15. The number of points in the base unit for is equal to 11. We have 4 additional points in the natural period for. These 4 points correspond to the area of the virtual points. Indeed, to determine whether these four points correspond to UNNP points, or to PUN points, it would be necessary to take into account the sequence. The fact of not take into account the sequence implies that the points considered as PUN points may be part of the points generated by the sequence. They can therefore be UNNP points. The figure below shows the distribution of UNNP points PUN and natural period for. For example : within the base unit and the - The number is considered as a point because it does not belong to sequences of points with et. However, if we consider the sequence j = 2, then there is a UNNP point because it belongs to the sequence of points with j=2. - The number is considered as a point. If we consider the sequence, this number corresponds to a point. So it is the index of a prime uneven number (PUN). All PUN points within a natural period are considered in this study, as virtual PUN points designated as. The UNNP points in the natural periods are necessarily UNNP points in the base units and vice versa. However, for consistency with points, UNNP points within a natural period are named. Figure 8: Determination of virtual primes PUNv for the sequence "jmax=1" Auteurs : François et Marc WOLF Page 31

32 We will calculate the number of Points generated by the sequences «j» within natural periods for the following values of sequences : Sequence «jmax» Number of points in the range of the natural period Intervals of measure on axis «k» 0 9 [3 ; 11] [12 ; 20] [21 ; 29] 1 15 [11 ; 25] [26 ; 40] [41 ; 55] [56 ; 70] [23 ; 127] [128 ; 232] [233 ; 337] [39 ; 144] [145 ; 249] [250 ; 354] Table 6: Number of UNNPv points in natural periods for sequences «jmax» number of NINPv Points generated by the sequences «j» Comments The values are stable The stabilization of the measurement of the density is obtained only if the measurement is performed in an interval in which the number of points corresponds to a natural period or a multiple of that period which corresponds to a harmonic of the period. The natural period of odd numbers thus evolves with the emergence of sequences «j» when the value of its period is equal to a prime number. This makes incompatible obtaining a stable measurement of the density of the UNNP points within the base units. This explains the fluctuation of measurements of density in the form of oscillations. The number of points within a base unit period of the base unit. natural number. is not proportional to the value of the natural ; natural number and p Conclusion : The density of UNNP points is measured within natural periods for each sequence «j» in order to obtain stable measurements, as confirmed by numerical computations [5] Density of primes We introduced the concept of natural period in the previous section. We will use this natural period to define sets of numbers. These sets are called «basic patterns». We will explain how the density of PIN points fluctuates within the basic units, and demonstrate why the number of primes increases on average within the base units. For this, we need the "basic pattern" which is the third structuring element. This basic pattern allows to define a structural schema of Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 32

33 prime numbers. This schema consists of a combination of basic pattern, hence the name of combined pattern. This scheme allows to understand the organization of primes Basic pattern Definitions : The patterns (or motives) consist of 3 consecutive sequences «j» whose the first is named. The first pattern is called basic pattern. This pattern uses the series of points generated by the three following sequences, and. This corresponds to the multiples of the first three prime numbers 3, 5 et 7. The natural period of a pattern corresponds to the multiplication of prime numbers that make up the periods of the three sequences of the pattern. Each first number is taken into account only once in the calculation of the natural period. This period begins at the remarkable point of the first sequence defined by : The extended period of a pattern corresponding to the multiplication of the periods of each sequence of the pattern. This period for the basic pattern is equal to. This period begins at the remarkable point of the sequence defined above. Each sequence "j" has a base unit. The base unit of a pattern is called consolidated base unit. This consolidated base unit starts to the remarkable point with the sequence as defined above. The interval of a consolidated base unit is the sum of the three base units that constitute the pattern. Thus the interval of the basic pattern is defined by the first point "3" and the last point hence the following interval which consists of 105 numbers:. This interval generates a pattern that repeats endlessly. For easier identification of the structure, an offset of one unit is realized. The study of the structure is then carried out in the interval. This shift of one unit does not affect the conclusions of the study. The first pattern is called basic pattern because it is the one that determines the initial organization of primes. The patterns (or Motif) are noted with the first sequence of the pattern. This first sequence always corresponds to a multiple of "3". The natural period of the pattern corresponds to the multiplication of natural periods of each of the sequences which are the pattern. The natural period of the pattern corresponds to the smallest common multiplier (SCM) of the periods of 3 sequences of the pattern. This value is determined by multiplying in a unique way the prime numbers which constitute the periods of the 3 sequences. A prime number is taken into account in the calculation of the natural period only once. For example, for the pattern Mt(135), the extended period of this pattern is to multiply the periods of the three sequences j=135, j=136 et j=137 hence. The prime number "5" appears 2 times. It should only be considered only once for calculating the natural period of the pattern. Hence the natural period of the pattern Mt(135) is equal to. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 33

34 Description of the table below: Parameters of the table : The parameter Ni corresponds to the values of odd numbers in space N. The indices of the numbers correspond to the parameter values «k». The parameter Y allows to position the values using indices from 1 to 105. These indices correspond to the position of points within the natural period. Description : The green numbers correspond to the composite numbers multiple of 3, 5 or 7. The other numbers in red correspond to the virtual primes. Indeed, these numbers are not all prime numbers. We only take into account the following sequences to determine whether the indices of the numbers in the interval belong to these sequences or not. If they do not belong to these sequences, then they are. But from we should also take into account the sequence, and from, we should also take into account the sequence in order to determine whether they correspond to primes. Indeed, these sequences reveal new composite numbers in the interval studied. In other words some of the points considered as points belong to sequences and/or. So they are actually composite. We have, among numbers, two distinct forms: - The numbers correspond to with the form equivalent to in space N, - The numbers correspond to with the form equivalent to in space N. In the table below, these two forms are accounted within the natural period to study their distribution and their distribution. This is the line of the table named «counting». Reminder : ; Ni odd number, and index of odd numbers. Description of the table :,with corresponds to the serie of points of the sequence. - The light gray colored boxes represent prime numbers. - The dark gray colored boxes represent virtual primes. But if one takes into account the following sequences and, then these points would be considered as composite numbers. Ni (Value) k (Index) Y=k ( + ) ( - ) Counting Start of the natural period: cycle of 105 numbers Start of symmetry SCMo Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 34

35 Ni k Y=k ( + ) ( - ) Counting Ni k Y=k (+) (-) SYM Counting *SCMo First point of symmetry within the natural period of the basic pattern Ni k Y=k ( + ) ( - ) Counting Ni k Y=k (+ ) ( - ) End of symmetry SCMo Maximum distance between two numbers PUNv equal to 5 = Table 7: distribution of points PUNv within the natural period of the basic pattern Numbers multiples of : End of the natural period *SCMo : The first common point multiple between the three sequences of the basic pattern. (See below for definition) Pod Second point of symmetry Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 35

36 Another representation consists in counting the distance between two points UNNP within the natural period and/or the extended period. The base interval is 3 because we studied the multiples of 3, 5 and 7. Thus, the table below shows how the intervals between these multiple are distributed. UNNP k (Index) Distance between two consecutive points UNNP NA Start of the natural period: cycle of 105 numbers Start of symmetry SCMo UNNP k (Index) Distance between 2 consecutive points UNNP SCMo the symmetry of the distances is observed around the point SCMo UNNP k (Index) Distance between 2 consecutive points UNNP End of symmetry SCMo Figure 9 : Distribution of points UNNP within the natural period of the basic pattern This representation allows to distinguish the structure of the pattern through the distribution of the values of distance between two points UNNP. We get 3 possible values for the distance 1, 2 and 3. Results : 1- Zones of density of points PUN Pod (point offset) We observe the different density zones in the base pattern. These areas are defined by an interval of five successive elements. Indeed, the maximum distance between two points is 5. This distance is used to measure the density of primes in the base pattern. The density is defined as the ratio of points PUNv on the sum of the odd numbers +. Note : this distance is also a measure of the density of points within the period of a sequence. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 36

37 We observe three zones with a high density equal to 80%. A high density corresponds to a higher density than the average density of PUN within the natural period. For example, in the case of pattern Mt(0), the average density is equal to 48 points PUN divided by 105 points which gives an average density of about 46%. It is fundamental to understand that the base unit always starts at the beginning of a natural period. The value of the interval of a base unit is far below that which characterizes the natural period. In other words, the base unit is still in a zone of a natural period with a high density of points PUN. In this example, the base unit has these intervals: for the sequence ; for the sequence and for the sequence. So the pattern has a consolidated base unit, consisting of the sum of the three previous intervals hence an interval equal to. This interval consisting of 44 numbers is to be compared to the natural period which is equal to 105 numbers. The pattern has a consolidated base unit of 72 numbers to compare to the natural period equal to. Measures of the density within these ranges for the basic pattern gave the following values: Sequence «j» Interval of the base units of the sequences «j» Density of PUN in % Comments 0 [3 ; 10] 62.5 Only points generated by the 1 [11 ; 22] 50 sequences are included in the calculations within the natural 2 [23 ; 38] period. 3 [39 ; 58] 40 4 [59 ; 82] [83 ; 110] [111 ; 142] [143 ; 178] The points of sequences taken into account. are not Table showing the densities in intervals defined by the basic units «j» within the natural period defined by the basic pattern. A variation in the density of the points PUN is observed within the intervals defined by the base units. This variation is sinusoidal, and fluctuates around the mean density of the natural period. This fluctuation is attenuated to reach the value of the average density when the value of the number k tends towards infinity. The average density within the natural period of the basic pattern is equal to. The density fluctuates between the values in the range. It is important to note that this type of fluctuation in the density of the points PUN in intervals defined by the basic units for the natural period of a pattern, is valid for all pattern. Each pattern has a natural period. Within this natural period, fluctuations in the density of points PUN in intervals defined by base units are oscillatory type. The unit of measurement of density is the maximum distance between two points PUN defined for each pattern within the natural period. For example, for the basic pattern, this distance is equal to 5. One can also define an average density within a base unit. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 37

38 The schema Figure 10 below represents the points of 3 consecutive sequences «j» of the basic pattern from the remarkable points. These remarkable points for each sequence «j» respectively correspond to points Point 0.1, Point 1.1 et Point 2.1. Each base unit «j» contains three points of its sequence «j». Par example, the bse unit contains this three points : Point 0.1, Point 0.2 et Point 0.3. Definitions : Intervals that are no sliding are consecutive intervals. They succeed each other one behind the other. The sliding intervals are shifted by one unit. Interval of 3 units: description of consecutive intervals and sliding intervals consecutive intervals k Unit sliding intervals 3 The density values represented in the Figure 10 are the following: Consecutive intervals Number Density consisting of 5 points of PUNv ]3 ; 8] 4 ]8 ; 13] 2 ]13 ; 18] 2 ]18 ; 23] 3 ]23 ; 28] 2 ]28 ; 33] 2 ]33 ; 38] 3 Comments ]3 ; 8] 4 First consecutive interval within the natural period with a maximum density ]98 ; 103] 1 * First consecutive interval within the natural period with a minimum density * The interval used is not a sliding interval. In the case of a sliding interval, the first interval having the minimum density is [41; 46]. This density is equal to. It is less than the minimum density found with the consecutive intervals. It should be noted that this point is beyond the base unit of the third sequence of the pattern. This means that the minimum density range is located within the natural period of the pattern but not within the basic units of the basic pattern as shown in the calculations given Figure 14 page 71. In addition, it was demonstrated in paragraph Distribution of primes: Brocard's conjecture and Legendre's conjecture page 82. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 38

39 Motif with : three consecutive sequences «j» Natural period of =105 Maximum distance between two PUN = Max :80% Base unit of the sequence Base unit of the sequence Base unit of the sequence Density in % Min :20% 5 40% 60% Fluctuation in density of Average :46% k 62.5% 50% 43.75% Density within a base unit Figure 10: Evolution of the density of the points within base units This schema shows that the calculation of densities within the patterns, taking the maximum distance between two points as a measurement unit for the density, generates density fluctuations of oscillatory type in the basic units. Note it is important to note that the density of points within the base units is greater than the minimum density of 20% within the natural period of the pattern. It is noteworthy that the interval of a base unit evolves in relation to «j» linearly using the following equation:, while the value of the interval of the natural period of the pattern evolves as a polynomial way. The value of the natural period of the pattern is equal to the multiplication of the periods of the 3 sequences that make up this pattern. The period of sequence «j» is equal to. The evolution of a natural period of a pattern is therefore of type polynomial because we multiply 3 numbers more and more larger. This is a evolution of polynomial type of degree three such as. Hence : This extended period may be reduced by taking into account only the prime numbers in a unique way because all the combinations between the three periods are present. This gives the natural period of the pattern: Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 39

40 If the number is prime then the function «FirstPrimes ( )» returns the value of the number. Si le nombre is not prime, then the function returns the value of the smallest prime number that composes le nombre, le plus petit. For example, if one has the three following sequences: jbase=81, j1=82 and j2=83 which correspond to the respective periods,,, the natural period of the pattern is equal to. When combining all the patterns (See section Combined patterns and Goldbach s conjecture page 58), we will combine the average densities of all natural periods taking into account the base unit of the greatest sequence «j». This explains why the density of primes oscillates within the base units. Note : the evolution of the magnitude of the interval of the natural period of a combined pattern is factorial type. This amplifies the oscillations of density because the number of combinations of points within an interval increases. 2- Points of internal symmetry within the natural period. Results for the pattern : The features of the pattern are : - The number of points with the form is equivalant to the number of points with the form within the basic pattern: the number of points is equal to 24. These numbers are distributed unevenly within the natural period. However, two intervals are distinguished within the natural period. The first interval may be subdivided into two sub-intervals, interval A and interval B, of identical size with the same number of numbers. Each of the intervals have the same number of numbers. However, there is not the same number of points of the form and form within each subinterval. The first sub-interval contains 12 numbers with the form and 11 numbers with the form. This number is inverted for the second sub-interval. It is important to note that these two sub-intervals are positioned around a point of symmetry named SCMo (smallest common multiple point) for the forms and. These forms are positioned symmetrically with respect with this central point. All numbers of a sub-interval has his equivalent point to a distance equivalent to that central point in the other subinterval, and vice versa. The second interval of the natural period named interval C has two numbers, one of each type ( et ). These points are at a similar distance to the second point of symmetry named (point offset). Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 40

41 The numbers of the two forms and of the interval C can be combined with the two numbers corresponding to 5 and 7. The symmetry with respect to point is thus extended in the interval C. This pattern is repeated «n» times with n that tends to infinity. This pattern is repeated indefinitely. Note : the number "3" is a wildcardit can be associated with any numbers. Indeed, it is neither of the form nor. A B C, *, + Figure 11: Schema of the structure of the two forms PUNv+ and PUNv- in the basic pattern This structure is fundamental to understanding the distribution of prime numbers. Moreover, it is the starting point for understanding the Goldbach's conjecture. We have three intervals that are repeated to infinity as shown in Figure 9: A, B and C. We have an infinite sequence of these intervals: A,B,C,A,B,C,A,B,C We have shown the existence of a point of symmetry SCMo between A and B, but also within point C with the point Pod. This structure shows the existence of an infinite number of symmetry points. A B C A B C A B C A B C A B C Symmetry SCMo Symmetry Pod Extents of the symmetry 0 n 2n The structure of prime numbers is based on the two internal symmetries of the basic pattern. All patterns have the same internal symmetry. Only the scale increases. The superposition of these patterns provides the combined patterns that also have this internal symmetry within their natural period. The 2 black points on the schema above shows an example of combined patterns with the internal symmetries. Prime numbers have symmetry that explains the Goldbach's conjecture. - In the interval, there are 15 points that can match the first prime number of a couple of twin primes. These points determine the position modulo 105 of the other twin Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 41

42 primes. It has been shown that couples of twin primes exist only at these locations. (Voir paragraphe 5- Method for determining the twin primes (Twin Prime Conjecture) page 104). - The PUNv numbers of the form (PUNv+ and PUNv-) equivalent to and in space W, are not uniformly distributed within the natural period. Their distribution is symmetrical with respect to 2 points previously defined: SCMo and Pod. These points defined by the following formulas depending on jbase The first common point multiple between the three sequences of the pattern is SCMo: The value corresponds to the index of the number obtained by multiplying the periods of the three sequences hence, hence : The point offset corresponds to the formula : ( (( ) *, For Mt(0), we get Pod= Generalization of this structure to all patterns All patterns have the same properties as the basic pattern namely: the two points of symmetry, density variation within the natural period and an equivalent distribution of PUNv + and NIPv-. Both forms PUNv + NIPv- and are identical in number within the natural period of the pattern. Why? The patterns consist of three sequences. The period of the first sequence is always a multiple of the prime number "3". To demonstrate that the structure of the patterns is always the same (indeed only the value increases), we will study the positioning of points of each sequence with the points of the sequence which follows within the base unit of a pattern and within the natural period of a pattern. Two cases exist: - Case 1 : study of the positioning of points of the sequence with the sequence regardless of the value. - Case 2 : study of the positioning of points of the sequence with the sequence regardless of the value. study of the relative positioning of the points generated by the sequences within a base unit To answer, we will study the position of the numbers associated with each sequence, and. The basic units of a sequence «j» is composed of three points corresponding to three numbers which are odd multiples of the period of the sequence «j» equal to. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 42

43 These three points are composites. Each sequence «j» consists of three points. These three points are compared to points of the previous sequence. We will therefore take four points from the previous sequence. These points are defined according to the parameter «j». The first sequence of the pattern, which has a «j» value which is a multiple of three, is named. CASE 1 : The study concerns the sequence and We define the values of points of each sequence within the basic units. a- Sequence Point 0.1 : ; Point 0.2 : ; These four points are compared with the following sequence: Point 0.3 : ; Point 0.4 : ; Point 0.5 : ; Point 0.6 : ; Reminder: The value of the parameter is a multiple of three. b- Sequence Point 1.0 : ; Ce point correspond au Point 0.2 Point 1.1 : ; Point 1.2 : Point 1.3 : ; What are the possible distance between the points of the sequence «sequence whatever the value? 1- Case of the first point «Point 1.1» of the sequence » and the points of the This point is within the interval bounded by the points «Point 0.3» and «Point 0.4» of the previous sequence The points «Point 0.3» and «Point 1.1» are separated by a distance equal to 2. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 43

44 The points «Point 1.1» and «Point 0.4» are separated by a distance equal to. 2- Case of second point «Point 1.2» of the sequence This point is within the interval bounded by the points «Point 0.4» and«point 0.5» of the previous sequence The points «Point 0.5» and «Point 1.2» are separated by a distance equal to. The points «Point 1.2» and «Point 0.4» are separated by a distance equal to «4». 3- Case of the third point «Point 1.3 of the sequence The points «Point 0.6» and «Point 1.3» are separated by a distance equal to. The points «Point 1.3» and «Point 0.5» are separated by a distance equal to «6». The point «Point 1.3» is a multiple of three. Indeed, this point is equal to. But is a multiple of three and «21» is also a multiple of three. Hence the point 1.3 of the sequence is a multiple of three. The schema below shows the relative positioning of the points of the sequences and. Pattern : 3 consecutive sequences «j» Distance = 2 Distance = 4 Distance = 6 Distance= +1 Distance = 2 Dist = 4 Dist = 6 Distance= - 1 Points multiple of three Distance= - 3 Base unit of the sequence Base unit of the sequence Base unit of the sequence k Moreover, this figure shows that whatever the value of the points of the sequences and are positioned within the base unit at fixed distances that are proportional to 2. The values of the difference between the points are: Point 1.1 Point 0.3=2 Point 1.2 Point 0.4=4 Point 1.3 Point 0.6=6 Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 44

45 CASE 2 : The study concerns the sequence and a- Sequence Point 1.0 : This point corresponds to Point 0.2 Point 1.1 : ; Point 1.2 : These four points are compared with the following sequence: Point 1.3 : ; Point 1.4 : Point 1.5 : Point 1.6 : b- Sequence Point 2.1 : ( ) ; This point corresponds to Point 1.2 Point 2.1 : ; Point 2.2 : ; Point 2.3 :. What are the possible distance between the points of the sequence «sequence? 1- Case of the first point «Point 2.1» of the sequence »» and the points of the This point is within the interval bounded by the points «Point 1.3» and «Point 1.4» of the previous sequence The points «Point 1.3» and «Point 2.1» are separated by a distance equal to «2». The points «Point 2.1» and «Point 1.4» are separated by a distance equal to. 2- Case of second point «Point 2.2» of the sequence This point is within the interval bounded by the points «Point 1.4» and «Point 1.5» of the previous sequence Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 45

46 The points «Point 1.4» and «Point 2.2» are separated by a distance equal to «4». The points «Point 2.2» and «Point 1.5» are separated by a distance equal to. The point «Point 2.2» is a multiple of three. Indeed, this point is equal to. But is a multiple of three and «30» is also a multiple of three. Hence the point 2.2 of the sequence » is a multiple of three. 3- Case of the third point «Point 2.3» of the sequence This point is within the interval bounded by the points «Point 1.5» and «Point 1.6» of the previous sequence The points «Point 1.5» and «Point 2.3» are separated by a distance equal to. The points «Point 2.3» and «Point 1.6» are separated by a distance equal to. The schema below shows the relative positioning of the points of the sequences and. Pattern : 3 consecutive sequences «j» Distance = 2 Distance = 4 Distance = 6 Distance = 2*jbase-1 Distance = 2 Distance = 2*jbase+3 Distance = 2*jbase+1 Base unit of the sequence Base unit of the sequence Base unit of the sequence k This figure shows that whatever the value of, the points of sequences and are positioned at fixed distances that are proportional to 2. The distance between two points generated by the sequences and within a base unit is always greater than «1» if the value is greater than «2». Ie the points 2.1, 2.2 and 2.3 are separated of the points 1.3, 1.4, 1.5, 1.6 by a distance greater than «1» for the sequences greater than 2. The three points which are a base unit are always distant from the points of the previous base unit by a fixed distance corresponding to a multiple of 2. Hence : - Point 2.1 Point 1.3 = 2 - Point 2.2 Point 1.4 = 4 - Point 2.3 Point 1.5 = 6 Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 46

47 We will demonstrate that within a pattern, a couple of points of the sequences « is never positioned in 3m+1 et 3m+2. There are three sequences per pattern. The first sequence corresponds to a multiple of the prime number 3. These points are then of the form 3m. The study therefore focuses on the other two sequences. We will consider only the points within the basic unit of the sequence. Indeed, previous points correspond to common points with points belonging to one or more previous sequences. For example, it is observed in Scheme that point 2.0 is identical to 1.2 as shown by the calculations. Within the base unit, three points are to be studied: Point 2.1, Point 2.2 and Point 2.3. We will compare these points to the points of the sequence «1.4, Point 1.5 and Point 1.6» et» which are Point 1.3, Point We showed that the points 1.3, 1.6 and 2.2 are multiples of three. They therefore can not position itself in 3m+1 or 3m+2. The points 2.1 and 1.4 are separated by a distance equal to. So whatever these two points can not position itself in 3m+1 or 3m+2. Result : no points of the sequences» and can position itself together in 3m+1 and 3m+2 within the natural period of the pattern. This result allows to state that the interval of the minimum PUNv density can not be located in a base unit. In other words, the maximum distance between two primes PUN within the base units is less than the maximum distance between two virtual primes PUNv within natural periods (Cf. paragraph Distribution of primes: Brocard's conjecture and Legendre's conjecture) page 61. We will study the differences between the sequences and for different patterns within the natural period of the pattern. That is, how changes the distance between the points of sequences and when the value increases. The schema obtained with each pattern is always the same. Only the scale changes. Indeed, we still have three forms: jbase=3m, j1=3m+1, j2=3m+2 with 3m and 3m+1 corresponding to odd numbers whose difference is always equal to the value 2. Indeed,. Each sequences j = 3m + 1 and j = 3m generates a series of points. The distribution of the distance between these two sequences is the same. A sequence «j» generates points according to the basic schema : We have, and. We investigate the distance between the points generated by the sequences et. The sequence of points generated by is described by the following formula : The sequence of points generated by is described by the following formula : Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 47

48 CASE 1 : The difference between and is equal to : CASE 2 : The difference between and is equal to : We have the difference between the points generated by the sequences j=3m and j=3m+1. What is the difference between two values of with the same values «n2» and «n1»? Regarding the distance between the two sequences of points, what change will it exists when a difference of "i" units occurs on the parameter? We have respectively and, hence and. The difference corresponds in the case 1 to the following value: The difference corresponds in the case 2 to the following value: The difference corresponds to a constant for a given value of "i". It is considered that one takes in each sequence, the distance between two points of each sequence with the parameter n2 for the sequences and and the parameter n1 for the sequences and. The distribution within the natural period is the same. The number of points within the natural period increases with the value of. But all the intervals between the points of the two sequences are spaced apart by a distance proportional to 6 times the distance... We will study the differences between the sequences and for different patterns within the natural period of the pattern. That is, how changes the distance between the points of sequences and when the value increases. The schema obtained with each pattern is always the same. Only the scale changes. Indeed, we still have three forms: jbase=3m, j1=3m+1, j2=3m+2 with 3m+1 and 3m+2 corresponding to odd numbers whose difference is always equal to the value 2. Indeed,. Each sequences j = 3m + 1 and j = 3m+2 generates a series of points. The distribution of the distance between these two sequences is the same. A sequence «j» generates points according to the basic schema : Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 48

49 We have, and, and We investigate the distance between the points generated by the sequences et. The sequence of points generated by is described by the following formula : The sequence of points generated by is described by the following formula : CASE 1 : The difference between et is equal to : CASE 2 : The difference between et is equal to : We have the difference between the points generated by the sequences j=3m+2 and j=3m+1. What is the difference between two values of with the same values «n2» and «n1»? Regarding the distance between the two sequences of points, what change will it exists when a difference of "i" units occurs on the parameter? We have respectively and, hence and. The difference corresponds in the case 1 to the following value: The difference corresponds in the case 2 to the following value: The difference corresponds to a constant for a given value of "i". It is considered that one takes in each sequence, the distance between two points of each sequence with the parameter n2 for the sequences and and the parameter n1 for the sequences and The distribution within the natural period is the same. The number of points within the natural period increases with the value of. But all the intervals between the points of the two sequences are spaced apart by a distance proportional to 6 times the distance... Result : The increase in the value of the parameter, named, generates within the pattern, gaps between the points of the sequences. These gaps are proportional to. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 49

50 Note : That gap helps explain why there are an infinite number of twin primes. Indeed, the basic pattern contains 15 couples of virtual twin primes. There is always at least one that is positioned in an interval equal to. The patterns are stacked in proportion as «j» increases. The new pattern, and thus the new structure begins at remarkable point. The interval between the start of two sequences, for example between sequence and the sequence is defined by this formula:. The interval between the beginning of each pattern corresponds to three times the previous interval, because each pattern is defined by three sequences. Hence we have a gap equal to. The stack of the structure of patterns always generates a gap proportional to between points of the sequences of patterns. This gap still allows to have an interval free of composite numbers between two numbers that are multiple of three, corresponding to a couple of prime numbers. In other words, there is always at least a couple of prime number in a base unit as shown in the chart page 124. Why? Each pattern allows to provide a maximum of two new primes. An offset of 6 units gives 4 spaces to fill up. This means that there is always at least one couple of twin primes within the natural period of a pattern. The high density of prime numbers in a base unit allows to understand that the probability of having a couple of twin primes in a base unit of a pattern is very high, as confirmed by the numerical results. A mathematical demonstration is proposed in this study to the section 5.9. The study of the positioning of the first point of each sequence "j" within a pattern: this point corresponds to a remarkable point «GW(j)» The three sequences "j" are always positioned in the same way within the natural period of a pattern. The initial point of a natural period is a remarkable point. The schema below shows the positioning of the remarkable points GW(j) of the sequences of a pattern:, et. Pattern : 3 consecutive sequences «j» Points that are multiples of the number 3 = Values which are multiples of 3 These values have two characteristics regardless of the value of : - They are odd (k is odd) - They are of the formk=3m+2 Note: k is the index of a number k Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 50

51 This schema shows that the remarkable points of the sequences of a pattern are positioned in the same way within the pattern regardless of the value. Results demonstrated in paragraph remarkable points in space N page We have demonstrated that the remarkable points GW(j) correspond to the numbers of the form 3m + 2 (6m + 1 in the space N) or 3m. That is, the points GW(j) never correspond to a number of the form 3m Moreover, the remarkable points all correspond to odd value of k whose value is given by the formula of the second degree corresponding to the points GW(j). The points can only take certain values in the base pattern. These points correspond within the basic pattern to GW(j) modulo 105. The list of 12 possible values with the form within the interval of the base unit ] is as follows: This comprehensive list, with a number of 12 values is obtained as follows with MAPLE: This gives a list of 24 values for all possible remarkable points within the basic pattern: We remove the values divisible by 3 to get the final list of possible points GW(j) of the form 3m + 2 within the basic pattern. The fact that there are even values is due to the modulo. The values GW(j) are all odd. This result is the number of odd numbers of the form 3m + 2 within the base pattern. So in the basic pattern, these points are always the same. This means that all the remarkable points are positioned in the pattern according to the schema GW(j) modulo 105. Results : - The first point of each sequence GW (j) has a defined position relative to the modulo 105 which corresponds to the basic pattern. The other two sequences of the pattern have a position relative to this starting point. This relative position is limited to 12 possibilities. - The initial position of each sequence in a pattern is therefore mathematically perfectly defined according to the value of the first sequence jbase. - The relative positions of other points between the different sequences, and between the different sequences of different patterns, are mathematically related in a similar manner. Virtual prime numbers appear in each pattern within the natural period with an identical structure. Only the scale increases, in other words the distance between the numbers increases due to the increase of the period of sequences «j». These elements help explain why the distribution of prime numbers is not random. The patterns offer the same structure. The stack of patterns preserves the properties of the structure. This explains the fractal dimension of primes and therefore the difficulty of finding them, and the difficulty of connecting them by a mathematical relationship. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 51

52 The table below shows the numerical results of the first patterns for studying the two forms of virtual primes PUNv + and NIPv- within the extended period and the natural period of the patterns. Patterns Mt(i) Type of period Sequences «j» The sequence jmax is in brackets * Initial Point = remarkable point GW(j) Period Pe(i) Number of numbers that are not multiples of sequences j <jmax in the interval defined by the natural period and which begins at the initial point. PUNv+ (jmax) PUNv- (jmax) total number (Nbt) % PUNv = Nbt/Pe(i) Basic pattern Mt(0) extended /natural 0, 1, (2) GW(0)=3 3*5*7= Mt(1) Natural period 0, 4, (5) GW(3)=39 3*11*13 = Extended period 3, 4, (5) 9*11*13 = Mt(2) Natural period Extended period 0, 7, (8) GW(6)=111 3*17*19 = 969 6, 7, (8) 15*17*19 = % Mt(3) Natural period Extended period 0, 10, (11) GW(9)=219 3*5*23 = 345 9, 10, (11) 21*23*25 = % Table 8: This table shows the percentage of points PUNv and the number of points PUNv+ and the number of points NIPvwithin the natural periods of patterns. The results were as follows: - The table shows that the percentage of points PUNv increases with the natural period Pe(i). The higher the value of the period, the higher the number of points is important because the distance between two consecutive points of a sequence «j» increases when j increases. - The number of point is equivalent to the number of points within a natural period or within an extended period. The study is completed with another example with the pattern. The pattern is characterized by the sequences 3, 4, 5 hence a natural period equal to (Cf. annexe I). Note : The period of the sequence j=3 corresponds to 9 that is to say at the prime number 3 squared. Another example is given for the pattern with defined sequences j=3, 4 and 5 corresponding to the first following numbers : 3, 11 and 13 below. Another representation consists in counting the distance between two points UNNP within the extended period. The base interval is 9 because we studied the multiples of 9, 11 and 13. Thus, the table below shows how the intervals between these multiple are distributed. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 52

53 UNNP k (Index) Distance between 2 consecutive UNNP NA Start of the natural period : cycle of 1287 numbers Start of the symmetry SCMo UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 53

54 UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP Symmetry is observed around the SCMo UNNP k (Index) Distance between 2 consecutive UNNP SCMo UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 54

55 UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP UNNP k (Index) Distance between 2 consecutive UNNP End of the symmetry SCMo UNNP k (Index) Distance between 2 consecutive UNNP Pod Table 9: distribution of points UNNP within the natural period of the pattern Mt(1) Symmetry is observed around the Pod This representation allows to highlight the points of symmetry of the structure of the pattern for the composites. Annexe I contains all of the distribution. Changes in the number of points are observed for the number of points and the number of points. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 55

56 Equivalence between forms PUNv+ and PUNv- : Annex I contains all of the distribution. A race was observed between the amount of numbers of the forms PUNv+ and PUNv-. The form PUNv+ run after the form PUNv-. The number of points PUNv+ and the number of points PUNv- are always equivalent in the period defined by Mt(i). In the case of patterns Mt(i) with i> 0, we observe that the number of points the largest among the number of points PUNv+ and the number of points PUNv- alternates regularly. The number of points NIPv- is often in front of because he starts the race. This helps explain the alternation of the form the more numerous between the form PUN+ and the form PUN-. The form PUNv- start the race. This explains why the number of points of the form PUNv+ is in front of that of the form PUNv- less often. The pattern has the same properties as the pattern namely : - The two points of symmetry SCMo and Pod - A variation of density at the beginning of the natural period, around the point SCMo and before the point Pod, as in the case of the pattern - A similar distribution to that of the pattern between the two forms PUNv+ and PUNv-. - The interval with the minimum density of points PUN is not located within the base units of the pattern. Indeed, the first sliding interval, within the natural period, with the minimum density corresponds to ] 419; 424]. The combined base units of pattern Mt(1) have the range [39; 110]. Note : The numbers are determined by taking into account the sequences j = 0, j = 4 and j = 5. The sequences "j" which are multiple of three is not taken into account for the determination of. The unit of measurement of the density of points PUN is the maximum distance between two numbers PUNv. This unit allows to study the evolution of the density of the points PUN within the base units. The figure below gives an example. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 56

57 Pattern with ; 3 consecutive sequences «j» Natural period of Mt(0) =429 Maximum distance between two PUNv= Base unit of the sequence Base unit of the sequence Base unit of the sequence Max :72.72% 11 k Density in % 63.63% 54.54% Average :56% Min :20 % Fluctuation in density of 65% 54.16% 53.57% Density within a base unit Interval composed of 11 Nombre Density of PUNv points of PUNv ]39 ; 50] 7 Comments ]50 ; 61] 6 [61 ; 72] 6 And so on ]633 ; 644] 8 First consecutive interval within the natural period with a maximum density ]419 ; 424] 1 First sliding interval within the natural period with a minimum density Note it is important to note that the average density of within the base units is greater than the minimum density of 20% within the natural period of the pattern. This figure above shows that the calculation of the density of points within the patterns, taking the maximum distance between two consecutive points as a measurement unit for the density, generates fluctuations of density of type oscillatory within the base units. We have the same behavior as in the case of the basic pattern. The patterns are therefore at the origin of the oscillation of the values of the properties of primes such that their density. These motifs accumulate itself to form combined patterns. What are the properties of these combined patterns? The same properties are retained as in the case of non-combined units? Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 57

58 3.6.2 Combined patterns and Goldbach s conjecture Definition : The combined patterns consists in superimposing patterns with. Par example :. The structures of patterns are stacked to form a combined pattern. a- Features of a combined pattern The stacking of structures of patterns led to the study of the following properties: - Property 1 : The variation of the density of points The schema below shows the impact of the accumulation of two patterns on the variation of the density of Points at local and global level and at the level of base units. To study the variation of the density of PUNv points at global level, we use a cumulative interval. To study the variation of the density of PUNv points at local level, we use a sliding interval. Combined Pattern with ; 3 consecutive sequences «j» Accumulation of patterns Mt(0) and Mt(1) Base unit of the sequence Max :45.45 Density PUNv in % Min : 9.09 % 27.27% 11 Base unit of the sequence 36.36% 36.36% 40% 37.50% 32.14% Base unit of the sequence Average :38.36% Global fluctuation of the density of points PUNv in a combined pattern Density within a base unit k Natural period of Mt(1) =429 local fluctuations Natural period of Mt(0) =105 The density of points within the base units of the combined pattern decreases relative to the pattern. This evolution is consistent due to the addition of a structure with a lower density of points. Indeed, the average density of points for is 56%, while the average density Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 58

59 of points for is 46%. This is a global fluctuation. We have not used a sliding interval with the maximum distance "11". With a sliding interval, we observed largest local fluctuations. Note : it is important to note that the average density of points within the base units of the pattern is much higher than the minimum density of 9.09% within the natural period of the pattern. This schema shows that the accumulation of several patterns modifies the fluctuations of the local density of points that are oscillatory type within the basic units. The accumulation of these patterns generates unpredictable fluctuations in the density of by analytical formulas. Indeed, this type of structure is characteristic of fractal structures. This stack of structure of the same nature explains the fractal dimension that is observed when studying the properties of primes such that their distribution with the Riemann's hypothesis (reference [1] page 180), or their density. - Property 2 : The position of the interval of the minimum density within the natural period. The first interval with the minimum density of points for the combined pattern is. It should be noted that this point is beyond the base unit of the third sequence of the pattern. This means that this interval of minimum density is located within the natural period of the pattern but not within the base units of the pattern as shown in the given calculations Figure 14 page 71. In addition, it was demonstrated in paragraph Distribution of primes: Brocard's conjecture and Legendre's conjecture page 82. Warning: do not confuse the minimum density measured using consecutive intervals, with that measured with sliding intervals. Indeed, sliding intervals give a minimum density less than that measured with consecutive intervals. In addition, the position of the first sliding interval having a minimum density is less than that obtained with consecutive intervals. However, only the consecutive intervals allow to measure a minimum density between two consecutive numbers of the PUNv form. For example, the first sliding interval with a minimum density of points for the combined pattern is The density is equal to. Note that this point also lies outside the limits of the base unit of the third sequence of the pattern. This point is beyond the combined units of the pattern. - Property 3 : The difference between the number of points PUNv+ and the number of points PUNv- within the natural period of a combined pattern The number of points PUNv+ and the number of points NIPv- are equivalent in a natural period of a combined pattern. This is because all the patterns have a natural period which contains an equal number of number of PUNv + and NIPv-. The fact that all the combinations between the different sequences of patterns are part of the natural period of the combined pattern explains why an equivalent number of forms PUNv+ and NIPv- exists within natural periods, regardless of the combined pattern. We can consider each sequence as a sinusoidal function (Cf.4- Mathematical characterization of odd primes - Riemann Hypothesis page 91 ). All combinations between this functions are in a natural period. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 59

60 Examples are given in the table below. Combined Pattern Mtc(i) Motif de base Mt(0) Sequences «j» The sequence jmax is in brackets * Initial Point = remarkable point GW(j) Period Pe(i) 0,1,(2) GW(0)=3 3*5*7=105 Mtc(1) 0,1,2,4,(5) GW(3)=39 105*11*13 = Mtc(2) 0,1,2,4,5,7,(8) GW(6)= *17*19 = Mtc(3) 0,1,2,4,5,7,8(10) GW(9)= *23 = Number of numbers that are not multiples of the periods of the sequences j<=jmax in the interval defined by the natural period and which begins at the initial point. PUNv+ (jmax) PUNv- (jmax) total number (Nbt) % PUNv = Nbt/Pe(i) % % % % Table 10: Equivalence between the number of points PUNv+ and the number of points NIPv- within the combined patterns The stacking of patterns and thus of their structure allows to keep the same properties. A combined pattern has within his natural period the following properties: - Two points of symmetry SCMo and Pod, - A density variation at the beginning of the natural period, around the point SCMo and before the point Pod, as in the case of patterns Mt(j), - A similar distribution of points PUNv between the forms PUNv+ and PUNv- within the natural period of the combined pattern. - The sliding interval of minimum density of points PUN is never located within the base units of a combined pattern. This means that the maximum distance between two primes in base units is lower by at least one unit compared to the maximum distance obtained in the natural period. This result is used to mathematically prove the conjecture of Legendre. b- Goldbach s conjecture. The following provides an explanation of the Goldbach's conjecture. The paragraph Distribution of primes: Brocard's conjecture and Legendre's conjecture presents a demonstration of the Legendre's conjecture, and shows that the maximum distance between two prime numbers evolves as predicted by Cramer's conjecture when «n» tends toward infinity. This means that the number of prime numbers within the two intervals and increases when the value of the number «n» increases. In addition, this study shows that the number of possible decomposition for an even number, that is decomposable into two prime numbers, also increases with the value of the even number (Cf. paragraph 6.2 Resolution of Goldbach's equation). The patterns and the combined patterns have an internal symmetry within their natural period. The internal symmetry of the pattern explains why an even number can be decomposed into two virtual primes. The conservation of the symmetry in the combined pattern explains why the decomposition of an even number is possible into two primes. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 60

61 Furthermore, this symmetry helps to explain why prime numbers of both forms 6m + 1 and 6m-1 are always present in any base unit. We call this symmetry, Goldbach s symmetry. The search of symmetries within the set of prime numbers has already been the subject of mathematical studies [14]. The evidence we have shown that there are always two types of prime numbers in a base unit. The distance between two primes follows a logarithmic evolution squared given by Cramer's conjecture (see next paragraph). This means that there is always prime numbers in the two intervals and. More «n» is large, the more there is prime numbers of the two forms. The Goldbach's conjecture has been validated numerically up to numbers greater than [9]. Beyond such values, the interval contains numerous base units. We are therefore certain to get an increasing number of primes of the form and in the range when "n" increases. This study helps to understand why Goldbach's conjecture is valid regardless of the scale of even number studied. Note : the composite numbers and the odd primes are complementary in the set of odd numbers. This means that the symmetries that exist in the set of odd primes also exist in the set of odd composites. This symmetry of odd composites is demonstrated in paragraph 5.5 Mathematical form of the composite numbers page 120. The paragraph The foundations of a mathematical proof page 205 laid the foundations of a mathematical proof for the Goldbach s conjecture Distribution of primes: Brocard's conjecture and Legendre's conjecture Bertrand's postulate, also called Chebyshev theorem [6] [7] [8] says that between the value of an integer and twice that value there is always a prime number. More formally, if "n" is a natural integer greater than or equal to 2, then there is always at least one prime number such that: We shall prove the existence of at least a prime number within the following interval] n ; ] and the existence of at least two prime numbers in the interval [ n ; ]. (Warning: the difference between the two intervals is due to the Hook open and closed). With, we will demonstrate that the Legendre's conjecture is valid. We always have a prime number between and. Brocard's conjecture is also proven. In addition, the study of the maximum distance between two points PUN in a base unit allows to validate the Cramer's conjecture. Indeed, the formula determined empirically to describe the evolution of the value of this maximum distance, leads to Cramér's formula when the "n" number tends toward infinity. The study of the evolution of the properties of prime numbers is possible because of the existence of the base units. These base units are mathematically determined using the formula to determine the boundaries of the interval of a base unit. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 61

62 In addition, the prime numbers within these base units behave according to the rule of the basic schema corresponding to the formula. The behavior of these numbers is the same regardless of the order of magnitude of the number, so whatever the base unit, even to infinity. The pattern, which combines 3 consecutive base units, allowed to highlight a local symmetry that exists within the natural period of the pattern. In addition, the patterns allow us to understand the oscillatory variation of the properties of primes. The combined patterns helped to understand the impact of the superposition of these patterns. Indeed, the accumulation of symmetry of patterns amplifies the variability of the values of the properties of odd numbers such as the maximum distance between two primes, the density of primes... This accumulation of structure corresponds to a fractal structure. This explains the aspect of the randomness of the occurrence of primes in natural numbers. But this is not random. The formulas for describing the evolution of the properties of prime numbers are explained using three structural elements that are: - The basic schema, - The base unit, - And the basic pattern and combined patterns. The formulas defined in this study are related to the base units. The properties of the odd numbers are treated only within the base units. The main parameter is the parameter "j" that matches the index of an odd number. The formulas thus determined allow to validate the behavior of the properties described by these formulas when the numbers tend to infinity. Why? Until today, mathematicians study the properties of primes within a single interval [0; infinity]. The theorems and conjectures therefore correspond to the behavior of the studied properties when the numbers tend to infinity (asymptotically to infinity). With the base units, studying the properties of prime numbers is valid within these units even if the numbers tend to infinity. The evolution of the properties of prime numbers does not change with scale. That is, if, for a given property, calculations show a logarithmic evolution per base unit, this evolution will not change even to infinity. The rules or structural elements evolve in the same way regardless of the scale. This allows to assert that the Cramer's conjecture is valid. This study will explain it by using the study of the maximum distance between two primes within base units and within the combined patterns. The demonstration is the following: Within the natural periods, we will define the maximum distance between two primes PUNv. We will study natural periods of combined patterns. Each pattern is a set of 3 consecutive sequences defined by, and. The first sequence is a multiple of three, and this sequence generates points corresponding to multiples of the number three. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 62

63 This maximum distance is used as a unit of measurement of the density of points within the natural period. The density is the number of divided by the number of total points within the interval defined by this maximum distance. Determining the optimal configuration to minimize the number of primes necesary to complete all the numbers of the form 3m, 3m + 1, m of a given interval. The first prime number is "3". The first sequence of points corresponds to multiples of three. The objective is therefore to fill the spaces left by the multiples of "3". We do not take into account that the sequences 'j' which correspond to a multiple of three. Indeed, these sequences are multiples of three and can not correspond to the numbers and.. Note : The index of an odd number is equal to with. There is a relationship between the index of an odd number and the index of its square root j. Why study the maximum distance between two prime numbers within a base unit? We have shown in section Primality test in space W, that only sequences "j", with a period corresponding to a prime number, in the range allowed to generate all composite numbers including the base unit This is equivalent to study the primality of a number "n" between [0; n]. It is therefore not necessary to take into account the sequences "j" whose the period correspond to prime number greater than " n" because these sequences generate identical points to those whose period is less than n, within the base unit "jmax". For each base unit, we have a fixed number of prime numbers to get the largest number of the consecutive composite numbers in a base unit. For example, to determine the maximum distance between two prime numbers in the base unit j = 5, we will use the primes that correspond to the periods of sequences "j" in the range [0; 5]. The prime numbers correspond to the following values: [3; 5; 7; 11; 13]. We therefore have a fixed number of 5 prime numbers to define the largest serie of composite numbers. The composite numbers must only be multiples of these five prime numbers. Why should we study the maximum distance between two primes in a combined pattern? To get consecutive numbers of composite numbers, it is necessary to complement the intervals between multiple numbers of three. The two numbers within this range are of the form and. They can not be multiples of a same prime number. It is therefore necessary to have two additional distinct prime numbers. For this it is necessary to study the maximum distance between two prime numbers within a pattern. The schema below shows why it is necessary to obtain two additional prime numbers to complete the interval between two sets of consecutive points for the basic pattern. The numerical results are given in a table on page 71 Figure 14. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 63

64 3m 3m+2 Multiple of m Multiple of 7 Multiple of 3 10 numbers k(j,n) To increase the maximum distance between two primes, 2 other primes are necessary to complete the numbers of the form 3m + 1 and 3m + 2 within this interval. Figure 12: Graph showing the maximum distance between two prime numbers for the first base unit j=0. Why take two consecutive sets of composite numbers in order to link them together to increase the maximum distance between two primes? It is impossible to obtain three consecutive prime numbers except the first interval with 3, 5 and 7. Each sequence generates sequences of points. We have a multiple of the number three each time we generate three consecutive points : 1 st point with n=0 : 2 nd point with n=1 : Multiple of three 3 rd point with n=2 : 4 th point with n=3 : 5 th point with n=4 : Multiple of three General case: We get : Consider that we obtain the largest series of consecutive composite numbers for a given sequence, how can we extend it? We can not extend it beyond two times its distance because we get multiples of three. This study shows that to connect two identical series, it is necessary to take prime numbers in the two series in order to combine these two suites. The Figure 13 page 69 shows that to combine the two series, it is necessary to take within each of the two series two numbers which correspond to the multiple of two primes. These two numbers are taken from the ends of the series to preserve a maximum number of consecutive composite. Taking these two numbers at the ends of series generates an hole at the end of the combined series that can not be filled. Indeed, we have a fixed number of prime numbers by combined pattern. The increase in the size of the series so can not exceed twice the distance of this series. We will study how to get the series of consecutive composite numbers the largest with a single multiple of every prime number except with the multiples of "3". Then, we will study how to combine this series with another series also containing one each multiple of prime number in order to increase the number of consecutive composite numbers. Study of the organization of odd numbers in order to get the largest serie of consecutive composite numbers. It is important to define the organization of the odd numbers according to their initial configuration: a- The schema below shows the natural organization of odd numbers Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 64

65 3m 3 We will position the odd numbers according to their order of appearance. When a composite number appears, it is replaced by the primes that compose it. For instance, the number 35 is composed by «5» and «7». This organization shows that the multiples of the new prime numbers are inserted between the multiples of previous prime numbers. 3m+1 3m+2 Multiple of 7 Multiple of 5 Multiple of Multiples of twin primes k(j,n) 25: this number is a multiple of 5. It must be associated with a multiple number of 5 as shown by This organization shows that a prime number appears when no multiples of the preceding prime numbers correspond to the number which must complete the interval. b- The schema below shows an inverted organization of consecutive odd numbers. The numbers that are multiples of three does not change position. For example, the number "5" appears before the number "7". This order is reversed. Positioning "7" before "5". Multiple of 3 Multiple of 7 Multiple of ? 17 Twin primes k(j,n) 25? : This number is a multiple of 5. It must be associated with a multiple of 5. The result: A VOID SPACE. An additional prime number is required to complete the serie of odd composite numbers. This organization shows that this initial configuration of odd numbers needs for the same interval of more primes to occupy all points of the interval. We swapped all twin primes. But swapping a single pair of consecutive odd numbers, eg 5 and 7, leads to the same result. For other couples, it is necessary to increase the visible range on the diagram to verify this result. c- The schema below shows a configuration having as a starting point a multiple of a prime number greater than or equal to five, which is also a multiple of three. We chose the number "15". The first position of multiple of 5 is identical to that of the first position of multiple of 3 Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 65

66 Multiple of 3 and 5 Multiple of 7 Multiple of ? ? Twin primes k(j,n) 25? : This number is a multiple of 5. It must be associated with a multiple of 5. The result: A VOID SPACE. Two prime numbers are required to complete the serie of odd composite numbers.. This organization shows that this initial configuration of odd numbers needs for the same interval of more primes to occupy all points of the interval. Why any change in the natural position of numbers requires more primes to complement numbers of the form 3 * m + 1 and m + 3 * 2 within a given range? This behavior is related to the organization of odd numbers. A prime number appears when the multiples of previous primes can not complete all the mathematical forms or. We know that there are infinitely many prime numbers. This means that no configuration of a limited number of prime numbers can fill in all the mathematical forms and. If this were not the case, then there would not be an infinity of prime numbers. Change the initial configuration of multiple primes is to move toward large numbers. But we know that the number of primes is infinite. We therefore need new primes to complete the range when the natural configuration is changed. Conclusion : The natural configuration, ie the order in which are positioned the primes, is one that requires the least of prime number to complete the mathematical forms and for a given interval. Determination of the interval of measurement of the density within a natural period of a pattern. We need an interval that allows us to measure the variations of the number of primes within a period of a sequence of a combined pattern. A pattern consists of three sequences "j" whose periods respectively correspond to three consecutive odd numbers with the mathematical form 3m, 3m + 1 and 3m + 2. Only forms 3m +1 and 3m + 2 can complete the space left by two consecutive numbers multiple of three: [3m; 3 (m + 1)]. The natural period of this pattern is calculated by multiplying prime numbers associated with each mathematical form. For example the combined pattern Mtc(0) consists of the following sequences j = 0, j = 1, j = 2, the corresponding periods are 3, 5, 7 and the associated prime numbers are 3, 5, 7 hence the natural period is equal to 105. For instance the combined pattern Mtc(1) consists of the following sequences j = 0, j = 1, j = 2, j = 3, j = 4 and j = 5, the corresponding periods are 3, 5, 7, 9, 11, 13 but the associated prime numbers are 3, 5, 7, 11, 13 hence the natural period equal to In the natural period of a pattern, what is the configuration of composite numbers that will get the maximum consecutive composite numbers without the need for a composite number related to a new Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 66

67 prime number? In other words, what is the range constituted by a maximum of composite numbers that we can obtain with prime numbers associated with a combined pattern? We have the combined pattern Mtc(0) with j=0. The primes belonging to Mtc(0) are : 3, 5 and 7. 3m 3 3m+1 3 3m+2 3 Empty space: no multiples of prime numbers 5 or What length should have a gap to be made up only of composite numbers? What positions must have the multiples of the numbers 5 and 7 to occupy the highest number of position of type "3m +1" and "3m + 2" in a given interval? To obtain the greatest possible interval, consisting of successive composite numbers, with the prime numbers of a combined pattern, it is necessary that there be the least numbers of the form 3m + 1 and 3m + 2 to be completed by composite numbers in the interval sought. What is the maximum distance that allows to obtain the fewest numbers with the mathematical forms 3m + 1 and 3m + 2 to complete when you have a fixed number of prime numbers? The extreme cases which correspond to the smallest maximum distance and to the maximum distance the largest, are defined respectively by a minimum configuration and a maximum configuration as follow. a- The maximum configuration that allows to obtain the maximum of numbers of the form and that are not multiples of prime numbers defined in a combined pattern, is the sum of the two highest numbers of the form and. So the interval has a value equal to the sum of values of the number and. The value 3m corresponds to the index «j» of the combined pattern Mtc(j) hence 3m=j. Hence the maximum interval (gap) is. Consider the case of combined pattern Mtc(0) with j = 0. The set of prime numbers within this combined pattern is the following: {3, 5, 7}. The largest value of 3m is «0» because and with. This value of "3m" is the index of the combined pattern Mtc(0). It means that 3m=j=0. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 67

68 3m 3m+2 Length of the interval : m This configuration allows to obtain 6 numbers with the mathematical form 3m + 1 and 3m + 2, which are not multiples of prime numbers 3, 5 and 7. It is therefore necessary to take other primes to be able to complete the interval. b- The minimum configuration that allows to obtain the minimum of numbers of the form and that are not multiples of prime numbers defined in a combined pattern Mtc(j), corresponds to an interval with a length equal to 3m+2 with. Indeed, insight this interval, we have all other intervals defined by all other sequences of the combined pattern: 3m + 1 is defined inside 3m + 2 ; 3m is defined inside 3m + 2. This interval corresponds to the distance equal to with. The schema below shows the case of the combined pattern Mtc(0). The prime numbers associated with the combined pattern are: 3, 5, 7. 3m 3m+1 3m+2 k(j,n) 6 other primes are necessary to complete the numbers of the form 3m + 1 and 3m + 2 in this interval which consists of 13 numbers. Length of the interval : other primes are necessary to complete the numbers of the form 3m + 1 and 3m + 2 in this interval which consists of 8 numbers. How to use this minimum configuration to increase the number of consecutive composite numbers? We know we can not have more than two successive numbers which are multiple of primes different of the mathematical form 3m. Indeed, there will always be among 3 successive numbers multiple of a prime number, a number that is multiple of 3. We can associate only two identical structures with a minimum of numbers with the mathematical form 3m + 1 and 3m +2 which must be completed. The structure that is taken into account is the natural configuration. The association of two natural configurations leaves a central gap that needs to be completed to extend the series of the consecutive numbers that are multiples of primes of the combined pattern Mtc(j). Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 68

69 Only multiples of prime numbers associated with the combned pattern are possible. It is therefore not possible to obtain an interval consisting of multiple consecutive prime numbers 3, 5, 7 whose distance is greater than ?? Ic : central gap Two natural configurations positioned symmetrically with respect to a central gap This schema shows that we have not enough primes to complete the central gap, and thus to combine both natural configurations to extend the number of consecutive composite numbers. There is a maximum of 4 consecutive composite numbers. There is therefore a theoretical maximum distance, between the first and last composite number, equal to 3. The schema below shows the case of the combined pattern Mtc(1). The prime numbers associated with the combined pattern Mtc(1) are : {3, 5, 7, 11, 13} consecutive composite numbers with 11 and Ic : central gap 10 consecutive composite numbers Distance theoretical maximum of consecutive composite numbers «Dmt =10-1=9» Figure 13: This figure shows how to combine two series of consecutive composite numbers This figure shows that by positioning the last two odd numbers belonging to the combined pattern in the central interval, then one increases the number of consecutive multiple. In the above case, one passes from 7 consecutive numbers to 10 consecutive numbers. There is therefore a theoretical maximum distance between the first and the last consecutive number multiple, equal to 9. This minimum configuration allows to keep a number of numbers of the mathematical form 3m + 1 and 3m + 2, which are not multiples of prime numbers belonging to the combined pattern, equal to two regardless of the combined pattern with. The schema below shows how, with this configuration, it is possible, like Russian dolls, to assemble the intervals. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 69

70 Two natural configurations positioned symmetrically with respect to a central gap The two numbers to complete the central gap must be multiples of twin primes Distance theoretical maximum of the number of consecutive composites «Dmt» Dmt = 9 This configuration is related to the appearance of twin primes. Indeed, if one or none of the two successive numbers of type 3m + 1 and 3m+ 2 is prime, then the maximum distance of successive composites will be lower than the theoretical distance because we could not associate the two natural configurations together. The theoretical maximum distance of the number of consecutive composites is equal to with * +, with * + with N is an natural number. The theoretical maximum distance between two prime numbers is Hence * +. hence with * +, with * + with N is an natural number. This distance is related to the appearance of twin primes. If the two numbers 3m +1 and 3m + 2 which must complement the central gap, are not multiples of twin primes, then the distance is necessarily less than the theoretical maximum distance. The results of the numerical calculations that provide the maximum distance between two primes within a natural period, are summarized in the table below. Combined pattern Mtc(t) Primes of the pattern Mt(t) Mtc(0) (5, 7) twin primes Mtc(1) Mtc(2) (11, 13) twin primes (17, 19) twin primes maximum theoretical distance maximum distance measured Position «k» of the maximum interval within the natural period Dmt = 15 Dmt < [98 ; 103] [3 ; 38] [4 718 ; 4 729] [39 ; 110] [ ; ] [111 ; 218] Dmt = 27 k(j,n) Dmt <33 25 and 35 are not prime numbers. Both natural configurations can not therefore be combined to form the theoretical maximum distance. Position of the interval of combined base units of the combined pattern Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 70

71 Combined pattern Mtc(t) Primes of the pattern Mt(t) maximum theoretical distance maximum distance measured Position «k» of the maximum interval within the natural period Mtc(3) 23 < [ ; ] Mtc(4) (29, 31) twin primes [ ; ] Mtc(5) 37 < [ ; ] Position of the interval of combined base units of the combined pattern [219 ; 362] [363 ; 542] [543 ; 758] Figure 14: evolution of the maximum distance between two primes within a natural period of a combined pattern. The calculations become impossible because the computing time would be excessive beyond Mtc(5). Supercomputers are then required. The important point to note is that the position of the interval corresponding to the maximum interval is not within a base unit, as predicted by theory. The unit of measurement for studying the density of primes is the theoretical maximum distance obtained within the natural periods. This unit of measurement depends on the parameter «j». Two numerical examples providing the composite numbers that constitute the maximum interval between two virtual primes: - the first combined pattern: Mtc(0) Table Figure 14 gives us the maximum interval following: [98, 103]. The bounds correspond to the indices of the two virtual primes. The indices of composite numbers correspond to the following values : 99, 100, 101, 102. Hence "Ni" is a multiple of the following primes k Ni the second combined pattern: Mtc(1) Table Figure 14 gives us the maximum interval following: [4 718 ; 4 729] k Ni We will study the maximum distance between two consecutive primes within each base unit. The graph below shows the evolution of this maximum distance from base unit. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 71

72 Maximum theoretical distance within a natural period. Maximum distance measured within base units. Figure 15: Evolution of the maximum distance between two primes PUN per base unit «j» This is a distance between two primes PUN in space W. We do therefore not account for even numbers. The Ceil() function is used to obtain an integer. As this is an average, we take the lower value of the real number obtained with the natural logarithm. (( ( ( * )), Auteurs : François et Marc WOLF Page 72

73 We multiply by two to account for even numbers between two odd numbers in the set of natural numbers N. Further, base units start with j = 0 which corresponds to the value n = 9. However, as corresponds to a maximum, this remains true even for n<9. With * +, * + ( ( ( ( * +)))) ( ( * +) ) (( ( When n approaches infinity, the function is approximated as follows : )) ) (( ( )) * hence ( ) hence The maximum distance within a base unit evolves as a natural logarithm at squared:. This is explained by the probability of getting a pair of twin primes. We have shown that for having a significant increase in the maximum distance between two primes, it is necessary to get a couple of twin primes. The Cramer's probabilistic approach suggests that events for each prime number in a pair of twin primes are independent. We then obtain the probability of having a pair of twin primes equals : empirically. In addition, we showed that there are an infinity of twin primes in section This is consistent with the results found The Cramer's idea to base its probabilistic theory on the appearance of the twin primes is valid. However, we do not believe that his conjecture is valid because his conjecture can only match the maximum average distance between two primes due to the probabilistic nature of his theory. The maximum absolute value between two primes less than. We have not obtained mathematically a better upper limit of this value. We believe that, given the numerical results, the maximum value follows a logarithmic squared law. The constant of this law remains to be determined mathematically. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 73

74 The graph shows that the interval in which the maximum distance fluctuates, can be approximated as follows: ( ( ( * )) This variation corresponds to the appearance of prime numbers that do not belong to a pair of twin primes. The variation does not correspond to an logarithmic evolution squared. Note : Maximum and minimum limits have been defined empirically as follows: The maximum limit corresponds to the red curve in the graph above. The formula is as follows: * + ( ( ( ( * +)))) ( ( * +) ) * + ( ( ( ( * +)))) ( ( * +) ) ( ( ) ) ( The minimum limit is the brown color curve in the graph above. The formula is as follows : ( )) * + ( ( ( ( * +)))) ( ( * +) ) * + ( ( ( ( * +)))) ( ( * +) ) ( ( ) ) ( ( )) Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 74

75 This formula highlights the base elements of the structure of prime numbers, which are at the origin of the organization of all the odd and even numbers. These base elements are: set of base. We defined a maximum average distance between two primes. We explained the evolution of this distance with the appearance of twin primes and the appearance of prime numbers. The appearance of twin primes is the most influential parameter on the evolution of the maximum distance between two primes. This influence is reflected in a changing. However, the occurrence of prime numbers that do not belong to a pair of twin primes also influences this distance. This change results in a change in this distance as. The figure 15 shows this influence. The blue curve corresponds to the addition of a positive term such as. The yellow curve corresponds to the subtraction of a term such as. Most of the distances between two primes are bounded by these two terms around the mean value. Conjecture Within a base unit, the property regarding the distance between two primes has the following characteristics in space N : - The maximum distance between two prime numbers fluctuates as a logarithm squared. This value is within the interval : With * + ( ( ( ( * +)))) ( ( * +) ) (( ( - The average distance evolves as - The minimum distance corresponds to twin primes. The minimum distance therefore corresponds to the value «2». )) ) Conclusion : The maximum distance can be approximated by a constant when «j» tends to infinity. This means that the number of primes increases within base units when the value «j» increases. Determination of the maximum interval between two primes in a natural period of a combined pattern. Definition : The sequence of base of a combined pattern Mtc(t) is equal to. Within a natural period associated with a pattern, the configuration which allows to obtain the maximum of consecutive composite numbers, is obtained when the periods and of the two sequences of the pattern correspond to prime numbers. The multiples of these two periods are not multiples of the previous primes. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 75

76 Definition: The distance corresponds to the maximum distance between two prime numbers in the space W within a combined pattern. It is this distance which serves as a unit of measurement for the density of primes in the base units as in the natural periods of a combined pattern. 1- Case of combined pattern, equivalent to the pattern Mt(0), Mtc(t) with t=0 :. The sequence of base of the combined pattern Mtc(0) is equal to. The sequences correspond to j=0 (period=2j+3=3) ; j=1 (period =5) ; j=2 (period =7). The maximum distance between two prime numbers is equal to 5 :. This distance corresponds to the sequence and therefore the period of this sequence is equal to. This distance corresponds to the minimum density of virtual primes. Indeed, we have only one virtual prime number (PUNv) on a total of 5 numbers. The schema below shows the organization of numbers which are multiples of prime numbers 3, 5 and 7. The numbers multiple of "3" are separated by three units. The numbers multiple of "5" are separated by five units. The numbers multiple of "7" are separated by seven units. All combinations of distances between these numbers multiple of primes are present within the natural period of the combined pattern. Definition : A configuration of numbers, eg the primes 3, 5 and 7, consists of placing the first point of each prime number on an axis. Then, we generate the multiples of these numbers. The points that do not correspond to multiples of prime numbers are considered as virtual primes that we call PUNv. In a natural period, there are two configurations which allow to obtain a maximum number of consecutive composite numbers that are multiples of this primes : 3, 5 et 7. Why two configurations? This is related to the symmetry of Goldbach. All numbers configurations are duplicate in a natural period. (See also the case of the combined pattern Mtc(1) below). Multiple of 3 Multiple of 7 Multiple of 5 Both configurations (Config 1 and Config 2) give the maximum number Config 1 Config 2 of successive numbers that are 7 5 multiples of prime numbers 3,5,and Previous interval: the density is equal to 3 PUNv out of 5 numbers. PUNv PUNv The configuration of the above numbers optimizes the maximum distance between two prime numbers PUNv in the case of the config 1 and config 2. This distance may not exceed "5". Indeed, all other configurations bring it to reduce the distance between two primes PUNv as demonstrated in the previous paragraphs. Note : Both configurations (Config 1 and Config2) differ in the order of the position of multiples of twin primes 5 and Case of combined pattern Mtc(t) with t=1 : +5 : interval of minimum density: 1 PUNv out of 5 numbers. 5 7 PUNv The sequence of base of combined pattern Mtc(1) is equal to. The sequences and their period correspond to j=0 (3) ; j=1 (5) ; j=2 (7) ; j=1 (11) ; j=2 (13). Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 76 3

77 5 It is necessary to place the numbers 7, 5 in this order, within an interval whose limits are multiples of three in order to obtain these numbers symmetrically to the central gap «Ic» as positioned in the below schema. Definition : The central gap consists of the periods of the last two sequences of the combined pattern, so the sequences are and. For the combined pattern Mtc(1), the central gap consists of multiple of twin prime numbers 13 and 11. Note : The order of these two sequences is not important within this configuration for this combined pattern but it has a significance for the following combined pattern. We have two configurations for multiples of primes (or composites) in order to obtain a maximum distance between two primes. The difference between the two configurations only comes from the positioning of the last two numbers which are 11 and 13 in the central gap. The order of this two numbers does not matter. The order of the other numbers is identical to the config 1 and to the config 2. Only the configuration 1 is taken into account in the study. Both configurations within the natural period exist, but they are equivalent with regard to the maximum distance between two primes. Only these two configurations provide a maximum distance between two primes within the natural period. All other configurations provide a strictly lower distance than that maximum distance. Indeed, if we are positioning any prime number: 5, 7, at a position other than those defined in the configuration 1, then we have shown in the preceding paragraphs that other points PUNv appear and they can not be multiple of "3" or "11" or "13". The maximum distance between two prime numbers is then strictly less than the maximum distance obtained in the case of the configuration 1. Note : this characteristic is true regardless of the combined pattern. The maximum distance is equal to 11 hence with. This distance corresponds to the sequence and so the period. Chart showing the configuration 1 PUNv 11 Ic : central gap Config Previous interval : the density is equal to 7 PUNv out of 11 numbers. PUNv Chart showing the configuration : interval of minimum density: 1 PUNv out of 11 numbers. PUNv Next interval : the density is equal to 5 PUNv out of 11 PUNv 11 Ic : central gap Config Previous interval : the density is equal to 7 PUNv out of 11 numbers. PUNv +11 : interval of minimum density: 1 PUNv out of 11 numbers. PUNv Next interval : the density is equal to 5 PUNv out of 11 Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 77

78 The maximum distance can not exceed "11" because all other configurations brings to reduce the distance between two primes PUNv. Hence the maximum gap is equal to. Note : The previous distance and the next distance between two points PUNv is "1". In addition, the density of points PUNv is higher before and after the interval of minimum density. 3- Case of combined pattern Mtc(t) with t=2 : period = 3*5*7*11*13*17*19 = 15015*17*19= The sequence of base of combined pattern Mtc(2) is equal to. The sequences and their period correspond to j=0 (3) ; j=1 (5) ; j=2 (7) ; j=4 (11) ; j=5 (13) ; j=7 (17) ; j=8 (19). It is necessary to place the numbers 13,11 and 7, 5 in this order, within an interval whose limits are multiples of three in order to obtain these numbers symmetrically to the central gap «Ic» as positioned in the below schema. The central gap is constituted with numbers whose the period correspond to the two latter sequences 19 and 17. Only two configurations of composite numbers provide a maximum distance between two primes within the natural period as in the case of the previous combined units. All other configurations provide a strictly lower distance than that maximum distance. The maximum distance is equal to 17 hence with. This distance corresponds to the sequence and so the period. 3 PUNv Ic : central gap PUNv PUNv +17: interval of minimum density: 1 PUNv out of 17 numbers. 3 The maximum distance can not exceed "17" because all other configurations brings to reduce the distance between two primes PUNv. Hence the maximum gap is equal to. Note : The previous distance and the next distance between two points PUNv is "1". In addition, the density of points PUNv is higher before and after the interval of minimum density. Indeed, in the case of the interval preceding the minimum density interval and in the case of the interval following the minimum density interval, the number of points PUNv present in these intervals whose value is equal to, is greater than 2 out of The sequence of base of combined pattern Mtc(t) with t=3 :. The sequence of base of combined pattern Mtc(3) is equal to. The sequences and their period correspond to j=0 (3) ; j=1 (5) ; j=2 (7) ; j=4 (11) ; j=5 (13) ; j=7 (17) ; j=8 (19) ; j=10 (23) ; j=11 (25). It is necessary to place the numbers 19, 17, 13, 11 and 7, 5 in this order, within an interval whose limits are multiples of three in order to obtain these numbers symmetrically to the central gap «Ic» as positioned in the below schema. The central gap will consist of periods of the last two sequences, therefore 23 and 25. The maximum measured distance is equal to 20.. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 78

79 The theoretical maximum distance should be equal to. wi Ic PUNv PUNv PUNv The optimal configuration of numbers to obtain a theoretical maximum distance can not be used in this case. This is because the number "25" is not a prime number. However all other configurations brings to reduce the distance between two primes PUNv. The calculations have given as maximum value "20". This value is less than the theoretical maximum distance equal to "23" as predicted by theory. +12 Conclusion: a- The maximum theoretical distance between two prime numbers can be obtained only if the periods of the sequences and correspond to prime numbers. This means that one must have twin primes. In all other cases, the maximum distance is less than the theoretical interval defined below by which corresponds to the set of natural numbers N to. b- We took the sequences 3 by 3 and the maximum distance was determined between two primes PUNv. The results show that: - The sequences j=0,1,2 (period=3,5,7) have a maximum distance equal to 5. Case of combined pattern Mtc(t) with t=0 - The sequences j=0,1,2+ 3,5,6 (period =3,5,7+9,11,13) have a maximum distance equal to 11 Case of Mtc(t), with t=1. Therefore, the maximum distance is linked to the integer part of the value We have : With Hence * + with * +. * + [ ] The distance corresponds to the distance between two points PUNv counting only odd numbers. In order to take into account the even numbers, the distance must be multiplied by two, because after each odd number there is an even number, where: With N=natural number. Hence Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 79

80 The integer parts are not included in the calculation. A maximum value is sought. The result obtained is above or equal to the desired value. Thus we obtain a value higher than the real value. So we obtain a maximum value. Indeed, an integer part decreases the final value. The fact of not take into account the integer part has the effect of either increasing the final result or being neutral in the operation. Hence Hence ( ( ) ) (( ( ) ) ) Hence Theorem : With Np, and Nps belonging to N, the maximum distance between a prime number «Np» and the next prime number «Nps» is equal at most to the distance: [ ]. The maximum distance corresponds to the interval between two primes. This translates to the following interval: ] ] Include the initial point which corresponds to a prime number, provides an average of two prime numbers within the following interval: [ ] It is an average because we can have only one prime number within this interval if shifting the interval in one direction or the other. However, if we take twice this maximum distance, then we get at least two prime numbers. However, the interval preceding the interval of minimum density and the interval following the interval of minimum density have a number of virtual prime numbers PUNv greater than two. Numerical calculations show that we are at least 4 prime numbers in this interval. Theorem : The interval defined by the following terminals contain at least four primes. [ ] If is a perfect square, then, hence we have the following interval : The proof is given below. Determining the maximum distance between two prime numbers in a «base unit» Each "j" sequence has a period equal to Pe_j = 2 * j + 3. The points generated by the sequence "j" are multiples of the number corresponding to that period. For example, the numbers generated by the sequence re multiples of its period is equal to "5". The study showed that by studying the sequences "j" by a set of three consecutive sequences starting with, the optimum configuration generating a maximum distance between two virtual prime numbers within a natural period of a combined pattern is linked to the position the last two numbers of the pattern, so and. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 80

81 These two numbers are located within a central gap. Their multiples are not multiples of the previous primes. These two numbers are distinct each others which means that they are coprime. We know that only two configurations can correspond to a maximum distance between two virtual prime numbers within a natural period of a combinated pattern. Question : do one of two configurations can be found in a base unit? To answer, we will study the position of composite numbers associated with each sequence and within the pattern. The base unit for a sequence "j" consists of three points corresponding to three numbers which are a multiple of the period of the "j" sequence. These three points are composite numbers. They are defined according to the parameter "j". The number "j" multiple of three is named. We will only study the sequences and. We define below the mathematical position of points within each sequence.we are going to compare these points to see if a point in the sequence «s located next to a point in the sequence at position 3m + 1 and 3m + 2 or vice versa. a- Sequence Point 1.1 : ; Point 1.2 : ; Point 1.3 : Point 1.4 : Point 1.5 : Point 1.6 : b- Sequence Point 2.1 : ; Point 2.2 : ; Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 81

82 Point 2.3 :. What are the possible distance between the points of the sequence «points of the sequence?» and the 1- Case of first point «Point 2.1» of the sequence This point is in a bounded interval by points " Point 1.3" and " Point 1.4" of the previous sequence The points «Point 1.3» and «Point 2.1» are spaced of «2». The points «Point 2.1» and «Point 1.4» are spaced of. 2- Case of second point «Point 2.2» of the sequence This point is in a bounded interval by points «Point 1.4» and «Point 1.5» of the previous sequence The points «Point 1.4» and «Point 2.2» are spaced of. The points «Point 2.2» and «Point 1.5» are spaced of «4». 3- Case of third point «Point 2.3» of the sequence This point is in a bounded interval by points «Point 1.5» and «Point 1.6» of the previous sequence The points «Point 1.5» and «Point 2.3» are spaced of. The points «Point 2.3» and «Point 1.6» are spaced of. Sequences of points «j» The schema below summarizes this analysis. Distance = 2 Distance = 4 Distance = 6 Distance = 2*jbase-1 Distance = 2* jbase +3 k Distance = 2* jbase +1 Figure 16: This figure shows that a minimal configuration can not exist within a base unit beyond the value. Conclusion : the distance between two points generated by the sequences and within a base unit is always greater than "1" if the value is greater than Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 82

83 "2". Ie the points 2.1, 2.2, 2.3 are spaced from points 1.3, 1.4, 1.5, 1.6 of a value greater than "1" for the sequences with greater than 2. We checked digitally for, the minimum configuration is not part of the base unit None of the two minimum configurations belong to the base units. The maximum distance between two prime numbers in a base unit is thus lower than in a natural period. The maximum distance between two prime numbers in a natural period can be reduced of a distance of 2 in the case of a base unit. Indeed, one can remove an odd number and so also an even number. Note : we decrease the distance of two units. However, numerical calculations show that we are well below this value. In other words, there are more prime numbers PUN in this nterval within the base units. Hence Theorem : The maximum distance between a prime number "Np" and the next prime number "Nps" within a base unit is equal at most to the following distance: [ ] The maximum distance corresponds to the interval between two primes. This translates to the following interval: ] ]. This interval is used to demonstrate the Legendre's conjecture. Include the starting point allows to get two primes in average within the following interval: [ ] If we take two successive intervals, we get = 4 primes in average. We have shown in this paragraph (3.6.3) that the number of primes increases on average in a base unit when "j" increases. Numerical calculations show that the number of prime numbers is always greater than 4, and this number is increasing on average when the base unit linked to the sequence «j» increases. The ratio between the number of composite numbers UNNP(j) and the number of prime numbers PUN(j) measured within a base unit is linked to the natural logarithm of the parameter "j" is a function that approaches the limit "1" when the parameter "j" tends to infinity. Furthermore, we showed 2 that the evolution of number of primes, with * +, is comparable to the function and therefore to that of the function of when «n» tends to infinity. 2 See Chapter II, «distribution of prime numbers» : determination of an exact formula and an empirical formula representing the evolution of the number of primes for a number less or equal to "n" [5] Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 83

84 Results of the evolution of the ratio : number of non odd prime (UNNP) / number of odd prime number (PUN) contained in the base units Ugw(j). This ratio corresponds to the density of points UNNP. ZOOM UNNP / PUN Figure 17 : Curve representing the evolution of the ratio UNNP / PUN based on the base unit j The graph shows a logarithmic growth of the density of points UNNP. In addition, the measured value of the density oscillates around an average whose evolution is logarithmic. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 84

85 The formula for the number of prime numbers in a base unit, corresponds to the number of odd numbers that are in a base unit «j» divided by the natural logarithm of this number. This formula provides the average number of prime numbers lying in a base unit «j» : is a function that approaches the limit "1" when the parameter "j" tends to infinity. Hence the following limit when "j" tends to infinity (The demonstration was conducted with MAPLE with b (j) = 1) ( * This formula was found empirically in Chapter II: Distribution of prime numbers [5]. This allows us to conclude that within a base unit, the number of prime numbers will always be greater than 4 on average and increases with the parameter «j». Theorem : The interval defined by the following terminals contain at least four primes within a base unit whose the interval is: [ ] If is a perfect square, then, hence the following interval : This interval demonstrates the Brocard s conjecture. Indeed, the distance between two consecutive odd numbers squared is equal to: With The number of prime numbers increasing within a pattern when "j" increases. Why? - The theoretical maximum distance corresponds to the period of the base unit within the combined pattern Mtc(t) avec. The maximum distance is the same for the three base units that is the combined pattern,. Cette distance est égale à. Warning : The sequence is a multiple of the prime number 3. It has the same maximum distance between two primes than. Why? Because the sequences that generates numbers that are multiples of three does not change the natural period of the pattern. Thus the maximum distance between two primes for this kind of sequence is the same than the maximum distance with the previous combined pattern. We will study this three sequences:. The base unit is composed of a number of points that increases with the parameter "j" as follows:. The base unit has 4 units more than the base unit, then the density of points PUN will be greater in the base unit than in the base unit. Indeed, we have the following inequality for the density of the primes PUN into the base units: Auteurs : François et Marc WOLF Page 85

86 The base unit has 4 units more than the base unit, then the density of points PUN will be greater in the base unit than in the base unit. Indeed, we have the following inequality for the density of the primes PUN into the base units: The density of points PUN in the base unit is greater than the density in the base unit which itself is greater than the density of the base unit. Indeed, the distance between two prime numbers is identical in all three base units, but the base unit increases by four units when "j" is incremented. There are therefore more prime numbers in the base unit than in the base unit than in the base unit on average. This is of course an average because the density of prime numbers oscillates as demonstrated in the case of the study of patterns. The schema below summarizes this analysis. Sequences of points «j» Base units Ugw(j) 4 = Maximum distance between two prime numbers PUN The density of points PUN in the base unit is greater than the density in the base unit which itself is greater than that of the base unit. Indeed, the maximum distance between two prime numbers is the same but the base unit increases by four units when "j" is incremented. k This figure shows that the density of points PUN increases between an regardless. It should be noted tha is a number that is a multiple of 3. The number of prime numbers increases when "j" increases within a combined pattern. - The curve Figure 15: Evolution of the maximum distance between two primes PUN per base unit «j», shows that the distance is much less than theoretical * + demonstrated page This evolution indicates that the number of prime numbers increases on average in the base units. The number of primes was computed, and the result is shown Figure 18: Representation of the evolution of the number of points PUN per base unit Ugw(j). Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 86

87 Number of primes PUN Figure 18: Representation of the evolution of the number of points PUN per base unit Ugw(j) This curve shows the increase in average of the number of primes per base unit. Conclusion : the number of primes increases on average within each pattern and also increases between each pattern when "j" increases. Validating the Legendre's conjecture The maximum distance between two prime numbers is studied within a pattern whose the sequence «j» maximum corresponds to. The value is a number multiple of «3». This value is equal to: * +. Definition : La distance correspond à la distance maximum entre deux nombres premiers dans l espace des entiers naturels. 1- The first sequence «j» is. The base unit for the sequence terminals : corresponds to the interval defined by the following This interval corresponds to the distance between the indices of two consecutive odd numbers squared in the space W. Note, only odd numbers are included in the interval. It is necessary to multiply by 2 this interval if one wishes to take into account the even numbers, in other words if it is desired to obtain the distance between two odd numbers squared in the set of natural numbers N. The initial point of the limit of the interval corresponds to the index of an odd number squared. For example, into space N we have the following number with. The index of this number in the space W corresponds to with * + * + and * + * * + + The last point of the terminal corresponds to the index of the odd number, that is consecutive to the odd number N1, squared minus 2, hence. Indeed, the odd number squared corresponds to the first terminal of the base unit. Note : This point can be removed from the interval without fear of removing a prime number of the interval because it is a composite number. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 87

88 Between the two odd numbers squared, we have an even number squared. The index of this number in the space W does not exist because only indices of odd numbers can be represented. However the index of the odd number that is before the even number squared corresponds to. We have the following upper bound: (EVEN NUMBER) The point corresponds thus to the following odd number: (ODD NUMBER) We define two intervals within the base unit (Space W): - Interval between which correspond to the following indices : - Interval between which correspond to the following indices : Note : It is important to note that the following points are always composite numbers regardless of the value of :. All these points are divisible by. 2- The second sequence «j» is. The same intervals as previously are defined with the base unit - Interval between - Interval between 3- The third sequence «j» is. The same intervals as previously are defined with the base unit - Interval between - Interval between The graph below shows the index of 3 consecutive numbers squared defined in the base units of a pattern in the space W in black, and their equivalent in the space N in red. Sequence of points «j» within a pattern k Interval between the indices of two consecutive numbers squared. This distance is equal to Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 88

89 A base unit exists for each "j" sequence. This figure shows the equivalence between the index of the three numbers squared which exist within a base unit and the value of these numbers into the space N. Proof : The maximum distance between two prime numbers is identical within a pattern. But the ratio of this distance to the period of one sequence is different for each of the sequences. Case : The sequence corresponds to a multiple of 3. This sequence of points has no influence on the value of the maximum distance between two primes. Therefore, the maximum distance is that obtained with the previous pattern therefore with. Hence Hence With * + and Hence (( ( ) ) ) Hence ( ( ) ) Hence the distance between two prime numbers is equal to : This result is consistent with the Legendre s conjecture. Indeed, if we consider N as a squared number: then a prime number is inside an interval whose value is equal to with. Legendre's conjecture says that this distance must be less than or equal to. We have. So the Legendre s conjecture is validated in this case. Case : We have a maximum distance between two prime numbers equal to : Check if the Legendre's conjecture is respected. N is considered as a squared number hence. Thus. It should be noted that the points generated by a sequence «j» such as these: GW(j) and, correspond to composites. These points can therefore be removed from the maximum distance in the case of the maximum bound of the interval. One can also remove the point even which is before the odd point that has been removed. So we have two cases : - If «a» is an odd number Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 89

90 The distance between two numbers squared is equal to hence. There is a difference of one unit between The number corresponds in the space W to the value. The number corresponds to an even number. To represent it in space W, it is necessary to remove «1» in order to obtain an odd number. Hence the number corresponds to the value. Hence The number of points between the minimum limit et and the maximum limit is equal to. These limits correspond to odd numbers. For accounting for even numbers, we must multiply by two, thus we get in space N a maximum distance equal to. Finally, we add the two even numbers which are juxtaposed to the limits of interval. The first even number is the number that precedes the odd number equivalent to in space W. The second even number corresponds to the number. It is after the odd number whose index is equal to. Thus we obtain a distance equal to. Even numbers are not of prime numbers (except for the number 2) we have one prime number in the range ] ]. Legendre s conjecture is validated for this case. - if «a» is an even number The distance between two numbers squared is equal to. This distance is greater than that corresponding to the distance between two prime numbers defined by. Inded, we have : So we have one prime number within the interval defined by Legendre s conjecture is validated for this case. Case : The period of the sequence is equal to, in space W. The period of the sequence is equal to, in space W. Reminder: in the space W, the maximum distance between two primes within a combined pattern Mtc(t) corresponds to with, which corresponds in natural numbers N to. In space W, the difference between the periods of the two sequences is equal to. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 90

91 In the space W, the maximum distance between two primes can be written as follows within the sequence in the pattern Mt(t): However, in the set of the natural numbers N, between two odd numbers there is one even number. The difference between the periods of the two sequences in the space N is equal to. Hence, we have in the set of the natural numbers N : Hence If we consider N as a number squared: thus we have : -, - and Legendre s conjecture is validated for this case. Indeed, the distance between two consecutive numbers squared, in this base unit is greater than or equal to the maximum distance between two consecutive primes. Conclusion : Legendre's conjecture is proved mathematically. Result: The study shows that the maximum distance between two virtual prime numbers PUNv within the natural period of a combined pattern, has for maximum value the period of the sequence equal to : This result allows to mathematically validate the Legendre's conjecture. 4- Mathematical characterization of odd primes - Riemann Hypothesis Number theory deals mainly with properties of integers. This theme is paradoxical in the sense that most of his problems can be stated so very basic, but the necessary tools for their resolution are generally very sophisticated. This is the case of the prime numbers and their distribution. Prime numbers are easy to define using a divisibility rule but their distribution is more elusive. For this, summatory functions are used and whether they are analyzed analytically, the Riemann hypothesis (formulated in 1859) that binds the non-trivial zeros of the zeta function with the distribution of prime numbers appears. This hypothesis remains a conjecture. We propose to test this hypothesis in the space W. Indeed, the combination of a sine function to each odd number helps make the connection between the Riemann hypothesis and space W Riemann Hypothesis Wilson's theorem provides a mathematical characterization of a prime number: «A number P greater than 1 is prime if and only if.» Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 91

92 This mathematical characterization is based on factorial functions. This characterization allows to write formulas which give all primes. However, this characterization is difficult to use for mathematical demonstrations. Moreover, these factorial functions are computationally inefficient. The calculation time increases rapidly when the value of the number p increases. The basic schema allows the characterization of an odd prime with sinusoidal functions. This characterization of prime numbers is used to prove conjectures such as the twin prime. The indices of odd numbers in the space W are described by sequences «j» so : characteristics:. These indices may also be connected by sinusoids with the following a period equal to twice the period of a sequence «j» so :, a phase offset corresponding to so equal to. The frequency is then equal to and pulsation in radians is equal to so :. The origin s phase of the sinusoid corresponds to the offset in radians so :. We expanded the definition of the basic schema to negative values for the parameter «n» and thus to the set of integers. The graph below shows the sequences «j» with the form of sinusoidal functions. The points correspond to the intersection of the functions with the axis «k». Note : Each point is associated with a sinusoidal function which can be likened to a dual representation "wave - corpuscle." This representation is equivalent to that of the matter at the microscopic level in the quantum mechanics. Note : the representation of a sequence «j» under trigonometric form allows to make the connection between prime numbers and trigonometry, especially with the number. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 92

93 Figure 19: Trigonometric representation of the indices of the odd numbers It is necessary to know what represent the following points in space N using the formula : - The point is equivalent to «0» in space N. It is equivalent to the position of Zero on the axis of the reals in the case of a complex number as shown below. - The point is equivalent to «1» in space N. - The point is equivalent to «2» in space N. This point corresponds the only even prime number. Auteurs : François et Marc WOLF Page 93

94 What is the peculiarity of the position of point? This point corresponds to zero in the space N. It is the only point where all sine functions associated with each odd number pass through a maximum. All these functions are in phase at this single point. What is the link between the parameter «n» and the numbers k generated by the sequences «j»? Each value of parameter «n» is used to connect one point of each sequence «j» using a formula corresponding to the equation of a line. Only integers are taken into account. The lines all converge at coordinates and. These lines have the formula, with : What is the connection between the numbers k generated by the sequence "j" and complex numbers? The sequences are represented by using sinusoidal formulas: Hence Another representation with cosine functions provides the following formula : With and, we get : ( ( *+ On the axis «j», we found a sine formula to connect all the points of a sequence «j» : ( ) If we consider this formula as representing the imaginary part of a complex number, then the real part of this complex number corresponds to ( ). The complex number Z whose imaginary part is the sine function and the real part corresponds to, we get : The so called formula De Moivre gives the following relationship : What is the particularity of the point of index in space W? The index corresponds to zero of the real part of the complex numbers in space W. It corresponds mainly to the single point where all the sine waves are in phase. Auteurs : François et Marc WOLF Page 94

95 What is the relationship between the space W and the Riemann hypothesis? This extract comes from the reference [10] : «If we write the zeros of the zeta function as, the Riemann hypothesis means that all numbers are real numbers. How to establish that a sequence of complex numbers is aligned with the real axis? The answer could come from methods developed for the study of physical phenomena. One of these methods is the functional analysis, that is to say the solution of equations whose unknowns are functions, and Euclidean space is replaced by the Hilbert space. Most vibration systems, sound, light, waves, brief any signal, is expressed as a basic signal superposition with coefficients, which are the amplitudes, and pulse,.,, which are real numbers. But Riemann himself observed that the explicit formula that he obtained shows that the deviations from the rule the density of primes are governed by a function in the form of a wave whose pulsations are numbers. As stated by M. Berryet J.P. Keating, Bristol, the numbers are harmonics of the music of primes! From there, it is tempting to see the numbers as the natural frequency of a physical system.» This is precisely what this study shows. For each occurrence of a prime number corresponding to the period of a sequence on the axis «j», we get a new natural period of the system and thus a new natural frequency or eigenfrequency. We get With a real number equal to : ( ( *+ And with ( * Hence if we sum the sine function, we get the following result: Then we get the harmonics of the music of primes in space W! What is the relationship with the Riemann hypothesis? Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 95

96 We take the following complex number : The said formula De Moivre gives the following relationship: Hence if we want to solve the following equation: ( * then we must have zero for the real part and zero for the imaginary part, hence the following result: And Axe réel ( ) Axe imaginaire ( ) The solution on the real axis corresponds to a constant «C» regardless of the value of the parameter «j». We obtain the following solution :. All solutions are therefore on the imaginary axis with abscissa. This corresponds to the problem of the Riemann hypothesis. All zeros of this complex function are on the imaginary axis with a unique value on the real axis. In space W, this value corresponds to. In space of Riemann, a conversion had to be made respect to the number k. This operation involves removing a constant value which corresponds to the number «1» hence axis.. So we get. We recover the desired constant value on the real Note : thus the sign of the value is irrelevant. What is the link between the distribution of prime numbers and the distribution of energy levels of atoms? A link between the Riemann hypothesis was done with quantum physics by Hugh Montgomery and Freeman Dyson. The function has two factors. These factors can be interpreted in quantum physics as two representations of the same element. This is the duality «Wave / corpuscle» : - The integer part may be associated with a particle, - The sinusoidal component can be associated with a wave associated with the particle. The eigenfrequency 3 would be the resonant frequency of the particule. (Cf. paragraphe 3.5- Natural Period of primes page 26). This is the sinusoidal component which establishes the link between the fundamental elements of mathematics (prime numbers) and the fundamental elements of physics (atoms). 3 See chapter II : «Distribution of prime numbers» : The oscillations of the measurements of the properties of the odd numbers are connected to the natural frequency of the system and the base unit [5] Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 96

97 We determined a natural frequency within the structure of the odd numbers. This frequency evolves with the magnitude of numbers. The energy levels of the atoms are equivalent to discrete frequencies that allow them to have a stable energy state. The natural frequency defined within the structure of odd numbers can not be reached because this frequency is evolving before this limit can be reached. Whether the atoms have such a structure, thus it helps to understand their stability. In order to destabilize such a structure, a significant energy requirement is necessary because the resonance frequency can not be reached. Note : Another result enables to consolidate the comparison with the Riemann hypothesis : In the study of the distribution of prime numbers to validate the conjecture of Legendre, the following results were obtained: Be the prime number of rank m, and the prime number of rank m+1, the maximum distance between two consecutive prime number corresponds to : However the Riemann hypothesis implies a constant C > 0 suitable The validation of the conjecture of Legendre consolidates the Riemann hypothesis. Theorem: The results of this study will allow us to validate the Riemann hypothesis. What is the connection between prime numbers and complex numbers? The absolute value of is in the range defined by [0 ; 1]. When the formula gives a value equal to 0, the value of k corresponds to a point of the sequence «j», ie the index of a composite number. For example, is equal to 0 when. This value is obtained when the parameter «n» is zero. When the point «k» belongs to the sequence of points of the sequence, hence :., thus this point is divisible by the period When the point «k» belongs to none of the sequences «j» which are in the interval [0 ; ], the number corresponds to a prime number. A sinusoidal function is represented with a complex number. The function is represented by the complex number «z» in space W with a sinusoidal function : with For j=0, we get the following formula : Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 97

98 For k=1, we have ( ) For k=2, we have ( ) For the general form of the formula, sinusoidal functions are superimposed. The position of the number on the axis «k» does not change. owever, on the axis «j», the sine waves are cumulative. This gives the following formula : All odd numbers which are not multiple of 3 and greater than 1 are represented by a complex number whose real part is equal to. All odd primes, except the number «3» has a non-zero imaginary part. The imaginary part is related to the following irrational number :. The numbers of the imaginary part are all numbers belonging to the set of real numbers How to characterize an odd prime number using trigonometric functions? The study of the primality of an odd number Ni is performed by checking the divisibility of this number by all odd numbers within the range [0 ; ]. In space W, each sequence «j» generates the indices which correspond to odd multiples of the period ( of the sequence «j». If the indice of the odd number Ni is equal to a value generated by the fonction, thus the number Ni is a composite number which is a multiple of the period of the sequence «j». This results in the value of the function equal to zero. To study the primality of an odd number Ni in espace W, it is necessary to test whether a sequence «j» in the range [0 ; ] generates by means of the formula a point equal to. If no sequence «j» generates a point equal to, then no function generates a value equal to zero regardless the value of «j». The multiplication of all functions then provides a nonzero value if and only if represents the index of an odd prime number. For a given value, the multiplication of all functions with «j» in the interval [0 ; ], with, leads to the following formula: Note : The only exception to this formula is the index that represents the prime number "3" for which the formula gives a value equal to zero. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 98

99 Theorem 5.1 : The following formula characterizes the odd prime numbers Ni except for the odd number «3» : With and * + With Hence the following theorem: Theorem 5.2 : Any odd number Ni greater or equal to 3 can be represented in the space W by a linear function described by with «j» and «n» which belong to the set of natural numbers N. This linear function can be represented as a sinusoidal function. This function has an natural frequency equal to and an initial phase equal to. The valuer corresponds to the inverse of the period of the sequence «j». We have the indice of odd number Ni, and the indice of the square root of the odd number Ni. We multiply the sine functions for «j» within the range [0.. ] and with the value. We get the following formula: With With, belong to the set of natural numbers N. With * + * + If If (* +), thus Ni is an odd composite number (UNNP). If (* +), thus Ni is an odd prime number (PUN). The Table 11 page 103 gives the values of calculated for values of k in the range [0 ; 28]. The graph Figure 20 page 100 shows the values of * + calculated with the values of Ni in the interval [0 ; 103]. The values of the function (* +) correspond to an irrational number proportional to. Indeed, we multiply all the sinusoidal from. However for, only three values are obtained irrespective of the value of «k» : (0,+, ). For and, by convention. This point corresponds to the first odd prime number «3». The formula allows to characterize every prime number strictly greater than «3». The numbers «2» and «3» are not characterized by this formula. These are the only two exceptions. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 99

100 Special cases: When is negative, the used value of is zero. For Ni= 5 or Ni=7 thus the formula is used with, hence Hence : ( * ( * If k=1 or k=2, thus Reminder: an odd number Ni is represented in this study by this formula Only odd primes have an index k for which the formula Sf(k) generates a nonzero value. Figure 20 : Graphical representation of primes determined with the formula Sf(k). The formula is used to define a primality function to detect the odd primes. With this primality function and with the following features: - if the indice k corresponds to a prime, thus the function is - if the indice k corresponds to a composite number, thus the function is the primality function is written as follows: With : that returns the absolute value of a number. : function that returns the integer part of the value plus one. Hence ( ( ( *), The following theorem allows to obtain a mathematical representation of an odd prime number strictly greater than «3». Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 100

101 Theorem 5.3 : Any odd prime number greater than «3» can be represented by two factors: - A factor corresponding to an integer :, - A factor characterizing the primality of the number :. The test of primality for an odd number greater than «3» is named Primes : ( ( ( *), With If the number Ni(k) is a prime number, the formula returns the value of the prime number. If the number Ni(k) is a composite number, the formula returns a zero value. Any odd number represented by only if the value of the formula and greater than «3» is a prime number if and is nonzero. Note 1 : The formula Primes(k) is not applicable for large numbers because of the number of calculations that increases in proportion to the square root of the number Ni(k). Note 2 : The formula can be implemented into computer language. Pour pallier aux To overcome the problems of accuracy in calculations, it is better to use this formula that provides for each sequence «j» a binary value «0» or «1» : ( ( ( **+ Note 3 : The introduction of sinusoidal formulas in the study of prime numbers helps to understand the origin of the number in many formulas that deals with probabilities such as the probability that a number is not divisible by any square is equal to solve the equation of twin primes.. For example, paragraph 5 uses trigonometry to We can also introduce Euler's formula that allows the link between the trigonometry functions and the exponential function. This equivalence relation allows to make the connection between prime numbers and the mathematical constants such as. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 101

102 k Sf(k) The odd number (Ni) corresponds to the value k: Ni=2 * k+3 Is a prime number? OUI 1 5 OUI 2 7 OUI Non OUI 13 OUI Non OUI 19 OUI Non OUI Non Non OUI 31 OUI Non Non OUI Non OUI Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 102

103 k 20 Sf(k) The odd number (Ni) corresponds to the value k: Ni=2 * k+3 Is a prime number? 43 OUI Non OUI Non Non OUI Non Non OUI etc Table 11 : Table giving the value of the sinusoidal component of odd numbers. Conclusion : The odd numbers Ni belong to the space of natural numbers N. We used the formula to connect the odd numbers Ni with their indices k to define a new workspace named space W. This space W helped to highlight the structure and the organization of the odd numbers by defining a basic schema and a base unit. This allowed the characterization of odd primes (PUN) using a formula consisting of sinusoidal functions. The primality function of the odd numbers Ni can characterize all odd primes except for the prime number 3. We have demonstrated that all odd primes, except the first prime number «3» can be defined using two components: a fixed part and a sinusoidal part. The space W introduced new mathematical tools to work on the odd primes. The tool for characterizing an prime odd number greater than the number «3» is one of them. This mathematical tool allows to link several prime numbers. It helps to equate odd numbers to determine the conditions under which these numbers are primes. This mathematical tool for the characterization of a prime number, is used to make the connection between two primes. This tool is used later in this study to solve the equation of twin primes (Cf. paragraph 5- Method for determining the twin primes (Twin Prime Conjecture) page 104, and Goldbach's equation (Cf paragraph 6.1 Determination of Goldbach's equation page 196). Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 103

104 5- Method for determining the twin primes (Twin Prime Conjecture) 5.1- Origin of twin primes Derivative Space W. Definition: the derivative space W corresponds to the points generated by the following formula:. We have therefore removed the first term of the formula. The offset "j" is removed from the formula. Function then describes the derived linear functions:. The points are shown in the graph below. Figure 21: Representation of the basic schema without the shifting «j», i.e. k'(j,n) Schema description: the value of the function is on the horizontal axis (abscissa) while the value of "j" is positioned on the ordinate. The points represented in blue on the graph are connected by the function for the values of n between 1 and 11. Consequences: 1- The points "derivatives PUN " correspond to values of k that are not generated by the formula with. 2- The points "derivatives PUN " are linked by the formula with the exception of the first point derivative PUN having k=0. The first points are: 1,2,4,8,16,32... This approach is consistent with the fact that no odd number can be described using the formula. Indeed, the powers of 2 are the only numbers that are not divisible by an odd number other than "1". Hence all points "derivatives PUN ", except for k = 0, are connected by the formula. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 104

105 (2a). Note that starts at «1» when. In the space N, the formula corresponds to hence 3- The number of points "derivatives PUN " generated by the formula is relatively low compared to the number of points generated by the function. 4- In this derivative space, there is no twin derivative PUN. 5- The twin primes are two prime numbers that differ by their value as a difference of 2. The difference between two odd numbers is obtained in the space W for two consecutive values of, i.e.: Two primes correspond to twin primes if and only if the index of the two prime numbers are separated by a value of 1. The twin primes thus have consecutive indexes in the space W. In the derivative space W, only the consecutive index derivatives PUN. correspond to twin The addition of "j" corresponding to an offset, is the factor behind twin primes Study of the first term of the basic schema Let's add the offset "j", and superimpose the points of the space W (red) to those points of the derivative W space (blue). Figure 22: Representation of 2 components of the basic schema: the offset "j" and the period What brings us the knowledge of this shift? Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 105

106 1- Formula provides index values which are either PUN or UNNP. So, only certain values of m generate prime numbers (PUN). For example, the following values of k: 1, 2, 4, 8, 32 correspond to a PUN, but the value 16 corresponds to a UNNP. The purpose of this Chapter is to determine a method to find the values of "m" that generate a prime number PUN. 2- PUN points are not interconnected by a single connection, as it is the case with the PUN derivatives points. The offset "j" complicates the relationship between all the PUN. 3- We note that many new PUN points appear. Indeed, when the points are shifted by a value "j", they leave a value which does not correspond to an other point which would in turn shifted. For example, the numbers generated by the values k = 5, k = 7, k = 14 correspond to the prime numbers with this shift. Twin primes then appear, for example, for the following k-values: 4 and 5, 7 and 8, 13 and 14 and so on. Value of k Odd primes 4 et 5 11 et 13 7 et 8 17 et et et This offset generates all twin primes, except the first 3 prime numbers (3, 5 and 7) that respectively correspond to the index. Conclusion : The origin of twin primes is related to the "j" parameter of the basic schema. 5.2 Determination of the equation of twin primes The twin primes are defined by two odd prime numbers consecutive. The twin prime conjecture predicts there are infinitely many twin primes. We will prove this conjecture in this paragraph. In the space W, the index «k» of a prime number belongs to a pair of twin primes if and only if the index also belongs to the pair of twin primes. An index «k» is a prime number if and only if the following condition is respected: With. If «k» and are odd prime numbers then the inequality below must be checked under the condition that the value of is the same for the indexes "k" and "k+1". In other words that the pair of twin primes is in the same base unit as shown in the section "3.4- Singular points". Indeed, the singular points which correspond to the limits of the basic units can not belong to a pair of twin prime numbers. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 106

107 ( ( * ) With Since So the inequality is written: ( ( ( ( *+ ( *)) This inequality is demonstrated if for all values of j between 0 and, the following relation is true: ( * If the value of the index "k+1" is greater than this one obtained with the index "k", then this value is the upper limit to be considered. The value is. This occurs at the upper limit of the range of the base unit. Indeed, the last point of the range must be associated with the first point of the next base unit. To take into account this limit, we have to do the calculations on the axis "j" in the interval [0 ; ]. At the upper limit of the base unit, the inequality to solve is: ( ( ** ( ) The same result is obtained in using the following inequality: ( ( * ( ** Why? We have shown in section 3.2 that, for a selected "J" sequence, there are some points generated by this sequence which are different of the points generated by the preceding sequence if and only if two conditions are met: The value of the period of the sequence «j» is equal to. This value is a prime number. The sequence «j» begins to densify the axis "k" only from the remarkable point k=gw(j). This means that some new points appear from the remarkable point GW(j) if and only if the period of the sequence "j" is a prime number. So the points "k" generated by the sequence and whose the value is lower than the value of the remarkable points are all of them existing points generated by the previous sequences. There are then not new twin primes. These points make not disappear some existing pairs of twin primes except for the singular point equal to. The singular points have been described to the section «3.4-». It has been shown that they can not belong to a pair of twin primes. So incrementing of a unit is a blank operation in determining the twin primes into the basic unit. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 107

108 Question : why in the formula below, we multiply the sinusoidal of the two points "k" and "k+1" by combining them in pairs with the same value? ( ( * ( ** Whichever combination chosen, if the value of one sinusoidal function is equal to 0, then the result is equal to 0. This choice was made for the sole purpose of simplifying the demonstration. Lemma: The indexes and correspond to the twin primes when the two following conditions are respected: Condition 1- For all values of «j» in the range [0 ; ], we get the following inequality: ( * with Condition 2- The values of k and are in the following range corresponding to the limits of the base unit : with and the base unit There are infinitely many twin primes if these conditions are met for at least one value of k in each base unit. Meet the condition 1 led to solve the inequality. The values of k not respecting the condition 1 are determined by solving the equation. The values of k searched are the values of k which does not respect the equation. 5.3 Resolution of the equation Definition of the condition 1: As known: and the value of, the condition 1 is written: As known:, the condition 1 is written: Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 108

109 Hence, after simplification, for all values of «j» in the range [0 ; ], the following inequality for a selected value of "k" is met if and only if the indices «k» and «k+1» correspond to twin primes: With [ ] Resolution of the condition 1: We will search for all values of "k" for which equality is respected. Solving the equation allows to find the values of k which can not correspond to the first prime number of a pair of twin primes. For instance, the value of an odd number «twin primes (11, 13) but it can take the value 13.» cannot be equal to the value 11 of the pair of The Euclidean division by 3 of the value of k can only take one of three forms: k = 3q, k = 3q + 1 and k = 3q + 2 equivalent to k= 3q -1 with q being a natural integer. In the space W, the indexes of the odd prime numbers can take two forms: 1- The formula which correspond to 2- The formula which correspond to It should be noted that such forms do not only correspond to primes. In the space N, these formulas correspond respectively to: 1- We note that. The value of is so in the form. 2- We note that. The value of is so in the form. with «m» belonging to the set N. The twin primes then have one of two forms: The first prime number is in the form: The second prime number is in the form: We will determine the solutions of the following equation: Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 109

110 The solution of this equation will provide us all odd numbers "Ni" which does not correspond to the first number of a pair of twin primes. We will use the following relation that was demonstrated on page 115 : ( + Example 1: Search the solution for the base unit, with. The half period of the sinusoidal function representing the sequence j=0 corresponds to the period of the sequence equal to. The set of points k in the sequence j = 0 is given by the formula. The set of solutions is given by the two following formulas: Hence This is equivalent to the following formulas if it is desired to normalize the solutions with the base schema: With j=0. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 110

111 Solutions for the sequence j=0: List of the values of «k+2» which do not belong to a pair of twin primes Figure 23: Chart depicting the values of and the values corresponding to «k+2». Ni Odd numbers Ni=2. k+3 k Value of k k Value of k+2 j=0 X X X X X The "X" correspond to the values of "k" that do not belong to a pair of twin primes Table 12 : Table of distribution of the numbers "k" for which we obtain a pair of twin primes in the base unit Ugw(0). In the range of k ϵ [3 ; 10] corresponding to the base unit Ugw(0), with which respect the inequality and generate the twin primes are as follow: The values k=4, 7 correspond respectively to the twin primes {11, 13} and {17, 19}., the values of k Note: the last point of the interval,, correspond to a singular point. It is not taken into account in the calculation because it was demonstrated that this point cannot belong to a pair of twin primes. This allows to restrict the calculation on the axis «j» to the range [0 ; ]. Example 2: Search the solution for the base unit, with. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 111

112 The half period of the sinusoidal function representing the sequence j=1 corresponds to the period of the sequence equal to. The set of points k in the sequence j = 1 is given by the formula. The set of solutions is given by the two following formulas: Hence With j=1. Solutions for the sequence j=1: List of the values of «k+2» which do not belong to a pair of twin primes Figure 24 : Chart depicting the values of and the values corresponding to «k+2». Ni k k j=2 X X X X j=1 X X X X X j=0 X X X X X X X X Table 13 : Table of distribution of the numbers "k" for which we obtain a pair of twin primes in the base unit Ugw(1). Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 112

113 We take into account the values of «j» in the range [0 ; must take the previous results. ] and so in the range [0 ; 1]. For j=0, we In the range of k ϵ ] corresponding to the base unit Ugw(1), with, the values of k which respect the inequality and generate the twin primes are as follow: The values k=13, 19 correspond respectively to the twin primes {29, 31} and {41, 43}. Note: the last point of the interval, k=22 (Ni=47 in black in the table), correspond to a singular point. It was displayed in this example in order to show that this kind of points cannot belong to a pair of twin primes. This was demonstrated in the section Using the values of k generated by the sequence j= jmax+1 do not modify the results obtained with the points of the sequences j < jmax+1. Example 3: Search the solution for the base unit, with. The half period of the sinusoidal function representing the sequence j=2 corresponds to the period of the sequence equal to. The set of points k in the sequence j = 2 is given by the formula. The set of solutions is given by the two following formulas: Hence With. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 113

114 Solutions for the sequence j=2: List of the values of «k+2» which do not belong to a pair of twin primes Figure 25 : Chart depicting the values of and the values corresponding to «k+2». Ni k k j=2 X X X X X j=1 X X X X X X j=0 X X X X X X X X X X X Table 14 : Table of distribution of the numbers "k" for which we obtain a pair of twin primes in the base unit Ugw(2). In the range of k ϵ ] corresponding to the base unit Ugw(2), with, the values of k which respect the inequality and generate the twin primes are as follow: The values k=28, 34 correspond respectively to the twin primes {59, 61} and {71, 73}. Note: the last point of the interval, k=38, correspond to a singular point. It was not taken into account because it was demonstrated that this point cannot belong to a pair of twin primes. This allows to restrict the calculation on the axis «j» to the range [0 ; ]. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 114

115 GENERAL CASE Corollary: The solution of the following equation: is : ( * Hence the following formulas: Equation 5: Solution for the twin primes et Proof : As known : hence but hence hence the 2 solutions: (1) Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 115

116 (2) But: So we get: ( * ( * Solutions : The half period of the sinusoidal function representing the sequence corresponds to the period of the sequence equal to. The set of points k in the sequence j=0 is given by the formula. The set of solutions is given by the two following formulas : Hence the following parametric solution with «j» corresponding to the parameter of the solution: The two parametric formula above give all the values of k in the interval given by the condition 1 with «j» in the range [0 ; ]. These values cannot correspond to the index of the first prime number of a pair of twin primes. The values of k searched are so the complema The values of k searched are therefore the additional values to those determined by the resolution of the equation in the interval [0 ; ]. The solutions searched are the values such as: And Note: there are two others solutions: These solutions give the negative values of «k» with. They are so excluded. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 116

117 5.3.3 Definition of the Condition 2 : The condition 2 is related to the limits defined: - on the axis «k» by the interval of the base unit, so [ ; ], - and on the axis «j» by the interval [0 ; jmax] We will show that the conditions to apply the condition 2 can be replaced with the sole condition. 1- Study the limits of the axis «k» The study of singular points in the section 3.4 shows that the upper and lower limits of a base unit do not belong to a pair of twin primes. a- Study the upper limit Removing this upper limit has no impact on the study of primality of a pair of twin primes. b- Study the lower limit Can we replace the value of the lower limit by zero? The sequences "j" in the range [0 ; jmax] generate the points k. Only points k in the range defined by the base unit are taken into account in determining twin primes. The study of the primality of an odd number in the section 3.2 showed that using a sequence of points with «j» greater than has no impact on the result if and only if. So the removal of this lower limit has no impact on the determination of twin primes. 2- Study the limit on the axis «j» The study of the primality of an odd number in the section 3.2 found that this limit can be removed if the following condition is met:. Conclusion: These results allow dispensing with the calculation of the upper limits on the axis «j» and the calculation of the lower limit of the axis «k» by using the condition. So we get the two following formulas by substituting «n» by «n+1» in the formulas (a) and (b): With 5.4 Mathematical form of twin primes Mathematical form of a pair of twin primes All the points on the axis which solve the equation are obtained with the formulas and with. The Euclidean division by 3 of the k value may only take one of three forms: k = 3q, k = 3q + 1 and k = 3q + 2 equivalent to k= 3q -1 with q being a natural integer. So we have a third form, different from the previous two: With Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 117

118 What do these formulas? The points «The points «one. The points «one.» correspond to the indexes of the odd multiples of the odd prime numbers.» correspond to the indexes of the odd multiples of the odd prime numbers less» correspond to the indexes of the odd multiples of the odd prime numbers plus Question : Is it possible from a value that for any given value greater than the value is always equal to at least one of two formula or? from? and/or from? Response : No, because if it was true then there would be no more prime numbers in the form in the space W from the limit value. This would mean that there would be no more prime numbers in the form in the space N from the limit value. Proof : In the space W, the prime numbers can be written in two forms: The formula which matches to The formula which matches to Note: these forms do not only correspond to the prime numbers. In the space N, the previous formulas correspond to these ones: The formula The formula which matches to which matches to with «m» belonging to the set N*. Hence the following conclusion: a) The equation requires that the odd numbers in the form are odd multiples of odd primes. This implies the following equality: This equality is not respected when the odd numbers of the form are prime numbers. However, the mathematical form of this term generates an infinitely many primes. The equation is not satisfied for whatever the values of j and n. b) If the following equation is respected whatever the values «j2, j3» and «n2, n3» then this equation is also respected: But the prime numbers of the form formula N3 and so. belong to the set of points generated by the Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 118

119 This means that the following equation is respected: But Hence Hence the following inequality: The following condition requires that the left term matches all odd multiples of the odd primes (NINP).This equality is not satisfied when the odd numbers of the form are prime numbers. However, the mathematical form of this term generates an infinitely many primes. The equation is not satisfied for whatever the values of j and n. The pairs of twin primes are so in the forms and. The value of «m» is the same for each form to get the two primes of the pair of twin primes Mathematical form of the set of prime numbers Proof that all prime numbers are in the forms or : Any prime number other than 2 and 3 can be written in the space N in the form. Demonstration: Any odd number "Ni" greater than 3 and divided by 6 can be written as follows: Ni = 6m + r where m is a positive integer and the remainder r is one of the following numbers: 0, 1, 2, 3, 4, or 5. If the remainder is 0, 2 or 4 then the number Ni is divisible by 2 and it cannot be a prime number. If the remainder is 3 then the number Ni is divisible by 3 and it cannot be a prime number. So if Ni is a prime number then the value of the remainder of the division r is either equal to : 1. In this case. Ni is a multiple number of six plus one. or 5. In this case. Ni is a multiple number of six less one. It has been shown that there is an infinitely many primes of the form 6m+5, equivalent to 6m-1, and an infinitely many primes of the form 6m+1. Proof with the form. Taking L={,,... } a finite list of primes of the form. Then is divisible by any of the existing prime numbers. But any prime number is in the forms or. If all the factors in the factorization of «q» were of the form, then the number «q» would be in the same form. The number «q» has a factor which is not in the L. The list L is so not complete; no finite list of primes of the form can be exhaustive. There are therefore an infinitely many primes of this form. About the form, Dirichlet's theorem states that for any two positive coprime integers a and r, there are infinitely many primes of the form a*m+r, where m is a non-negative integer. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 119

120 The proof that there are infinitely many primes of this form is related to the Schur(1912 [3])- Murty(1988 [4]) theorem which states that there is a proof of the Dirichlet's theorem on infinitely many primes in the arithmetic progression (with (a,m)=1 ) if and only if. We deduce that there is a Euclidean proof of infinitely many primes of the form:. 5.5 Mathematical form of the composite numbers The composite numbers correspond to NINP points. The composite numbers in the form and, corresponding to the points generated by series, are obtained by the following formulas: 1- Study of the form Hence the equality: Both parameters «j» and «n» may take the forms: So there are 3 separate cases for the value j and 3 cases for the value n, so 9 separate cases: - Case with because Hence Whatever the form of the parameter «n», there are no solutions. Indeed, the number «2» is not divisible by «3». - Case with Hence with Case with because There are no solutions. Indeed, the number «5» is not divisible by «3». Case with There are no solutions. Indeed, the number «5» is not divisible by «3». Case with because Hence with With, we get the following formula: with The parameter «m» depends on two parameters «r» and «g» belonging to the set of natural integers. - Case with because Hence with because «j» is greater or equal to 0. Case with There are no solutions. Indeed, the number «1» is not divisible by «3». Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 120

121 Case with There are no solutions. Indeed, the number «2» is not divisible by «3». Case with Hence with With and, we get the following formula: with The parameter «m» depends on two parameters «r» and «g» belonging to the set of natural integers. 2- Study of the form Hence the equality: Both parameters «j» and «n» may take the forms: So there are 3 separate cases for the value j and 3 cases for the value n, so 9 separate cases: - Case Hence Whatever the form of the parameter «n», there are no solutions. Indeed, the number «1» is not divisible by «3». - Case Hence Case with There are no solutions. Indeed, the number «2» is not divisible by «3». Case There are no solutions. Indeed, the number «7» is not divisible by «3». Case Hence with because. With, we get the following formula : with The parameter «m» depends on two parameters «r» and «g» belonging to the set of natural integers. - Case Hence with because «j» is greater than or equal to 0. Case There are no solutions. Indeed, the number «1» is not divisible by «3». Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 121

122 Case There are no solutions. Indeed, the number «1» is not divisible by «3». Case Hence with because. With et, we get the following formula : avec The parameter «m» depends on two parameters «r» and «g» belonging to the set of natural integers. All composite numbers of the form and are obtained with the parameter "m" using the following formulas: - Case The solutions are : and with by factorizing, we get: and This gives a factorization of the odd number of the form : and The parameters «r» and «g» play a symmetrical role. - Case The solutions are : and with By factorizing, we get respectively: and This gives a factorization of the odd number of the form : or The parameters «r» and «g» play a symmetrical role. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 122

123 The formulas of the parameter "m" are symmetrical in the forms and but also between both forms et. These formulas have been explained in the article [11]. Conclusion: Taking "Ni" an odd integer greater than or equal to 3. Will this number be the first prime number of a pair of twin primes? The first step is to calculate to position this point in the space W.. If or with, then the odd number «Ni» is not the first prime number of a pair of twin primes. However, in the opposite case, if and with, then the odd number «Ni» is the first prime number of a pair of twin primes. The graph below illustrates changes in the number of twin primes in each base unit when the value of «j» increases. This evolution can be represented by an function increasing on average with the parameter "k". It is observed that most the base unit increases the greater the number of twin primes increases. There are an infinitely many twin primes. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 123

124 It is observed that there are infinitely many twin primes. Moreover, this number grows linearly on average with base unit Ugw(j). Each base unit has at least a pair of twin primes. ZOOM Figure 26 : Graphic representing the number of twin primes in each base unit. Auteurs : François et Marc WOLF Page 124

125 5.6 Formula to count the pairs of twin primes We will use the formula to count the primes, π (N), in order to determine the number of prime numbers between two consecutive multiples of the composite number «6». The intervals are: pair of twin primes. When «case possible. The formula range with. If, then it is necessary to add the pair [3 ; 5] :. If the number is equal to «2», then it is a», the formula provides the number «3». It is the sole allows getting the number of twin primes in the * + * + If and, then we get the pair [3 ; 5]: If, then. The function corresponds to the formula described in the reference [5]. The function is then also an exact formula. Does this formula provide an infinitely many twin primes? 5.7 Enumeration of twin primes We will explain and determine the evolution of the number of pairs of twin primes in base units and in natural periods to validate the existence of infinitely many twin primes. The numerical calculations and charts are made using an algorithm performed with MAPLE tool and a specific program in C / C Formula of the twin primes. Based on heuristic considerations, a law (the twin prime conjecture) was developed, in 1922, by Godfrey Harold Hardy ( ) and John Edensor Littlewood ( ) to estimate the density of twin primes. According to the prime number theorem the probability that a number n is prime is about, therefore, if the probability that is also prime was independent of the probability for n, we should have the approximation: Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 125

126 But a more careful analysis shows that this model is too simplified (an argument is given in [12]). In fact we have the following and more accurate conjecture (called conjecture B in [13]) With C2 which is the twin prime constant and is defined by In 2004, WU Jie showed, by sieve methods, that for relatively large numbers, the following inequality applies: These formulas do not demonstrate the existence of infinitely many twin primes. Indeed, the measurement interval used corresponds to [0; infinity]. It is therefore not possible nor to extrapolate or to explain the behavior of the number of twin primes to infinity. We will introduce in the calculations the measurement intervals of our model: the basic units and the natural periods Determination of the evolution of the number of twin primes in base units. We have explained that between two odd numbers which are multiple of «3», we have a pair of odd numbers of the form 3m+1 and 3m+2. If this pair consists of two primes, then we have a pair of twin primes (Prime number=nip). Otherwise, we have a pair of odd numbers consisting of at least a composite number called NINP. We will define four elements within a base unit «j» : - The number of pairs of odd numbers consisting of at least one composite number:. These couples are named composite couple. - The number of pairs of twin primes: - The density of composite couple is equal to the ratio between and, so : - The total number of pairs of odd numbers is equal to the number of odd numbers divided by three, so * +. Only the integer part is used. The number of odd number in a base unit is equal to (See the section 3.3- Structure of a "base unit" page 20). So the total number of pairs of odd numbers within a base unit is equal to: Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 126

127 [ ] The number of pairs of twin primes in the interval [0 ; N], is denoted. Numerical study of the density of the number of pairs of twin primes. An algorithm was developed with the MAPLE tool for studying the evolution of the density of the pairs of twin prime, the pairs and the density. The results are presented in graphical form. NOTE: The measurements are not cumulative. The measurements are performed within each base unit independently of each other. The graph below shows that the number cninpj growing faster than the number cnipj in a base unit. The number of pairs of twin primes (cnipj) decreases in comparison to the number of other pairs (cninpj) when the value of "j" tends to infinity. However, the number of pairs of twin primes cnipj increases when the "j" increases. It increases on average around a value that increases according to a logarithm function squared. This number cnipj tends to infinity as the basic units tends to infinity. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 127

128 Analysis of the results presented in the chart above Figure 27: Evolution of the density in the base units Ugw(j) of the seqences «j» This analysis leads to the approximate calculation of the number of pairs of twin primes in the interval with set of the natural integers. Authors : François WOLF and Marc WOLF mathscience.tsoft .com Page 128

129 In the space W, for an odd number Ni given, we know the values of and : [ ] The maximum value for the calculations on the axis is: * + The following function provides the value of remarkable point : with j 0. The ratio between the number of pairs of odd numbers consisting of at least one composite number and the number of pairs of twin primes for a value of given is represented by a logarithmic formula: Equation 6: density of the airs of odd numbers On the graph is observed, "Erreur! Source du renvoi introuvable.", the density changes between. This fluctuation is due to the fact that the function represents the density of composite odd numbers (NINP). We measure the density of a pair of odd numbers. We consider that the composite couple is composed of two composite odd numbers. So the overall variation is twice the variation of the density of the composite odd numbers, hence. The section 3.5- page 26, explains the origin of the oscillations in the measurements of the properties of the odd numbers. This oscillation phenomenon can also be applied to measures of the number of pairs of twin primes in a base unit. Hence Equation 7: fluctuation of the density of the pair of odd numbers The parameter has a value which is related to the base unit and so depending on the value of j. The value of this parameter increases when the value of j increases. The upper limit is 1, hence with. The number of odd numbers in a base unit according to the parameter "j" is given by the following formula: The number of intervals with 3 units is obtained by taking the integer part of the division of by 3, with * + Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 129

130 and the integer part of the density [ ] Hence * + is a function with a limit equal to «1» when the parameter «j» tends to infinity. Hence the following limit for when «j» tends to infinity: ( * + ) The number of pairs of twin primes is infinite in a base unit «j» when the value of «j» tends to infinity. Enumeration of the pairs of twin primes in the interval. Counting the pairs of twin primes in the interval is done by an addition : - of the pairs of twin primes with j belonging to the interval - and the number of pairs of twin primes with. The determination The result is: of the last value with j = jmax is done from a rule of three. [ ] So : [ ] [ ] * * + + This formula does not take into account the following values: 2, 3, 5, 7. For, corresponds to the value. The formula does not take into account the numbers which match respectively the odd prime numbers 3, 5, 7. The pair of prime numbers {2 ; 3} is not counted in our formula. It is so necessary to add the 2 pairs of twin primes that are {3 ; 5} and {5 ; 7} to get the number of pairs of twin primes less than or equal to N. Hence the following formula: Hence [ [ ] ] * * + + With Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 130

131 * + and The parameter «b» belongs to the set of real values R. It increases as a logarithmic function: ( ( )*. How to explain the evolution of this parameter? This phenomenon is related to the occurrence of prime numbers. When a prime number appears on the axis, then the number of composite odd number increases. The value of «b» increases also because the ratio increases. However, the scarcity of prime numbers makes increase of the value of the parameter «b» very slow. Hence there is an evolution inversely proportional to a function composed of with 3 nested logarithmic functions: ( ( )*. The formula "Equation 6" page 129 leads to the following relation: ( ). To avoid division by zero, we get the following equation: ( ( )* The study refers to a pair of prime numbers that leads to a formula depending on 2 nested logarithmic functions. We use another nested logarithmic function to get a formula written with 3 nested logarithmic functions. In a way empirical, the formula of is: ( ( ( ))+ ( ( ( ( ( *)+) ) Numerical results in the basic units show that there are infinitely many twin primes couples. In addition, we evaluated empirically the number of pairs of twin primes up to a value less than or equal to N. We calculated the number of pairs of twin primes for the following values of N: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 131

132 N Interval related to the fluctuation of the density [ ; ] Table 15: Numeric results: determination of the number of pairs of twin primes less than or equal to N. The real number of pairs of twin primes is in the interval is in the measurement interval. This difference is related to the fluctuation in density. The measurement accuracy is. We determined an approximate formula for the enumeration of pairs of twin primes. The adjustment of the curve is obtained with a regression coefficient of The formula is as follows: [ ( ) ] With ( ) ( ) The axes of the graph below correspond to logarithmic scales. Figure 28: Comparison of the approximate formulas of the enumeration of the pairs of twin primes with Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 132

133 The values of the formulas shown in the chart above are displayed in the following table: Note : The values in the shaded boxes have not been determined numerically. They are the initial values matching the pairs {3 ; 5} and {5 ; 7}. N Interval related to the fluctuation of the density [ ; ] Table 16: List of values obtained with the approximate formulas and The numerical results show that the number of pairs of twin primes is in the range of measures related to the fluctuation of the measurement of the density. The number of pairs of twin primes increases regularly as the square of a logarithmic function. Why are there always pairs of twin primes in base units? We will study the number of pairs of twin primes within natural periods Determination of the evolution of the number of twin primes in the natural periods. We calculated the number of pairs of twin primes within natural periods. Within the natural periods, we defined virtual numbers, and, to calculate the virtual pairs and the virtual pairs. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 133

134 We calculated the values of, and the density for the following values of «j». j Number of virtual pairs of twin primes: Table 17: Evolution of the number of twin primes within natural periods. The results show that density increases regularly. The density increases as a logarithmic function. The number of pairs of twin primes therefore increases with the index j of the sequence of points when the value of the period equal to is a prime number. There are an infinitely many prime numbers. So, there are an infinitely many twin primes. The number of pairs of twin primes and the density do not oscillate around a mean value. The values are stable as explained in section 3.5- Natural period of the prime numbers Page 26 The number of virtual pairs of twin primes is infinite. Why? We have demonstrated the stability of the structure of prime numbers within the patterns. This structure is the same regardless of the base unit selected. The evolution of the properties of odd numbers such as the density of twin primes remains stable within natural periods. The number of virtual pairs of twin primes is growing. Its growth is infinite Conclusion We have shown within natural periods that the number of virtual pairs of twin primes increases with the base unit «j». The numerical results show that the number of pairs of twin primes increases in the base units. This increase oscillates around a mean value that increases logarithmically. This means that there is an infinity of twin primes. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 134

135 The following section provides a mathematical proof of these results. 5.8 Representation of prime and composite numbers by binary values. We will show in this paragraph, how to determine the pairs of twin primes, prime numbers and composite numbers in each basic unit. The factorization of composite numbers into prime factors is also found. This determination is performed by analyzing the table below, the odd numbers are represented by binary values 0 and 1. For instance, the odd number Ni=15 is equal to the product of 2 factors 3 and 5. In the table, k is the index of the odd number Ni. They are related by the formula: Ni=2*k+3. The study of the values k+2 (Third column) is related to the resolution of the equation of each sequence of points j (See section 5.3). For each sequence of points j, an odd factor DPej is defined such as DPej=2*j+3. In the table, each sequence of points j is represented by a column. The solutions of the equation allow determining the points k which correspond to the first prime number of a pair of twin primes. For each factor DPej, the resolution of the equation attributes a binary value 1 or 0 to each point k. A value 1 is associated to the points k to which the odd number Ni cannot belong to a pair of twin primes. The value 0 is applied in the opposite case. However, a point k is the index of the first prime number of a pair of twin primes only if for each factor of the base unit to which belongs to the odd number studied the value associated is equal to 0. For Ni = 29, k is equal to 13. The base unit is j = 1 that is DPej = 5. The binary value associated with each DPej factor of between 3 and 5 is equal to 0. The number 29 is the first prime number of a pair of twin primes (29, 31). The line corresponding to the odd number Ni=29 is highlighted in yellow. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 135

136 Half period Dpej j=0 and j=1 Ni=2*k+3 Remarkable point - GW(j) A base unit Singular point - GW(j+1)-1 A base unit first prime number of a pair of twin primes A base unit consists of a set of k points which the first point is a remarkable point GW(j) and the last point is a singular point GW(j + 1) -1. These 2 points, as demonstrated above, do not belong to a pair of twin prime numbers. The number of points in a base unit UGW(j) equals the number of points located in a period of the sequence "j" plus the 2 points mentioned above. Only the points of the period can contain one or more pairs of twin prime numbers. Each base unit is a sequence of points j. This point sequence corresponds to the numbers which are odd multiples of the value of the half-period of the sequence j equal to DPej. The study to determine the points that form a pair of twin prime numbers is performed for all the points in the period of the base unit. The value of the points studied is that of k + 2. For each sequence j, a value 0 or 1 is set to each point k + 2. When the value of j is higher than 0, the analysis of the points k + 2 should be performed for each sequence of points in the range [0; j]. For a value of k + 2, the value 0 existing for each sequence j leads to determine the first prime number of a pair of twin prime numbers. The above table shows the binary values for each point k + 2 according to the value of the sequence j. Each row of the table, an odd number N corresponds to a point k + 2 by the formula Ni = 2 * k + 3. Each column of the table, a sequence of points j provides the k values which are multiples of an odd number. The value of the odd number is the value of the half-period Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 136

137 (DPej) of the sequence j: DPej = 2 * j + 3. The table also shows the binary values 0-1 associated with the half period Dpej for each odd number Ni. The lines of the table which the value k corresponds to a remarkable point are highlighted in bright red. The lines of the table rows which value k corresponds to a first prime number of a pair of twin prime numbers are highlighted in yellow Study of the sequence of points j=0 This sequence defines the odd multiples of odd number 3. Thus, we obtain the odd numbers [9,15, 21, so on] respectively for the following k + 2 values [5,8,11, so on]. The binary values 0-1 associated with k + 2 values are distributed, for j = 0, in the following sequence: The period of the sequence is while the half-period is 011. This set of binary values 011 is equated to the value of the half-period DPej = 3. In the table, the Ni and k+2 values, linked to the binary representation 011, are found by reading the last line where the value 1 is set. Thus, the first binary representation 011 of the sequence of points j = 0 corresponds to the number Ni = 3 that means the value k + 2 = 2. The second binary representation 011 corresponds to the number Ni = 9, multiple of 3, and so the value k + 2 = 5 and so on. To simplify the determination of an odd number multiple of 3, we will associate the number 3 with the binary representation "11" in the column j = 0 of the table. The search for the first prime number belonging to a pair of twin prime numbers is performed in a base unit. The base unit begins at the remarkable point and ends at the singular point. The remarkable point is the index of odd number which is the square of the half period DPej. We obtain for the remarkable point: Ni = 9 and K+2 = 5. The singular point is the remarkable point of the next sequence of points j minus one. The following values are then obtained for the singular point: Ni = 23 and k+2 = 12. Only points between the 2 points mentioned above may contain one or more pairs of twin primes. All of these points corresponds to a period Pej=2*Dpej of the sequence j. For j = 0, the base unit is represented by the k+2 values in the range [5; 12] or the Ni values in the range [9; 23]. In the last range, the binary values "11" are presents for the numbers 9, 15 and 21. These numbers are then odd multiple of the value of half period DPej, ie the value 3. There are no binary values "11" for the following odd numbers 11, 13, 17, 19 and 23. These numbers are not multiple of 3. They cannot be decomposed into prime factors. These numbers are primes. The pairs of numbers (11,13) and (17,19) form pairs of twin primes. This is reflected in the table by the presence of the value 0 for the first prime number of pairs of prime numbers: 11 and 17. The number 23 is a limit point of the range. This is a singular point which cannot belong to a pair of prime numbers. It is the same for the remarkable point. It is observed in the table that the value 1 exists for the number 23 when j = 1. This value comes from the next remarkable point Ni = 25, k+2 = 13. For j = 1, the value 1 does not exist for the other 2 numbers 11 and 17. The determination of twin primes only takes place in the base unit excluding the remarkable point and the singular point; i.e. in the range ]9; 23[. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 137

138 Demi période Dpej j=0 et j=1 For Ni=23 and j=0, the binary values "11" does not exist (Witness in green). The number 23 is a prime number. The value "11" only exists for j=10 and Dpej=23. The prime number 23 can only be divided by itself and 1. For Ni=37 and j=0, j=1, the binary values "11" (Witness in green) does not exist. The number 37 is a prime number. The value "11" only exists for j=17 and Dpej=37. The prime number 23 can only be divided by itself and 1. The value Ni=45 is written as the prodcut of the factors (3,15), (5,9) and (45,1). The factors 3, 5, 9, 15 and 45 are represented by the pair of binary values "11" vertically aligned with the second value "1" on the line Study of the sequence of points j=1 This sequence defines the odd multiples of odd number 5. Thus, we obtain the odd numbers [5,15,25,35, 45, so on] respectively for the following k + 2 values [3,8,13,18,23, so on]. The binary values 0-1 associated with k + 2 values are distributed, for j = 1, in the following sequence: The period of the sequence is while the half-period is This set of binary values is equated to the value of the half-period DPej = 5. In the table, the Ni and k+2 values, linked to the binary representation 00110, are found by reading the last line where the value 1 is set. Thus, the first binary representation 011 of the sequence of points j = 1 corresponds to the number Ni = 5 that means the value k + 2 = 3. The second binary representation corresponds to the number Ni = 15, multiple of 5, and so the value k + 2 = 8 and so on. To simplify the determination of an odd number multiple of 5, we will associate the number 3 with the binary representation "11" in the column j = 1 of the table. For j = 1, the base unit is represented by the k+2 values in the range [13 ; 24] or the Ni values in the range [25 ; 47]. In the last range, the binary values "11" are presents for the numbers 25, 35 and 45. These numbers are then odd multiple of the value of half period DPej, ie the value 5. There is no binary values "11" for the following odd numbers 29, 31, 37,41, 43 and 47 when j=0 and j=1. These numbers are not multiple of 3 nor 5. They cannot be decomposed into prime factors. These numbers are primes. The pairs of numbers (29,31) and (41,43) form pairs of twin primes. This is reflected in the table by the presence of the value 0, when j=0 and j=1, for the first prime number of pairs of prime numbers: 29 and 41. The number 47 is a limit point of the range. This is a singular point which cannot belong to a pair of prime numbers. It is the same for the remarkable point. It is observed in the table that the value 1 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 138

139 exists for the number 47 when j = 1. This value comes from the next remarkable point Ni = 49, k+2 = 24. For j = 2, the value 1 does not exist for the other 2 numbers 29 and 41. The determination of twin primes only takes place in the base unit excluding the remarkable point and the singular point; ie in the range ]13; 24[. This result is true into each base unit Study of the sequence of points j. This sequence defines the odd multiples of odd number Ni= DPej. The binary values 0-1 associated with k + 2 values are distributed, for j, in a sequence depending on the sequence of the half-period of the first sequence 011 and the j value. Thus, the value 0 is added j times in front of and behind the first sequence. For j=3, the half-period of the sequence j is This set of binary values is equated to the value of the half-period DPej = 2*j+3 = 9. For j, the base unit is represented by the k+2 values in the range [GW(j) ; GW(j+1)-1] or the Ni values in the range [2*GW(j)+3 ; 2*(GW(j+1)-1)+3]. The determination of twin primes only takes place in the base unit excluding the remarkable point and the singular point; i.e. in the range ] 2*GW(j)+3 ; 2*(GW(j+1)-1)+3 [. For a value of k + 2, the value 0 existing for each sequence j leads to determine the first prime number of a pair of twin prime numbers. Moreover, an odd number Ni with the index k is a prime number if no binary value "11" is present in the range [0; j] with j dependent on the odd number Ni. The odd number Ni is then not multiple of any odd number Dpej Decomposition of an odd number into prime factors. For an odd number Ni with an index k smaller than the remarkable point of the base unit j = 1, we note that the binary values "11" are presents when j = 0 and j = 1. The number 15 is an odd number multiple of DP ej=0 = 3 and DP ej=1 =5. The pair of numbers (3,5) corresponds to the factorization of the odd number Ni=15=3*5. In the table, the binary values "11" only exist when j = 0, j = 1 and j = 6. The last binary values "11" forms a pair of numbers with the value 1 : (1.15). This pair of numbers is not searched. For j = 6, the value of the half-period Dpej of the sequence of points has the value of the odd number Ni: DP ej=6 = Ni = 15. The binary values "11" of the sequences j found for a number Ni determines the pairs of numbers (Dpej1, Dpej2). Thus for Ni = 45, the pairs of numbers are (3,15) and (5,9). When the number of pairs of binary values "11" is odd, such as for Ni = 81, one of the values of Dpej is the square root of the odd number Ni. So for Ni = 81, the pairs of numbers are (3.27) and (9.9). The pairs of numbers which factorizes an odd number Ni follow these rules: - the first odd number is associated with the last odd number Dpej MAX such as Dpej MAX < Ni - the second odd number DPej2 is associated with the next to last number Dpej MAX-1. When only one number exists, the number is associated with itself. This number is the square root of the odd number Ni: DPej =. - so on. A description of the distribution of these pairs of numbers is given in paragraph Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 139

140 81 = 3*27 81 = 9*9 The first number of a pair of numbers is always located for a value j in the range [0 ; jmax]. The jmax value is equal to the integer part of the formula. It is written jmax=. The value of the second number is found in the table for a value j higher than jmax. When one of the pair of odd numbers is not a prime, the factorization of this odd number is searched in the table in the same way. The research of the factorization of each composite odd number can be done. Using the table allows to find the prime factors of any composite odd number Ni without any compute Odd Primes and twin primes. An odd number Ni is a prime number when there is no binary values "11" in the table except to the half-period Dpej=Ni. For instance, for Ni=23 and Ni=37 that are at lines k+2=12 and k+2=19, the binary values "11" does not exist. These odd numbers are primes. The binary values "11" respectively are only presents for j=10 and j=17; ie DPej=23 and DPej=37. These numbers are divisible by 1 and by themselves. To determine if an odd number is a prime number, the absence of binary values "11" for the sequences j in the range [0; jmax] is sufficient. The jmax value is equal to. For a composite number Ni, the first odd number Dpej which factorizes Ni is always get for j in the range [0 ; jmax]. If no odd number DPej is present in the last range, the odd number Ni is a prime number. When two consecutive odd numbers Ni and Ni+2 are composite numbers, the binary values "11" for Ni and Ni+2 are offset by one unit k. In the table, the second value "1" of the binary values "11" for Ni is located on the same line as the first value "1" of the binary values "11" for Ni+2. When two consecutive odd numbers are prime numbers, these binary values "11" do not exist. Instead of a common value "1", a value of "0" is present. An odd number Ni is the first prime number of a pair of twin primes when the value "0" exists for each sequence j in the range [0; jmax + 1]. Example of first prime number of a pair of twin primes: 11 (13), 17 (19), 29 (31), 41 (43), so on. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 140

141 Prime numbers and pairs of twin primes are determined in the table without doing any calculations. In the table, the first prime number of a pair of twin primes is highlighted in yellow Description of the distribution of binary values "11". The purpose of this section is to describe the distribution of binary values "11" in the table. This distribution is related to the value j of each sequence of points. When the value j increases, the first pair of binary values "11" is offset by one unit k relative to the binary values "11" of the previous column. In the table, an alignment of the first binary values "11" for each column j is present according to a linear function as Ni=f(j): k+2=2+j or k=j. As Ni=2*k+3 and DPej=2+3*j then Ni=DPej with j>=0. This linear function represents all odd numbers. It does not belong to the study of the odd numbers because it contains both the composite numbers and prime numbers. The second pair of binary values "11" of each column is offset by 3 unit k relative to the binary values "11" of the previous column. In the table below, an alignment of the second pair of binary values "11" for each column j is present according to a linear function as Ni=f(j): k+2=5+3*j or k=3*(j+1). As Ni=2*k+3 and Dpej=2*j+3 then Ni=3*DPej=3*(2*j+3) with j>=0. This linear function represents the set of odd numbers which are odd multiple of 3. For j=0, the same odd numbers Ni are found when the half-period of the sequence j, that is DPej, is multiply with an odd number Mi defined by 2*m+3: Ni=DPej*Mi=3*(2*m+3) with m>=0. The first odd number Ni of the 2 linear functions is gotten for j=m=0. It is a common value Ni=9 which corresponds to a common point k=3. This number 3 is the remarkable point of the sequence j = 0. When the two linear functions starting to Ni = 9 and j = 0 are plotted on a chart, a half inverted V appears. A composite odd number can always be factorized with two odd numbers. The value of these factors is determined to be the half period Dpej of the points of each linear function. The points of the function Ni=3*(2*m+3), only correspond to one factor, that is the remarkable point; i.e. in the example above, j=0 and Dpej=3. Each point of the second function Ni=3*(2*j+3) gives a different factor Dpej=2*j+3. The odd number Ni is factorized with the following pair of odd numbers: (3, DPej). It is written Ni=3*DPej=3*(2*j+3). Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 141

142 K=J and Ni = DPej = 2*J+3 k +1 Ni=3*(2*m+3) Ni=3*(2*J+3) +3 Ni=5*(2*m+3) Ni=5*(2*J+3) +5 Ni=7*(2*m+3) Ni=7*(2*J+3) +7 The third pair of binary values "11" of each column is offset by 5 unit k relative to the binary values "11" of the previous column. In the table below, an alignment of the second pair of binary values "11" for each column j is present according to a linear function as Ni=f(j) and so on: k+2=8+5*j or k=5*(j+1)+1. As Ni=2*k+3 and Dpej=2*j+3 then Ni=5*DPej=5*(2*j+3) with j>=1. This linear function represents the set of odd numbers which are odd multiple of 5. For j=1, the same odd numbers Ni are found when the half-period of the sequence j, that is DPej, is multiply with an odd number Mi defined by 2*m+3: Ni=DPej*Mi=5*(2*m+3) with m>=1. The first odd number Ni of the 2 linear functions is gotten for j=m=1. It is a common value Ni=25 which corresponds to a common point k=11. This number 11 is the remarkable point of the sequence j=1. When the two linear functions starting to Ni=25 and j=1 are plotted on a chart, a half inverted V appears. A composite odd number can always be factorized in two odd numbers. The value of these factors is determined to be the half period Dpej of the points of each linear function. The points of the function Ni=5*(2*m+3), only correspond to one factor, that is the remarkable point; i.e. in the example above, j=1 and Dpej=5. Each point of the second function Ni=5*(2*j+3) gives a different factor Dpej=2*j+3. The odd number Ni is factorized with the following pair of odd numbers: (5, DPej). It is written Ni=5*DPej=5*(2*j+3). Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 142

143 From each remarkable point GW(j) whose odd number 2*GW(j)+3 is the square of the half period DPej of the sequence j, some new odd numbers Ni appears. For j=n, the half-period of the sequence of points is DPen=2*n+3. New composite odd numbers Ni are then displayed in the table and represented by the two following linear function: Ni= * (2*j+3)=DPen*(2*j+3)=(2*n+3)*(2*j+3) with j>=n and Ni= j=n * (2*m+3)= DPen*(2*m+3)=(2*n+3)*(2*m+3) with m>=n and The odd number Ni matching with the remarkable point is got for j=m=n. For each sequence j a remarkable point exists. From this point in the table, a structure as a half inverted V is displayed. A regular structure composed of half inverted V nested into each other appears. A composite odd number can always be factorized with one or more pair of odd numbers. When Ni is a remarkable point, a pair of odd numbers with the same value exists Formulas connecting prime numbers It is known that the primes are of the following mathematical form: 6*m-1 and 6*m+1. These results can be found by studying the binary values "11" of the table. The first odd numbers are odd multiples of odd number 3. They are obtained for j=0. The half-period of the sequence of points is DPej=3. We have shown that an odd number Ni is a prime number in the table when the binary values "11" do not exist. That is, when this number Ni is not a multiple of 3. When j=0 the study of odd numbers is in the range [9 ; 23]. For odd numbers inferior to 9, the value "0" is present for the number 5. The numbers 5 and 7 are primes and then form a pair of twin primes. In the table, the odd number 3 is represented by only one binary values "11", the value of the factor (DPej=3) is the number itself. This is a prime number. This number does not belong to the range for the sequence of points j=0. For the sequence of points j>=0, we study binary values "11" for the points connected by equations Ni=f(j) different from that representing the set of odd numbers: Ni=DPej. For j>0, an odd number is a prime number when there is no binary values "11" in the range [0 ; j]. The odd number Ni is then no multiple of any Dpej. This condition must exist at first for j=0 that is DPej=3. Odd numbers which respect this condition are located in k+2 = 3+3*m and k+2 = 4 + 3*m with m>=0. Due to Ni=2*k+3, the odd numbers Ni which respect the condition are linked by the two following formulas: - For k+2 = 3+3*m and k = (Ni-3)/2, we get: k = 3*m+1 = (Ni-3)/2 Hence Ni = 2*(3*m+1) + 3 = 6*m+5 with m>=0 in natural integer space N. In the space W, the formula is k = 3*m+1 with m>=0. - For k+2 = 4+3*m and k = (Ni-3)/2, we get: k = 3*m+2 = (Ni-3)/2. Hence Ni= 6*m+7 with m>=0 in natural integer space N. In the space W, the formula is k=3*m+2 with m>=0. These last results are the same as respectively Ni=6*m-1 and Ni=6*m+1 with m>0. In space N, the prime numbers Ni and their index k in space W are represented by formulas which exclude the values multiples of the prime number 3. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 143

144 k = 3*m+1 k = 3*m+2 Location of the primes higher than 3 k=3*m+1 so k=1 for m=0 k=3*m+2 so k=2 for m=0 A singular point can be a prime number A remarkable point cannot be a prime number For m=0, two primes are created and form a pair of twin primes: (5,7). However the first twin primes is (3,5). For recap the number 3 is the origin point in the space W and also the table. The first sequence of points j=0 is responsible for the creation of twin primes. It always exists two odd numbers between two consecutive odd multiples of 3. For instance, between 9 and 15, numbers 11 and 13 are present. These pairs of odd numbers are twin primes when none of these odd numbers are odd multiples of an odd number higher than 3. This translates to the first prime number Ni of a pair of twin primes, by a value "0" for each value DPej with j>=0, highlighted in yellow in the table. The sequences of points obtained for j> 0, generate odd numbers that are multiples of the new odd primes or composite numbers. These odd numbers are multiples of the value of the halfperiod of the sequence DPej. When the value of DPej is a composite number, odd numbers generated correspond to a subset of odd numbers previously created by another sequence of points. The value of the DPej of the new sequence of points is a multiple of the value DPej of a previous a sequence of points. These points can therefore not generated new odd numbers. The new odd numbers are created from multiple of prime numbers. For j <= 0, the primes 5,7,11,13,17,19 and 23 are created. The numbers that are odd multiples of these primes are new odd numbers. In the table, the binary values "11" of these numbers, to be called prime factor, are positioned for a part of them between the values which are odd multiples of 3. For j=0, between the values which are odd multiple of 3, it exists all pairs of odd numbers that can give birth to the pairs of twin primes. When the value of j increases, a part of these pairs of odd numbers is converted into pairs of composite numbers and mixed pairs consisted of a prime number and a composite number. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 144

145 5.8.8 Formulas connecting the pairs of twin primes. An odd number Ni is a prime number when there is no prime factor which factorizes it. In the table, this translates into the lack of binary values "11. The presence of two consecutive primes is translated into the table for the first prime by a value "0" existing for each value of j in the range [0; jmax] with jmax= DPej is inferior to Ni.. This is true until the value of j whose half-period For Ni=59 and k+2=30, jmax is equal to 2. The value "0" is got for j=0, j=1 and j=2. It is then the first prime of a pair of twin primes (59,61). The half periods DPej of the sequences of points j=0,j=1 and j=2 are respectively equal to 3, 5 and 7. It is the three first odd primes. These numbers are unique because the analysis of their primality realized in the preceding paragraph is not made in a base unit. It is realized when j<0, in the range [Ni =3 ; Ni =7] by using the binary values "011" of j=0 and DPej=3. When jmax>2, a pair of twin primes can appear if for the first prime a value "0" exists for j=0, j=1 and j=2. In the table, distribution of binary values "11" from j=0 to j=2 is repeated endlessly, from the point k+2=5, that is k=3, and then every 105 points (105=3*5*7). The point k=3 is the first point of the sequence j=0. It exists 15 points between the points k=3 and k=108 which can give birth to a pair of twin primes except to the following pairs of twin primes: (3,5) et (5,7). In the table, the first prime belonging to a pair of twin primes is got for the point k=4. This point is repeated every 105 points: k=4+105*x with x a natural integer. We get then the following equation: k=4+105*x=(ni-3)/2. The result is: Ni=11+210*x. The mathematical form of a first prime number Ni of a pair of twin primes is as follows: Ni=6*m-1. With the equality Ni, the relationship between the parameters m and x is: m=2+35 * x. In appendix 2, a representation of the points generated by the 15 formulas giving birth to the first prime number of a pair of twin primes is given. The table below displays for the 15 points of k the formulas which generate all odd numbers Ni where the first prime of a pair of twin primes can be found. 15 values of k First prime Ni of a pair of twin primes Ni=6*m-1 Relationship between m and x 4 Ni=11+210*x m = 2+35*x 7 Ni=17+210*x m = 3+35*x 13 Ni=29+210*x m = 5+35*x 19 Ni=41+210*x m = 7+35*x 28 Ni=59+210*x m = 10+35*x 34 Ni=71+210*x m = 12+35*x 49 Ni= *x m = 17+35*x 52 Ni= *x m = 18+35*x Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 145

146 67 Ni= *x m = 23+35*x 73 Ni= *x m = 25+35*x 82 Ni= *x m = 28+35*x 88 Ni= *x m = 30+35*x 94 Ni= *x m = 32+35*x 97 Ni= *x m = 33+35*x 103 Ni= *x m = 35+35*x The second prime of a pair of twin primes corresponds to the point k of the first prime plus 1. The first point is k=5. This point is repeated every 105 points: k=5+105*x with x a natural integer. We get then the following equation: k=5+105*x=(ni-3)/2. The result is: Ni=13+210*x. The mathematical form of a first prime number Ni of a pair of twin primes is as follows: Ni=6*m+1. With the equality Ni, the relationship between the parameters m and x is: m=2+35 * x. The relationships between m and x are the same as those found for the first prime number. The m values that generate the pairs of twin primes in the mathematic form 6*m-1 and 6*m+1 are given by the 15 formulas displayed into the table below. The table below displays for the 15 points of k the formulas which generate all odd numbers Ni where the second prime of a pair of twin primes can be found. 15 values of k Second prime Ni of a pair of twin primes Ni=6*m+1 Relationship between m and x 5 Ni=13+210*x m = 2+35*x 8 Ni=19+210*x m = 3+35*x 14 Ni=31+210*x m = 5+35*x 20 Ni=43+210*x m = 7+35*x 29 Ni=61+210*x m = 10+35*x 35 Ni=73+210*x m = 12+35*x 50 Ni= *x m = 17+35*x 53 Ni= *x m = 18+35*x 68 Ni= *x m = 23+35*x 74 Ni= *x m = 25+35*x 83 Ni= *x m = 28+35*x 89 Ni= *x m = 30+35*x 95 Ni= *x m = 32+35*x Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 146

147 98 Ni= *x m = 33+35*x 104 Ni= *x m = 35+35*x 5.9 Twin prime conjecture In this paragraph, we are going to demonstrate the infinity many twin primes. The number of pairs of twin primes is named Ncnpj. The value of this number increases to infinity according to a second degree polynomial for which parameter is the index of the sequences of points j:. We are going to demonstrate also that the number of primes increases to infinity according to a second degree polynomial for which parameter is the index of the sequences of points j: Détermination of the parameters In this section, three parameters (A), (B) and (C) are determined. They are used to get the formula representing the number of pairs of twin primes: Ncnpj. Two of these parameters (A) and (B) define the formula representing the number of prime numbers: Nnp. The study of the sequence of points for j <= 0 showed that the prime numbers are located only between the odd multiples of odd number 3. There is at maximum two primes Ni between two consecutive odd multiples of the odd number 3. The first prime number is written as 6*m-1 while the second one is written as 6*m +1 with m>0 except for the first pair of twin primes (3,5). When two prime numbers exist for the same value of m, a pair of twin primes appears. There are potentially an infinite number of pair of twin primes. Indeed, for n consecutive multiples of the odd number 3, the number of possible pairs of twin primes is equal to n-1. The number of prime numbers is equal to 2*(n-1). In the table, the number of possible pairs of twin primes is determined by the number of value "0" obtained for j=0. For instance, when Ni is between 9 and 25, there are three possible pairs of primes and therefore 6 primes. However, the result is only 2 pairs of twin primes and 5 prime numbers. This result is related to the existence of an odd number which is multiple of a prime number greater than 3. In this case, the number Ni=25 is multiple of the prime number 5. The number Ni=23 being a prime number, the third potential pair of twin primes becomes a pair of odd numbers composed by a single prime number. It is a mixed pair of odd number. Pairs of twin primes Mixed pair 25 is a composite number. It is factorized by the prime factor Dpej=5 of j=1 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 147

148 The odd numbers multiple of a prime number greater than 3 are responsible for reducing the number of pairs of twin primes. A pair of consecutive odd numbers which are not multiple of 3 can only exist in 3 forms: - a pair of twin primes - a pair of composite numbers - a mixed pair of odd numbers composed of a prime number and a composite number. The composite numbers are factorized into one or several pair of odd numbers which are prime numbers or composite numbers. For instance, the numbers Ni=203 and Ni=205 are located between two odd numbers which are odd multiples of 3: 201 and 207. The number 203 can be written as the product of 7*29 and the number 205 as the product of 5*41. The pair of odd numbers [203,205] is a pair of composite numbers [7*29 ; 5*41]. This pair of odd number can be described by its prime factors: 7, 29, 5 and 41. The pair of odd number [203,205] can be represented by the pairs of the prime: [7,5], [29,41], [7,41] and [29,5]. The pair of odd numbers which are not multiple of 3 is written as [Ni=6m+5, Ni=6m+7] in the space N. It is written as [k=3m+1, k=3m+2] in space W. Composite number 203 as k=3m as k=3m+2 Prime Nombres numbers premiers and composite et composés numbers prime factors: 5, 7, 29, 41 Pairs of prime factors 3m+1 = 7 and 29 3m+2 = 5 and 41 (7,5), (29,41), (7,41), (29,5) - When one or several prime factors are located in the table in k=3m+1 or in k=3m+2, one of both odd numbers of the pair of odd numbers is a composite number. The second odd number is a prime number. It is a mixed pair of odd numbers. - When two or several prime factors are located in the table in k=3m+1 and in k=3m+2, one or several pairs of prime factors exist. The pair of odd numbers is a pair of composite numbers. The number of pairs of odd numbers (A) not multiple of 3 is the sum of: - the number of pairs of twin primes (Ncnpj). There are no prime factors. - the number of mixed pairs of odd numbers. When several prime factors exist in 3m+1 or in 3m+2 for a same pair of odd numbers, only one prime factor is accounted for counting the number of composite numbers. - the number of pairs of composite numbers (C). When several prime factors exist in 3m+1 and in 3m+2 for a same pair of odd numbers, only one pair of prime factors is accounted for counting the number of pairs of composite numbers. The number of pairs of odd numbers is written (1) (A) = Ncnpj + + (C). The number of composite numbers in a mixed pair is equal to 1. The number of composite numbers in a pair of composite numbers is equal to 2. There is 0 composite numbers in a pair of twin numbers. The number of composite numbers (B) located in 3m+1 and in 3m+2 is: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 148

149 (B) = 0* Ncnpj +1* +2*(C) hence +2*(C)=(B) We get: +(C)= (B) - (C) The formula (1) can then be written: (A) = Ncnpj + (B) - (C). The number of pairs of twin primes is: Ncnpj = (A) - ((B) - (C)) = (A) - (B) + (C) The number of pairs of twin primes is equal to the number of pair of odd numbers which are not multiple of 3 less the number of composite numbers located in 3m+1 and in 3m+2 plus the number of pairs of composite numbers located respectively in 3m+1 and in 3m+2. The number of prime numbers Nnp is equal to twice the number of pair of odd numbers (A) less the number of composite numbers (B): Nnp = 2 (A) - (B) Determination of the formulas for the parameters (A) and (B) In this section, we will show that each of the two previous parameters (A) and (B) is dependent on the sequence of points j. The formulas are written in the form of a second degree polynomial f(j) Formula f(j) for (A) The number of pairs of odd numbers [3m+1, 3m+2] which are not multiple of 3 is the number of points k in the following mathematic form: 3m+2. For k=3 and m=0, it exists one pair of odd numbers [k=1,k=2]. The number of pairs of odd numbers depends on the parameter k. The formula (2) is: (A)=[(k+1)/3] with k>=0 and [ ] is the integer part of the division. Each sequence of points j owns a remarkable point k=gw(j) = replacing k in the formula (2), we get: (A) = [. After ]. There are 2 remarkable points out of 3 alternating regularly for which the result of the formula is an integer value, for instance j=5,7, 11,13,17,19,23,25 and so on. The result is not an integer value when j is equal to 9, 15, 21 and so on. If we only take into account the values of j whose the result is an integer value, (A) is dependent of j such that. The result of both formulas increases to infinity in the same way: and. When the value of j increases to infinity, the number of pairs of odd numbers increases to infinity such as the highest rank monomial that is. The number of pairs of odd numbers is therefore related to the formula [ ] and the formula when using the value of j whose the result of the formula is an integer value. These integer values are obtained when. Then there exists an infinity of j points leading to an integer value of. The number of pairs of odd numbers increases according to the formula when j tends toward infinity. A remarkable point GW(j) cannot get a value in the form 3q+1. Proof: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 149

150 k = GW(j) = 3q+1 and k= For that GW(j) can be written in 3q + 1, it needs that q =. It means that must be divisible by 3. It has been shown in paragraph "3-4- Singular points" that this was impossible. When k=gw(j) is in the form 3q+2, the number of pairs of odd numbers is an integer value: (k+1)/3 = (3q+2+1)/3) = 3 (q+1)/3 = q+1. There are 2 remarkable points out of 3 alternating regularly with this result. When k=gw(j) is in the form 3q, the number of pairs of odd numbers consists of an integer part and a fixed fractional part equal to 1/3: (k+1)/3 = (3q+1)/3 = q + 1/3. When q approaches infinity, the decimal part 1/3 becomes negligible. When k tends towards infinity, the number of pairs of odd numbers is written such as (A) and (A) Formula f(j) for (B) The number of composite numbers (B) is the number of values of k which at least one prime factor exists in the factorization of the associated odd number Ni. The values of k multiple of 3 are not taken into account. Only the k values in the form 3m+1 and 3m+2 are counted. When several prime factors exist for a same value of k, only one prime factor is counted Enumeration of prime factors In this section, the formula that compute the number (B) of composite numbers which are not multiple of 3 is determined. The number of k values which are not multiple of 3, N K3, is obtained with the relation N K3 =[k/3] and [ ] is the integer part of the division. Each number whose a prime factor is equal to 3 is excluded of the enumeration. The k values that are multiples of 5 are related to the following formula k= 5*m+1 because the half period Dpej of the sequence of points j=1 is 5 and the first point is equal to k=1. This sequence of points starts with an offset of 1 unit of k. The number of k values that are multiple of 5 is given by the following formula: N K5 =[(K- 1)/5]. This formula does not count the point got for m=0 because the first point corresponds to the prime factor itself that is a prime number. It is not a multiple of the prime factor. Moreover the points obtained for the sequence j with n=0 are not taken into account for the enumeration and the study of the odd numbers (See the section "2- Building the space W"). So the first point is for K=6 that leads to the result N K5 =1. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 150

151 Dp = offset of one unit of k to get the first point of each new sequence of points j when j=1, the prime factor is Dpej=5 and the first point is k=6 k=5*m+1 when j=2, the first point is k=9 k=7*m+2 The values of N K7 et N K11 are N K7 =[(K-2)/7] and N K11 =[(K-4)/11] with a respective offset of 2 and 4. An odd number Ni is written Ni = 2*n + 3 with n being the sequence number of occurrence of odd numbers in the range [0 ; N]. For the value 3, n is equal to 0 while for 5, 7 et 11, n is respectively equal to 1, 2 et 4. The offset Dp is the half of the difference between the prime factor P and the first odd number 3, hence Dp = (P - 3) / 2. Since P is an odd number, we get P = 2*n+3 hence Dp=n. The sequence number of occurrence of odd numbers is equal to the value of offset. We also note that the value of the offset is equal to the value of j. n Ni = P = 2n + 3 Dp = (P - 3) / 2 = n j The number of values of k which are multiple of a prime factor P, called N KP, is written according to the following formula N KP = [(k - (P-3)/2)/P] = [(2*k+3-P)/(2*P)]. This formula can also be written dependent of the odd number Ni matching the value k: N NiP = [(Ni - P) / (2*P)]. The test of primality in the paragraph 3.2.3, demonstrates that an odd number Ni is a prime number when this number is not divisible by any prime odd number P in the range [3 ; ]. In space W, this is translated by taking into account all prime numbers P matching the half period DPej=P, of the sequences of points j belonging to the range [0 ; [ with Kmax = [(Ni - 3)/2]. In the range [0 ; Ni], we defines (B1) which is the number of the composite numbers that are found from each prime factor P such as Dpej=P for the sequences of points j belonging to the ]] Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 151

152 range [1 ; [ ]]. The same composite number is counted several times because each prime factor of the composite numbers is counted. (B1) matches the number of composite numbers which are not unique. The sequence of points j=0 is excluded because Dpej is equal to 3. The first value in the range j starts to the value j=1 so Dpej=5. The prime factors used for the enumeration are located in the range [P 1 ; P Max ] with P 1 =5. For an odd number Ni, the number of composite numbers (B1) is the sum of the values of k which are multiple of a prime factor P, called N KP, in the range [P 1 ; P Max ]. The parameter n is now defined as being the index of a prime number. Prime numbers are ranked in order of increasing value in the range [P n=1 ; P n=nmax ]. For an odd number Ni, the number of composite numbers (B1) that are not unique in the range [0 ; Ni] is equal to (B1) = = and (B1)=. The number NKPn is the number of values of k which are multiple of the prime factor Pn. This number includes the common values of k obtained by the prime factor Pn and the previous prime factor Pn-1. That is the number of values of k which are multiple of the factor F=P n *P n-1. These values are counted twice for (B1), once for the factor P n and once for the factor P n-1. To compute the number of composite numbers (B) in a unique way the common values must be removed. The first odd prime number is named P0=3. For instance, with P1=5 the common values with P0 obtained for the factor P1*P0 = 3*5=15 must be removed. For P2=7, The common values obtained for the factors P1*P0, P2*P1 and P2*P0 must be removed. But with the factor P2 some new common values appears with the factor P2*P1*P0. The counting of values leads to add these values to (B1). Indeed, the calculation is performed taking into account only the values of k that are multiple of P1 and P2 in a unique way and that are not multiple of P0: - we add the number of values found with the prime factors P2 and P1. We get (B1). - we remove the number of values found with the factors P2*P1, P2*P0 et P1*P0 - we add the number of values found with the factor P2*P1*P0 because these values have been added twice due to P1 and P2 and deleted 3 times due to P2*P1, P2*P0 and P1*P0. After adding the number of values got with the factor P2*P1*P0, we only get the number of values of k which are multiple of P1 and P2 in a unique way and are not multiple of P0. The enumeration for the first Pmax prime factors is done in a same way. The table below gives an example of the value of factors existing when Pmax=P4=13. The factors are composed of 1 to 5 prime factors. The parameter Nprimes is the number of prime factors. These prime factors generate by their multiplication the factors F. This factor F depends on N primes and the prime factors: F(N primes, P) = F Nprimes (P j ) = Table with the factors F which are composed of 1 to all prime factors existing in the range [3; P4=13] N primes (Number of prime factors) Add/Delete the values Factors F for P1=5 Factors F for P2=7 Factors F for P3=11 Factors F for P4= (5) (7) (11) (13) 2 - (3*5) (3*7) (5*7) (3*11) (5*11) (7*11) (3*13) (5*13) (7*13) (11*13) Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 152

153 3 + (3*5*7) (3*5*11) (3*7*11) (5*7*11) (3*5*13) (3*7*13) (3*11*13) (5*7*13) (5*11*13) (7*11*13) 4 - (3*5*7*11) (3*5*7*13) (3*5*11*13) (3*7*11*13) (5*7*11*13) 5 + (3*5*7*11*13) Note: the number of factors F for the addition of the values is equal to that for the removal of the values. That is true for each prime factor P and for the set of the factors F. Addition and deletion of the number of values of k which are multiple of a factor F i depends on the number of prime factors that composes it, i.e. N primes. The sign of the addition (+) or the deletion (-) is computed with the value (-1) to the power of (N primes +1). The sign + or is computed with the following formula SignF = (-1) (Nprimes+1). The number of factors F depends on the number of prime factors Nprimes and the number of prime numbers Nmax in the range [0 ; Ni] with the prime number P0=3 excluded. When a number of prime factors Nprimes is selected, the number of factors F is the sum of the factors F existing for each prime factor P up to the prime factor equal to P Nmax. The enumeration of the number of factors F depending on the number of prime numbers Nmax is available in the table below for the first four prime numbers (P1 to P4) and for the Nmax prime numbers (P1 to P Nmax ). The prime number P0=3 is included in the list of the prime numbers when Nprimes>=2 in order to remove the number of values which are multiple of 3. The list is then composed of the primes from P0 to P Nmax. N primes (Number of prime factors) Add/Delete the values SignF P4 Number of factors F P4 Number of factors F - Combination P4 - Combination on primes [P0 ; P4] General case P Nmax. Number of factors F Combination ( * The number P0 = 3 is excluded. Search of 1 element among 4 => [P1 ; P4]. These factors are used to compute (B1). ( * Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 153

154 ( * ( * ( * ( * Search 2 elements among 5 to compute F Search 3 elements among 5 to compute F Search 4 elements among 5 to compute F The 5 elements are used to compute F ( * ( * ( * ( * X (-1) (x+1) ( * Nmax (-1) (Nmax+1) ( * The enumeration of the number of the factors F is found with mathematic combinations. When a number of prime factors N primes is selected, the number of factors is equal to the following combination: N primes elements among all elements Nmax. The element P0=3 is included in the elements of the combination when Nprimes>=2 to remove the number of values of k which are multiple of 3. In the range [0 ; Ni] the set of composite numbers to be removed of the set of composite numbers (B1) that are not unique is called (B2). This set is composed of the sum of the number of values of k, called NF that are multiple of each prime factor F for which Nprimes is greater than or equal to 2. Each prime factor is composed of i number of prime numbers. The number of values of k which are multiple of a prime factor F is associated with the SignF formula which is depending on the value of i. The number of values of k for a prime factor F is either added (+) or removed ( - ) to the value (B2). The formula to get the value (B2) is as follows: (B2) = NF = with that is the number of values of k for the factor which is composed of i prime factors (Nprimes) belonging to combinations. We will determine the number of values of k called. The values of k are multiple of a factor F. The factor F is equal to the multiplication of the i prime factors which composes the factor F. The number of prime factors i (Nprimes) is between 2 and Nmax. For the factor F 2 (P 1,P 0 )=P1*P0, the first point is obtained with the odd number Ni=P1*P0. The value of k is then k P1-P0 = (Ni - 3) / 2 = (F 2 (P 1,P 0 ) - 3) / 2. The next points are multiple of the factor Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 154

155 F 2 (P 1,P 0 ) starting to k P1-P0. The points are related by the following formula: k= k P1-P0 + F 2 (P 1,P 0 ) * m. When k is equal to k P1-P0, the number of the values of k which are multiple of the factor F 2 (P 1,P 0 ) called NF 2 (P 1,P 0 ) must be equal to 1. The number NF 2 (P 1,P 0 ) of values of k which are multiple of the factor F 2 (P 1,P 0 ) is then computed due to the following formula: NF 2 (P 1,P 0 ) = [( (k - k P1-P0 ) / F 2 (P 1,P 0 )) + 1] NF 2 (P 1,P 0 ) = [(2*k F 2 (P 1,P 0 )) / (2* F 2 (P 1,P 0 ))] NF 2 (P 1,P 0 ) = [(Ni + F 2 (P 1,P 0 )) / (2* F 2 (P 1,P 0 ))] This formula can also be written in a general case in the following way: NF i (F) = NF i (F) = We get the generic formula for (B2): (B2) = with F i (P j ) =. The prime factors (Nprimes) belong to each combination. This formula can be written depending on the odd number Ni: (B2) = The formula (B) which enumerates the number of composite numbers which are not multiple of 3 is defined as follows: (B) = (B1) + (B2) (B) = + ou encore en en remplaçant 2*k+3 par Ni: (B) = + Example enumeration for j=2: The maximum value of the odd number Ni for which it is possible to do an enumeration is the value of the remarkable point k= GW(j+1) = = 39 hence Ni=2*39+3=81. There are 2 prime factors inferior to 81 and not multiple of 3: P1=5, et P2=7 with Nmax=2 and P Nmax =7. The prime number P0=3 belongs to the combinations which compose the factors F when i>=2. It allows computing the value (B2). The enumeration for j=2 is given in the table below: i P1 = 5 Result P2 = 7 Result Tota l 1 [(81-5) / 2*5)] 7 [(81-7) / 2*7)] * [(81+3*5) / 2*3*5)] -3-1* [(81+3*7) / 2*3*7)] -1* [(81+5*7) / 2*5*7)] 3 [(81+3*5*7) / Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 155

156 2*3*5*7)] Total For j=2, the number of composite numbers (B) is equal to +6. This value can be easily found manually from the table below. P1=5 and i=1 7 prime factors P2=7 and i=1 5 prime factors (1) à (5) (1) P1=5 P0=3 and i=2 3 common values (2) P1=5 P2=7 and i=2 1 common value (3) P2=7 P0=3 and i=2 2 common values (4) (5) The formulas (B1) et (B2) are built as the sum of the integer part of a linear function dependent of k: and. The linear function is denoted as being the term of (B), (B1) and (B2). When k tends toward to infinity, the fractional part of the division by P or belonging to in the range [0 ; 1[ is negligible. Moreover, when k increases the value of the integer part of a function [ ] evolves towards infinity identically as the value of the function and in particular when taking into account the values of k giving a numerator that is multiple of the denominator: (2*k+3-P) = X (2*P). The study focuses on the evolution of the number of composite numbers when k tends to infinity. The integer part is then removed from the formula (B): (B) = + Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 156

157 1- The fractional part of the division is between 0 and. For the prime factor 5, the fractional part is in the range [0 ; 9/10]. The result of the sum of the fractional parts for the set of the terms of (B) is called St(B). This result St(B) is greater than or equal to 0. When k tends to infinity this result is negligible in comparison with the integer part of the division. However, the value of St(B) can be estimated in the case of the largest value St for each term of (B): St(B) = + St(B) = + There are as many positive values as negative values as showed in the table " Table with the factors F which are composed of 1 to all prime factors existing in the range [3; P4=13]". The value 1 of the terms St(B) = + cancel. The formula St(B) is then: St(B) = -1/2*( + We note a B = + St(B)= -1/2 * a B 2- It is known that the evolution towards infinity of a polynomial of degree n is the same as the evolution of monomial of the highest degree. For (B), when k tends to infinity, we only retain the monomial of highest degree. That means and (B) = + We get (B) = a B * k k is related to the value of j by the following formula: k= ). The formula (B) is then: The evolution of the value (B) according to the parameter j is shown in appendix 9. When j tends to infinity, the formula (B) keeps only the monomial of highest degree: (B) = 2 * a B * Determination of the coefficient ab. We will demonstrate that the coefficient ab is written as an alternating series. The convergence of this series is then demonstrated and its value is estimated a Alternating series: ab. A alternating series can be represented by one of the following formulas: or. When the series respects the two rules of Leibnitz, the series converges. The 2 rules are: - - Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 157

158 The coefficient a B can be written as an alternating series. The value (B1) = can also be written with the prime factor P0=3 which is excluded of the elements of the combination only for i=1. The factor is equal to. is the prime factor of the combination. The coefficient a B is then equal to a B = number of prime factors tends to infinity: a B = ) ) or when the After replacing the letter i by n, we get an alternating series: a B = with and with Pj corresponding to each prime factors of the combination b Convergence of the alternating series: ab. Each element of the alternating series consists of the sum of the inverse of the factors Fn. A Fn factor is the product of the prime factors that compose it. The number of prime factors increases for each element of the series when n increases. n Number of combination The divisor dn is the product of the prime factors. The number of the prime factors increases when n increases. 1 ( * P1 2 ( * P1*P2 X ( * P1*P2*...*PX The term of the element Un is the function 1/Fn. The last element of the alternating series has only one term because the number of combination is equal to = 1. When n tends to infinity, the product Fn whose the number of prime factor is infinite tends to infinity. The inverse of this product that is the function 1/Fn is the term of the element and also the element. The term tends to 0. Hence. The first rule of Leibniz is respected. The second rule of Leibnitz is respected when whatever the value of n. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 158

159 The distribution of the number of combinations is close to a normal distribution (Gaussian function). The graph of a Gaussian is a characteristic symmetric "bell curve" shape. There is a growing phase followed by a decreasing phase. The first point is equal to n-1 and the last point is equal to 1. It is known that the points and ( ) are identical which generates a symmetry in the layout of the distribution of the number of combinations. In our study, this symmetry is valid for because the value p=0 is not taken into account. More over for p=1, the prime factor P0=3 is removed, which leads to take one factor among n-1 and so. Graphical representation of the distribution of the number of combinations when Nmax =8 and Number of combinations Nmax=8 Nmax= n There is symmetry in the distribution of combinations. This symmetry is located to [(Nmax+1)/2)] with [] being the integer part. From this point, the number of combinations of is less than or equal to the one of Un. The prime factors used to compute Un and come from the same set of prime factors, ie [P0=3 ; Nmax]. To each product of prime factors of, a product of prime factors of Un can be associated to it such as F n+1 = F n * P with P belonging to [3 ; Nmax] and different of one of the factors composing Fn. The inverse of the product F n+1 is then inferior to the inverse of the associated product Fn: 1/ F n+1 < 1/F n. The sum of the inverse of the products of is then always inferior to the sum of the inverse of the products of U n from the point of the symmetry:. The table below shows the previous result for the combinations of the two last elements of U n, ie U4 and U5 with Nmax=4 and so n= N primes =5. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 159

160 Un with Nmax=4 and n= N primes =5 Number of factors Combinations of factors Product of prime factors = F Remarks U4 5 ( * F1 = (3*5*7*11) F2 = (3*5*7*13) F3 = (3*5*11*13) Each of these factors F is inférior to the factor F1 of U5. F4 = (3*7*11*13) F5 = (5*7*11*13) U5 1 ( * F1 = (3*5*7*11*13) F1 of U5 = 13 * F1 of U4 Remarks Number of factors of U4 is > to this of U5 1/F1 of U5 < 1/F1 of U4 so 1/F1 of U5 < of U4 The value of U5 is inforior to this of U4 U5 < U4, < When the number of terms 1/F n of is greater than the one of U n, it must be determined if the number of terms of allows to get a value greater than the value. The value is the sum of a smaller number of terms. But the value of the terms is higher. We will study the evolution of the values of U n when the number of factors increases with n and when the number of terms of is greater than the one of U n. For the sequence of points j = 2, we will calculate and compare the values of U1 and U2. The prime factors for U1 are P1=5 and P2=7 while for U2, they are P1, P2 and P0 = 3. U1 whose the number of combinations of the factors [P1 ; P2] is *U1= U2= U2 whose the number of combinations of the factors [P0 ; P2] is * In order to get the same denominator as U2, the multiplicative factor P0 was added to the numerator and the denominator of U1. if The value of prime factors increases with n. The value P0=3 is regarded as the first prime number. All next prime numbers can be written according to the order Js in which they appear as being equal to P0 plus a number depending on Js:. After replacing by its formula in the previous inequality, we get: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 160

161 Hence that is always true because. The numerator of Un is written as a polynomial function with as the parameter P0. However P0 is a constant and is a variable. The previous result is also got in comparison the coefficients of the elements and in particular for the inequation their sum. Thus since P0 is greater than 1 and the constant is greater than the constant 3 * P0, the inequality is true. The values of the coefficients depend on P0 and the number of combinations of prime factors. The number of prime factors is equal to Nmax. If we write the number of combinations of k elements among M as follows: ( ) then the coefficients of the polynomial functions of the numerator of Un are written as follows: if k=1, we get if k>1, we get with the following parameters: k is the index of the element Un If k=1, x is a parameter whose the value belongs to the range [0 ; Nmax-1]. The degree of the monomial is in the range [Nmax ; 1]. The number of monomial in the polynomial function is Nmax. If k>1, x is a parameter whose the value belongs to the range [0 ; Nmax+1-k] that is also [0 ; Nmax+1-n]. The degree of the monomial is in the range [Nmax+1-k ; 0]. The number of monomial in the polynomial function is Nmax-k+2 that is also Nmax-n+2. The number of monomial for k=n=2 (U2) is then the same than for k=n=1 (U1) that is Nmax. The previous inequality is written: And also The comparison of the values of U n is done through the values of the numerator for an identical denominator for both elements to compare U n and U n+1. The values of U n, for the sequences j belonging to the range [2 ; 5], are given in the table below. These values depend on P0 and. We define the function T as the function that returns the js index based on the prime number returned by the combination. This prime number is represented in the formulas by a value between 1 and jsmax. Sequence J and Nmax U n U n+1 2 and 2 Combination of prime factors = Denominator: P0*P1*P2 Numerator of U1: Combination of prime factors = Denominator: P0*P1*P2 Numerator of U2: ( * ( * ( * ( * Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 161

162 4 and 3 Combination = Denominator: P0*P1*P2*P3 Numerator of U1: Combination = Denominator: P0*P1*P2*P3 Numerator of U2: ( * ( * ( * ( * With 5 and 4 Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Numerator of U1: With Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Numerator of U2: ( * ( * ( * ( * With 5 and 4 Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Numerator of U2: With Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Numerator of U3: ( * ( * ( * ( * With With Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 162

163 For a sequence of points j selected, a number of prime numbers Nmax is associated therewith. When n is defined as the index of the element Un, we get: for n = 1, U1 = for, Un = with and when x=0, If we write the previous formulas become: for n = 1, U1 = for, Un = with for x=0, Ax=A 0 =1 U n is represented by a polynomial function of Ax. For n=1, x is in the range [0 ; Nmax-1] whereas for n>1, x is in the range [0 ; Nmax+1-n]. In the table below, Un is given as a function of Ax. Because A 0 =1, A 0 is not indicated. Sequence J and Nmax U n U n+1 2 and 2 Combination of prime factors = Denominator: P0*P1*P2 Numerator of U1: Combination of prime factors = Denominator: P0*P1*P2 Numerator of U2: 4 and 3 Combination = Denominator: P0*P1*P2*P3 Numerator of U1: Combination = Denominator: P0*P1*P2*P3 Numerator of U2: 5 v 4 Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 163

164 Numerator of U1: Numerator of U2: 5 and 4 Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Numerator of U2: Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Numerator of U3: For n=1, the comparison of the elements U n and U n+1, i.e. of U1 and U2, is done by removing Ax with the highest value of x to the two elements. The number of monomial of the polynomial U n is then always greater of 1 than this one of the polynomial U n+1 whatever the value of n. The combinations which the value is equal to 1 are no more written. We then get the table below. Sequence J and Nmax U n U n+1 2 and 2 Combination of prime factors = Denominator: P0*P1*P2 Numerator of U1: Combination of prime factors = Denominator: P0*P1*P2 Numerator of U2: 4 et 3 Combination = Denominator: P0*P1*P2*P3 Numerator of U1: Combination = Denominator: P0*P1*P2*P3 Numerator of U2: 5 and 4 Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 164

165 Numerator of U1: Numerator of U2: 5 and 4 Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Numerator of U2: Combination of prime factors = Denominator: P0*P1*P2*P3*P4 Numerator of U3: The polynomial U n has one monomial more than the polynomial U n+1.this monomial has a degree greater than the monomial with the highest degree of Un+1. The polynomials U n and U n+1 are compared by pairs of monomial with the highest degree:. The value of U n is higher than this one of U n+1 when by pair the monomial of U n is greater than this one of U n+1. The first monomial (A 0 ) of U n is not taken into account because it does not change the comparison. Thus U1 U2: for Nmax=2, si for Nmax=4, if so and so and so Thus for n=1, U1 U2 if and Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 165

166 In the same way, for n > 1, Un Un+1 if and Ax is the sum of product of i js. This value i js corresponds to the value added to P0 to create the prime numbers. The graph below shows the value of i js based on the order number js in which appears the prime numbers up to js= For, the index js is in the range [1 ; Nmax]. Empirically, we observe that this value i js increases in the form of a polynomial with one monomial of degree b: i js = with the following values: 5,4195 * N 1, The parameter N is equal to js in the range [1 ; Nmax]. i js = 5,4195 * N 1,07625 i js N When the ratio respects the previous inequality, the coefficient a B converges. We will study the value of the previous ratio with the following formula: i js = with N which tends to infinity. The value of the degree b of the monomial is close to 1. We will use the value 1 for b. We will search the conditions on the value of for the respect of the previous inequalities and therefore the convergence a B. When i js =, is a common factor hence Ax is written: A 1 is the sum of all natural integers N from 1 to Nmax that multiply the factor. A 2 is the sum of the product by pair of all natural integers from 1 to Nmax that multiply the factor a 2. A x is the sum of the product by x values of all natural integers from 1 to Nmax that multiply the factor a x. The results given hereinafter are determined empirically. The details on Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 166

167 the results are presented in appendix 3. Empirically, we observe in the section B) of the appendix 3, that the ratio evolves in a similar way of the ratio of the partial sum of the products computed with the highest value Nmax. This ratio decreases when x increases for a same value of Nmax. This ratio is written:. The letter P of ANP refers to the word "Partial". We then get the following ratio: The formulas obtained for the ratios, and are : The strongest constraint on the value. So we get: for n=1 and, U1 U2 if when Nmax tends to infinity is obtained for the ratio and That is if and so respectively and n>1 and, Un Un+1 if and Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 167

168 That is if and so respectively and The strongest constraint on the value when Nmax tends to infinity is:. In appendix 3, the section C) displays the minimum values of for n=1 and n>1 according to X for Nmax equal to 11. These minimum values are computed according to the ratio et. For, the conditions on to get are respected with respect to the stronger condition >3. For and n>1, The strongest condition on a previously determined, i.e., is higher than the minimum values of to get. However for x=nmax-1 and n=1 (U1/U2), the minimum value of is greater than the strongest constraint on. In fact, we get a value of 5.63 for Nmax = 11 against a value of 3 for Nmax equal to infinity. The graphic 1 in section D) of the appendix 3 shows that the coefficient of the formula i js = increases with N according on the formula =0,4106*ln(1+N). For N=Nmax=11, we get =1,02. It appears then that the minimum value of a for the highest values of X is higher than the value =1,02. That shows the restriction of the use of the comparison of the coefficients by pair to validate when the number of terms of is greater than this one of. This condition on the comparison of coefficients by pair is too restrictive. The condition must be done on the polynomial. However, this shows that it is possible to obtain, whatever the value of n, with the formula i js = if the value of increases enough according on N. The convergence of the series a B is mathematically demonstrated by the method of the absurd. We have demonstrated that the number of primes is written Nnp = 2 (A) - (B). Euclid's theorem says that there are infinitely many primes. This means that Nnp is always greater than 0. We can write: > 0 If the value of a B is greater than or equal to 2/3, this means that there are no infinitely many prime numbers. But this is false. Therefore the value of a B converges to a value strictly less than 2/3. This means that the alternating series converges. The inequality would be then true. This inequality is discussed in the following paragraph with the calculation of a B. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 168

169 c Alternating series: estimation of the value of ab The value of a B is equal to with. Nmax is the number of primes which the value is greater than 3. with and with Pj which is a prime factor of the combination This alternating series has a particularity. The value of each element Un of the series increases when Nmax tends to infinity. The value of the element U 1 increases when the number of primes Nmax increases because the number of combination also becomes greater. For Un with n>1 the number of combination is because the prime factor P0=3 is taken into account. Fn is the product of n prime factors. When Nmax increases the product Fn which the value is the highest is this one which corresponds to the multiplication of the n prime factors with the highest values among the Nmax+1 values. Thus for Nmax=25, U 26 is the last element of the series. Its value is the inverse of the product of the first 26 prime numbers. These numbers are in the range [3 ; 103]. The value of this product is about 10 to the power 40 so about. If we want to calculate the value of the alternating series with a great value of Nmax, it is necessary to use a mathematic library for the great numbers and a supercomputer. For Nmax between 1 and 499, the numerical values of a part of the elements Un are given in the table below. The table in appendix 4 and section A contains more elements. In addition, a graphical representation of the evolution of the elements Un depending on the value of Nmax is given in this appendix section B. The algorithm to compute a B and the enumeration of the composite numbers (B) is available in section C and D of the appendix 4. Nmax +U1 -U2 +U3 -U4 +U5 1 0,2 2 0, , ,52E , , ,82E-02 2,60E-03 6,66E , , ,02E-02 7,67E-03 4,74E , , ,97E-02 0, ,18E , , , ,63E-02 3,10E , , , ,85E-02 5,35E , , , ,00E-02 7,71E , , , ,52E-02 1,35E , , , ,59E-02 1,89E , , , , , , , , , , , , We observe that is inferior to whatever the value of n for the first values of Nmax. When the value of n increases, the ratio between and becomes lower by a power of 10. The accuracy of the value a B then depends on the number of elements calculated. For Nmax=16, if a precision ε with 2 decimals (10-2 ) is searched, the calculation of Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 169

170 a B is stopped to U5, i.e. when <. However when the value of Nmax increases, the values of increase up to that the same precision is got for a higher index n of, i.e. with n>5. For a value of Nmax greater than 200, the value of U1 becomes lower than the value of U2. The inequality is then no more valid. The coefficient a B is written: a B =U1-U2+U3-U4+U5-U6+U7-U8+U9-U10... but also a B =U1-(U2-U3)-(U4-U5)-(U6-U7)-(U8-U9)-(U10... In a convergent alternating series, the first element U1 has always the highest value. This first element is called U1Max. When the number of prime numbers Nmax increases the numeric values show that the value of U2 becomes lower than this one of U1 but that the value of U1- (U2-U3) is positive. If we write U1Max= U1-(U2-U3) and the following conditions are true U1Max > U4 > U5>... then the alternating series is written a B = U1Max-(U4-U5)-(U6-U7)- (U8-U9)-(U10... and then a B = (U1Max-U6-U7)-(U8-U9)-(U10... and so on. Thus the first element U1max is modified by an addition with the 2 next elements in order to always have a positive value which is higher than the value of the fourth element in absolute value. The theory is: when the value of Nmax increases, as the first element U1Max plus the 2 next elements is greater than the fourth element and so on, the inequality is true. The table and graphic below show the evolution of the value of a B with a precision of 6 decimals depending on the value of Nmax. The coefficient a B is computed respectively up to the 4th and the 5th elements of Un: U1 to U4 and U1 to U5. The real value of a B is between these 2 last values. Nmax coeff a B (U1-U4) 0,2 0, , , , , , , , , , , coeff a B (U1-U5) 0,2 0, , , , , , , , , , ,5 0,45 a B 0,4 0,35 0,3 coeff a (U1-U4) coeff a (U1-U5) 0,25 0,2 0, Nmax The evolution of the value of coefficient depending on the value of Nmax converges to a value greater than However, the use of a personal computer is insufficient to get the Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 170

171 New theory of odd numbers 8th problem of Hilbert 2013 value to which the coefficient converges when Nmax tends to infinity. It is necessary to use a supercomputer. Another method is therefore needed to get an estimation of the value of this coefficient. We know that the numerical value of (B) evolves according to a second degree polynomial depending on the value of the index of the sequence of points j. We have then calculated the number of composite numbers for sequences of points j up to a value of j equal to The formula which computes the value of (B) determined to the paragraph " Enumeration of the prime factors" (B) = + is not used for big numbers because it requires too many calculating with numbers too big. We then used a primality function to make this calculation. The values which are not multiple of 3 have been then removed to get the value of (B). The result is shown in the graphic below. 1,4E+09 (B)=f(j) 1,2E+09 1E+09 (B) = 1,1575 j 2-483,87 j R² = (B) Poly. ((B)) The value of (B) is depending on the value of j such as: (B) = j j When the value of j tends to infinity, the formula (B) evolves with the monomial of the highest degree of j: The estimated value of a B is then equal to divided by 2 and so a B = Infinity of prime numbers Euclid's theorem says that there are infinitely many primes. We will demonstrate in which way the number of primes Nnp evolves towards infinity. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 171

172 New theory of odd numbers 8th problem of Hilbert 2013 The number of prime numbers is calculated by the formula Nnp = 2 * (A) - (B). The graph below shows the evolution of the formulas 2 * (A), (B) and Nnp when the sequence of points j increases up to j= ,5E+09 2E+09 1,5E+09 2*(A) (B) Nnp Poly. (2*(A)) Poly. ((B)) Poly. (Nnp) 2*(A) = 1,3333 j j + 2,4446 R² = 1 1E (B) = 1,1575 j 2-483,88 j Nnp=2*(A)-(B) = 0,1758 j ,88j R² = R² = 1 0-5E+08 We have demonstrated that the number of pairs of odd numbers (A) which are not multiple of 3, evolves according to a second degree polynomial depending on the value of j. A similar result was also found for the formula (B) representing the evolution of the number of composite numbers which are not multiple of 3. The formula Nnp = 2*(A)-(B) evolves according to a second degree polynomial depending on the value of j if and only if the coefficient of the monomial which has a degree of 2 is greater than zero. When the value of j tends to infinity, we get: Hence The graphic shows that the number of primes Nnp increases according to a second degree polynomial: Nnp = j ,88 j When j tends to infinity, the number of primes evolves according to the following formula: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 172

173 5.9.4 Determination of the formula f(j) for the parameter (C) The number of pairs of composite numbers (C) is the number of values of k in the form 3m+2 whose at least one prime factor is present in the factorization of the associated odd number Ni. In addition, the value of k in the form 3m + 1 must also have at least one prime factor present in the factorization of the associated odd number Ni. The values of k which are multiple of 3 are therefore not taken into account. There is then at least one pair of prime factors that is written (3m+1, 3m+2). When several pairs of prime factors exist for a same value of k and k+1 such as (3m+1, 3m+2) then only one pair of prime factor is counted Enumeration of the pairs of prime factors We will determine the formulas to count the number of pairs of prime factors. These formulas are obtained by solving linear Diophantine equations. The values of k which are multiple of 5 are defined by the formula k= 5*m+1 because the half period DPej of the sequence of points j=1 is equal to 5 and the first point is obtained when m=0 that is k=1. As demonstrated in the paragraph " Enumeration of the prime factors", the value of k of the first point of a sequence of points j corresponds to the sequence number of occurrence of odd numbers n that is equal to the value of j. A pair of prime numbers (P1, P2) is composed of 2 prime factors P1 and P2. The values of k which are multiple of these factors give rise to pairs of prime factors. Each prime factor is an element of the factorization of the associated odd number Ni. So a pair of prime numbers gives rise to infinity of pairs of prime factors. With the pair of prime numbers (5, 7), the values of k that are multiple of these factors are respectively: { To get a pair of numbers with the values of k in the form 3*m+1 and 3*m+2, an offset of 1 must be introduced. This offset can be applied either to the first prime factor or the second one. There are then 2 pairs of prime numbers distinct (5,7) and (7,5). The first prime factor of the pair is always taken into account in the form 3*m+1 and the second one in the form 3*m+2. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 173

174 Pair of prime numbers New theory of odd numbers 8th problem of Hilbert 2013 Pair of prime factors (7,5) => k=105*m+101 Pair of prime factors (7,5) = (3m+1,3m+2) 3m+1 => k=105*m+100 3m+2 => k=105*m+101 Pair of prime factors (5,7) => k=105*m+107 If we take into account the pair of primes (5,7), the values of k searched must meet the following equations: { We get the equation: 5*x-7*y=0 This is a linear Diophantine equation. The solutions of the equation are x=7*m and y=5*m. Hence: k = 35*m+1 and k+1= 35*m+2 It is also needed to exclude the values of k that are multiple of 3 like k=5*3= 15. There must have an offset of 1 between the values of k searched for the factor 5 and the values of k that are multiple of 3. These values of k are obtained by the equation k=3*y+1. A new linear Diophantine equation must be solved : { Hence 35*x-3*y=0 The solutions of the equation are: x=3*m and y=35*m. The values of k that are multiple of 5 and not multiple of 3 in pairs with the values of k that are multiple of 7 are connected by the following formulas: for 5, k=105*m+1 for 7, k+1=105*m+2 The pairs of prime factors obtained from the pair of primes (5,7) are written (k, k+1) and also (105*m+1, 105*m+2). We will define that a pair of factors is represented by the second factor. In the present example, it is the values of k which are multiple of 7 and not multiple of 3 and therefore the pair (k-1,k). The values of k that identify the pair of factors are: k=105*m+2. The first pair of factors is obtained for m=0 which corresponds to the pair of values (k=1, k=2) that is the pair of odd numbers (Ni=5, Ni=7). These points are excluded from the enumeration. As explained in the preceding paragraphs, these values are the prime numbers themselves and not multiples of prime numbers. The points obtained for the sequences j with n = 0, are excluded from the enumeration (See paragraph "2- Building of Space W"). The first pair of values of k is obtained for m=1 that is the pair (106,107). In order to obtain, for m=0, the first pair of prime factors, m is replaced by m+1 and the values of k searched are: k=105*m+107 avec. In the general case, when at least one of the points of the pair Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 174

175 (Ni=2*(k-1)+3,Ni=2*k+3) corresponds to the value of one of the two prime factors, the values of k are obtained by replacing m with m+1 in formula. We get a formula in the following form: k= a*m+b - with a the product of the prime factors: a=3*5*7. For recap, the prime factor 3 allows excluding the values of k that is multiple of 3. Thus a=3*p1*p2 with P1 and P2 that are the prime factors. - and b the value of k where the first pair of prime factors is located. The values of k for the pair of primes (7,5) are got with the previous method: k=105*m+101 Thus the pairs (5,7) and (7,5) are represented respectively by the 2 following formulas: Some examples of pairs of prime factors created by the pairs of primes (5,7) and (7,5) are given in the table below. Pair of primes Pairs of prime factors for (5,7): (k=105*m+106, k=105*m+107) Pairs of prime factors for (7,5): (k=105*m+100, k=105*m+101) m =0 m =1 m =2 m =3 m =4 (5,7) (106,107) (211,212) (316,317) (421,422) (526,527) (7,5) (100,101) (205,206) (310,311) (415,416) (520,521) The prime numbers 5 and 7 are twin primes. The coefficients b of the previous formulas that are respectively 107 and 101 can be found using a system of two equations with two unknown. The sum of the coefficients b1 and b2 plus 2 is equal to twice the value of the coefficient a. The difference of the coefficients is equal to the sum of prime factors divided by 2. Hence { {. and { The formulas of k=f(m) are presented for several pairs of primes in the table below: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 175

176 Pair [5,7] [5,11] [7,11] [5,13] [7,13] [11,13] [5,23] k *m *m *m *m *m *m *m Inverse pair [7,5] [11,5] [11,7] [13,5] [13,7] [13,11] [23,5] k *m *m *m *m *m *m *m When the prime numbers are not twin primes, the coefficients b1 and b2 of the formulas of k for the pairs (P1, P2) and (P2, P1) cannot be computed with the formulas (α). The formulas (α) are written according to the prime factors P1 and P2. For the sum of the coefficients, two formulas were found. When for the first pair of factors, at least one of the points of the pair (Ni=2*(k-1)+3,Ni=2*k+3) corresponds to one of two factors, the sum of the coefficients is written either: or However, the formulas relating to the difference of the coefficients are very different. Many instances exist. The values of a and b of the formula of k are then obtained by using the extended Euclidean algorithm. The pairs of factors for the pair of prime numbers (5,7) are determined by the k= *m with. The number of pairs of factors is then equal to. For k=107, there is one pair of factors. It must be added the value 1 to the formula to take into account the offset related to the value of k of the first pair. The number of pairs of factors is then given by the following formula * +. For a pair of primes (P1,P2), the values of k of the pairs of prime factors are got by the formula k=a*m+b with a=p0*p1*p2. The number of pairs of factors for the pair of primes (P1,P2) is. The value of b is a constant related to the prime factors P1 and P2: b(p1,p2). The number of primes is Nmax. The number of pairs of primes is then equal to with Nmax which tends to infinity. For 2 primes, 2 distinct pairs of primes exist: (P1, P2) and (P2, P1). The order of prime numbers must be taken into account. The number of distinct pairs composed of 2 primes among the Nmax primes corresponds to an arrangement. This arrangement is equal to. The sum (C1) of the pairs of prime factors of each pair of primes is equal to: with Pj which corresponds to the primes of the arrangement A( the first pair of prime factors. ) and b(pj) that is a constant related to This sum (C1) also includes the common pairs of prime factors to two or several pairs of primes. For instance, for the 2 pairs of the primes (5,7) and (5,11), the common pairs of primes are related to the common values of k obtained from the 2 following formulas of the pairs (5,7) and (5,11):, Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 176

177 We must solve a linear Diophantine equation: The solution of this equation is : X=3+11*m and Y=2+7*m The common values searched are:. This values corresponds to the combination of the 2 following pairs of primes (5,7) and (5,11). For instance, for m=0, we get the same value of k with X=3 and Y=2 that is k=422= *3=92+165*2 The standard formula is in the form: k= a*m+b with the coefficient a equal to 3 times the product of all unique prime factors which compose the 2 pairs of primes. For the pairs (5,7) and (5,11), the value of a is: a=3*5*7*11=1155. This value is the least common multiple (LCM) of the values 105 and 165. These 2 last values are respectively the coefficients a of the formulas got for the pairs of primes (5,7) and (5,11). The value b=422 is a constant that is the position of the first common pair of prime factors with the factor 5 in the form 3m+1 and the factors 7 and 11 in the form 3m+2. The number of common pairs of prime factors that are (5,7) and (5,11) is found by this formula * +. The standard formula of the number of common pairs of prime factors for a number of combinations of pairs of prime numbers greater than or equal to 2 is: [ ] with a and b which are 2 constants that depend on the prime numbers which compose the pairs of primes:. These common values have already been counted twice: - once as the pair of prime factors of the pair of primes (5,7) - once as the pair of prime factors of the pair of primes (5,11) We must therefore remove the number of common pairs of prime factors for a number of pairs equal to 2. However, there are also the common pairs for a number of pairs equal to 3. They are counted 3 times, i.e. once per pair, and removed 3 times, i.e. once per common pair for a number of pairs equal to 2 because there are 3 ways to take two pairs of prime numbers among three pairs. So the number of common pairs for a number of pairs equal to 3 must be added to count in a unique way the number of pairs of prime factors that are not a multiple of 3. The number of common pairs for a number of pairs equal to n is added or removed depending on the value of n. When n is even the number of common pairs is removed. This operation is repeated until the maximum number of pair of primes that is NCmax is reached. The value NCmax is lower than the arrangement A. All combinations of pairs of prime numbers are not possible. The value of a factor cannot be in both forms of a pair (3m+1, 3m+2). Thus, the pairs of factors (5,7) and (7,11) cannot exist because the factor 7 cannot be in both forms 3m+2 and 3m+1. These combinations are not valid. Note that the order of pairs does not matter: (5,7) (5,11) is identical to (5,11) (5,7). The common pairs must also be counted once. The table below gives an example for the combinations of 2 and 3 pairs of primes. Number of pairs of primes NC Add/Remove the values Pairs of primes 1 + (5,7) (5,11) (7,11) (5,13) 2 - (5,7) (5,11) (5,7) (11,7) (5,7) (5,13) Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 177

178 (5,11) (7,11) (5,11) (5,13) (7,11) (5,13) (7,11) (13,5) 3 + (5,7) (5,11) (5,13) (5,7) (5,11) (5,13) (5,11) (7,11) (5,13) The addition or the deletion of the number of combinations of the pairs of factors is related to the number of pairs into the combinations, i.e. the value of NC which is between 2 and NCmax. The addition (+) or the deletion (-) is defined by using the value (-1) to the power (NC+1). The sign + or is got by this formula SignNC = (-1) (NC+1). We denote (C2) the number of pairs of prime factors which are commons to all existing combinations between 2 pairs of primes up to NCmax pairs of primes. This number (C2) must be added to the value (C1) to get the number of unique pairs of prime factors that are not multiple of 3. Hence (C)=(C1)+(C2). (C2)= ( ) with that is the set of valid combinations of NC pairs of primes among pairs of primes. A combination with NC pairs of prime factors is composed of n prime numbers with at most n=2*nc primes. (C2)= ( ) The formula (C1) is similar to the formula (C2) with: - NC=1 and so =(-1) 2 = 1 - n is equal to 2 - ( ) Thus (C)=(C1)+(C2) is written: (C) = ( ) The formula U NC is the sum of integer part of a linear function related to k:. The linear function is denoted as the term of the element U NC. The fractional part of the result of the division by a is in the interval [0; 1 [. When k tends to infinity, this fraction is negligible compared to the integer part of the result. Moreover when the value of k increases, the Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 178

179 evolution of the value of the integer part of the function [ ] is the same as this function when we only take into account the values of k for which the value of (k-b+a) is multiple of the divisor a. The study focuses on the evolution of the number of pairs of composite numbers when k tends to infinity. The integer part is then removed from the formula U NC : ( ) It is known that the evolution towards infinity of a polynomial of degree n is the same as the evolution of monomial of the highest degree. For (C), when k tends to infinity, we only retain the monomial of highest degree. The formula U NC is then: ( ) ( ) with ( ) and (C) = and so (C) with ( ) k is related to the value of j by the following formula k= The evolution of the value (C) depending on the parameter j is similar to that of the value (B) shown in appendix 9. When j tends to infinity, the formula (C) keeps only the monomial of highest degree: (C) Determination of the coefficient ac. We will demonstrate that the coefficient a C is written as an alternating series. The convergence of this series is then demonstrated and its value is estimated a Alternating series: ac. A alternating series can be represented by the following formulas: The coefficient a C can be written as an alternating series when the value of NCmax tends to infinity and that n is replaced by NC. To simplify the formula, U NCa is renamed U NC. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 179

180 with ( ) ( ) When the series respects the two rules of Leibnitz, the series converges. The 2 rules are: b Convergence of the alternating series ac. For a number of primes equal to Nmax, there is a number of pairs of primes equal to the arrangement. We will study the evolution of the number of combinations of pairs of prime factors depending on the number of prime numbers Nmax. The particularity of this enumeration is related to the position of each factor inside the pair of prime factors (P1,P2). The first factor P1 is in the form 3m+1 whereas the second one is in the form 3m+2. A same factor cannot be in both positions that are 3m+1 and 3m+2 with a same value of m. As shown previously, the values which are multiple of a prime number P are written: k=p*m+(p-3)/2. The gap between two values of k obtained for m and m + 1 is equal to P that is greater than 1. This implies that the combinations of pairs must be created with prime factors whose each of these factors uses only one of these forms: 3m+1 or 3m+2. If a factor is present in many pairs, this prime factor is either at the first position or the second position into the pairs of the combination. The combination is said valid. When Nmax=2, there are only two pairs of factors: (5,7) and (7,5). The pair (7,5) is inverse of the pair (5,7) because the position of the factors is inversed. The pair (7,5) is the inverse pair of the pair (5,7) that is named main pair. The number of pain pair is calculated in the following way: = 1. A prime number among =2 primes is taken to get the first prime factor in the form 3m+1. The second factor is selected among the remaining primes that is in this example the second prime number because it stays only 1= prime number. A division by 2 is done to take into account only the main pairs. This result corresponds to take 2 primes among 2 prime numbers. There is then one combination:. The number of pairs of main factors is equal to the number of pairs of inverse factors. The enumeration of the main pairs of factors is performed to get a main enumeration. The inverse enumeration is obtained by inverting the position of the prime factors in the pairs. The complete enumeration is the sum of the main and inverse enumerations. When Nmax=3, there are =3 main pairs. This corresponds to take 2 primes among 3 primes. The number of combinations is =3. From the value Nmax=3, the combinations with 2 pairs of factors appear. We denote NC as being the number of pairs which compose a combination. In the present example, the value of NC is equal to 2. The number of combinations composed of 2 pairs of factors is calculated in the following way: ( ) ( ). This corresponds to take 1 primes among 3 primes, that is, to get the first factor in the form 3m+1. The 2 pairs of factors are built in taking the 2 remaining primes as factors in the form 3m+2:. There are then 3 distinct Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 180

181 combinations: (5,7)(5,11), (7,5)(7,11) and (11,5)(11,7). They are the main combinations. There are also 3 inverse combinations : (7,5)(11,5), (5,7)(11,7) et (5,11)(7,11). For the factors in the form 3m+1, the number of primes used for building the combinations must be inferior to [Nmax/2]. In the present example, [Nmax/2]=1. The inverse combinations are built by taking 2 primes among 3 primes to get 2 factors in the form 3m+1: (5,11)(7,11), (5,7)(11,7) and (7,5)(11,5). The calculation of the number of combinations is stopped when the number of primes used to build the factors in the form 3m+1 is equal to [Nmax/2]. The first table below shows the 3 cases to take 1 prime number among 3 primes for the form 3m+1 and the pairs of factors and the combinations with 2 pairs of factors. The second table shows an example for k= *m with m=0 that is k=422. This example corresponds to the first case that is the first cell. Prime factors in the form Cell Cell Cell pairs (5,7) (5,11) (7,5) (7,11) (11,5) (11,7) 3m+1 3m+2 Cell 1 (5,7) and (5,11) k=3m+1 k=3m+2 When Nmax=4, there are =6 main pairs. This corresponds to take 2 primes among 4 primes. The number of combinations is =6. From the value Nmax=3, the combinations with NC=3 and NC=4 pairs of factors appear. The maximum number of primes used as prime factors in the form 3m+1 to build the combinations is i max = [Nmax/2]=2. The combinations are determined according to the number i of prime factors in the form 3m+1 and the number (Nmax-i) of prime factors in the form 3m+2. The calculation of the number of combinations composed of NC pairs of factors is realized by doing an addition of the number of combinations for each value of i. The table below gives the number of combinations for Nmax in the range [2, 5]. When the number of primes is even and i=imax, there are as many prime factors in the form 3m+1 as prime factors in the form 3m+2. The number of combinations is then equal to : - 2 primes among 4 primes are selected to form the prime factors in the form 3m+1: - the prime factors in the form 3m+2 are selected in the following way: One prime number among the 2 primes remaining is selected to get the first pair:. To get the second pair, the last prime is selected:. The set of these combinations also contains the inverse combinations. This number of combinations must be divided by two to get only the main combinations. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 181

182 Table: number of main combinations of pairs of prime factors according to the number of primes Nmax in the range [2, 5] and the number of pairs of prime factors NC. NC => i Nmax 2 1 ( * ( * ( * ( * ( * ( * ( * ( * ( * ( * ( * ( * ( ( * ( *) ( ( * ( * * ( ( * ( *) 5 1 ( * ( * ( * ( * ( * ( * ( * ** 2 ( * ( * ( * ( * ( * ( * ( * ( * ( ( * ( * * ( * ( * ( * ( ( * ( * * ( * ( ( * ( *) ( * ( * ( * ( * ( * ** The formulas are explained in appendix 5 The maximum number of combinations obtained for a number of primes equal to Nmax is NCmax=(Nmax- i max ) * i max. The value of i max is the maximum number of factors in the form 3m+1 multiplied by the maximum number of factors in the form 3m+2 which is complementary of Nmax and so equal to (Nmax- i max ). When the value of Nmax is even, i max =Nmax/2 and NCmax is equal to. One value of NCmax exists for each value of i: NCmax=(Nmax - i)*i. When the number of primes increases the enumeration of the combinations is related to the sums of products of combinations. The number of sums and products increases. Some of formulas of enumeration can be written with a generic form depending on the number of primes. Here are some examples: - for i=1 and NC=1, the number of pairs of prime factors is - for i=1 and NC>1, the number of combinations is - for i>1 and NC=NCmax, the number of combinations is ( * ( * - for i>1 and NC=NCmax-1, the number of combinations is Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 182

183 ( * ( * ( * ( * - for i>1 and NC=NCmax-2, the number of combinations is (( ) ( ) ) - for i>1 and NC=NCmax-x, the previous formula evolves in a similar way when x increases and as the coefficient is present. That means that all factors in the form 3m+2 are present in each combination. The number of factors in the form 3m+2 is equal to Nmax-i. The previous formula is true as x is inferior to i. The criteria x < i is obtained as follows: - a combination is composed of a number of pairs of prime factors equal to NC - the number of pairs of prime factors in the form 3m+1 is equal to i. To each prime factor in the form 3m+1 is associated a number of prime factors in the form 3m+2 to create some pairs of prime factors. This number of pairs of prime factors is called Z. The value of Z corresponds to the number of prime factors selected in the form 3m+2. - For each prime factor in the form 3m+1, a value Z greater than or equal to 1 exists. The sum of all values of Z is equal to NC. Hence: NC=. The number of combinations composed to NC pairs of prime factors is related to: - the number of prime factors in the form 3m+1 equal to i - the number of prime factors in the form 3m+2 equal to Nmax-i - for each value of i, a combination of NC pairs is built by selecting for each prime factor in the form 3m+1 a number Z of prime factor in the form 3m+2 such as NC=. - the number of combinations is related to the number of prime factors in the form 3m+2 selected for each prime factor in the form 3m+1. This number is then related to i times the value of with Z greater than or equal to 1 for each prime factor in the form 3m+1. The number of combinations is related to the value of i which is in the interval [1 ; i max ]. The factor corresponds to that a prime factor in the form 3m+1 creates a number of pairs of prime factors equal to the number of prime factors in the form 3m+2 that is equal to Nmax-i. The factor is no more common to all combinations when the value Z is equal to Nmax-i-1 for each factor. The value of NC is then equal to NC=i*(Nmax-i-1). The value of x is equal to x=ncmax-nc=(nmax-i)*i - i*(nmax-i-1) = i. Thus the above formulas are respected as the value of x is less than the value of i. The number of combinations composed of NC pairs of prime factors is computed as the sum of combinations obtained for each value of i. The coefficient is common to all combinations when the value of NC is in the interval [NCmax - i max ; NC=NCmax]. - for the others enumerations of combinations when x is greater than i the formulas are complex. This result is related to the existence of inverse combinations which must be removed of the main enumeration. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 183

184 The value of the last element of the series a C is calculated with a number of pairs of prime factors equal to NC=NCmax and i=i max. For each combination of pairs, the number of prime factors in the form 3m+1 is equal to i max whereas the number of prime factors in the form 3m+2 is equal to Nmax- i max and then NCmax=(Nmax- i max )* i max. The formula which determines the number of combinations of NC pairs of prime factor is written k=a*m+b. The multiplication of each prime number belonging to the interval [P0=3 ; P Nmax ] allows to get the coefficient a of the formula. The value of a is equal to: a=. The number of main combinations of pairs of prime factors is equal to with an even value of Nmax, i.e. with i max =Nmax/2. The value of the last element of the series a C is: Each factor of the factorial Nmax! is respectively less than each factor of the product of the first Nmax primes. Thus When Nmax tends to infinity, the value of NCmax tends to infinity and tends to 0. The first rule of Leibniz is respected: We will study the second Leibniz rule that is if whatever the value of NC. The enumeration of the main combinations of NC pairs of prime factors is done from an algorithm which builds all the combinations and removes the inverse combinations. The inverse combinations are found from the main combinations by reversing the factors in the pairs. The number of main combinations is equal to this one of the inverse combinations. The main enumeration is done in two parts. The first part calculates the number of combinations composed of NC=1 pair of prime factors. This calculation uses the formula with Nmax equal to the number of primes. The second part calculates the number of combinations of NC pairs of prime factors with. The table below shows the parameters used to do the calculation of the second part of the enumeration. The main parameter is the number of primes Nmax. It allows to compute the maximum number of prime factors in the form 3m+1: i max =[Nmax/2]. The number of prime factors in the form 3m+1 is in the interval [1 ; i max ]. The number of prime factors in the form 3m+2 is then found through the formula t=nmax - i. The number of pairs of prime factors NC which forms a combination has a value between NCmin and NCmax = i*t. The value of NCmin is equal to i except for i=1 it is equal to NCmin=2. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 184

185 Number of primes Nmax Number of prime factors in the form 3m+1 = i i max =[Nmax/2] Number of prime factors in the form 3m+2 = t =Nmax - i Number of pairs of prime factors NC NCmax=i*t Number of combinations composed of NC pairs of prime factors NC= NC is in the interval [NCmin ; NCmax] with NCmin=i except for i=1 it is NCmin= = i max 2 2 = NCmax = = 4 3 = NCmax = 4 2 = i max 2 2 = with z1=z2=1 and NC = z1+z2 = = = NC = z1+z2 = z3+z4 = = 3 Each value Z is greater than or equal to 1. z1 =1 and so z2=nc-z1=2 hence z1 =2 and so z2=nc-z1=1 hence 4 4 = NCmax = NC = z1+z2 = = 4 z1 =2 and so z2=nc-z1=2 hence The parameter Z is the number of prime factors in the form 3m+2 associated to one prime factor in the form 3m+1 to form Z pairs of prime factors. The number NC of pairs of prime factors existing inside a combination is equal to the sum of the values of Z which depends on the number of prime factors in the form 3m+1 which is equal to i. The prime factors in the form 3m+2 are spread over the prime factors in 3m+1 to form the pairs of prime factors. There must be at least one prime factor in the form 3m+2 for each prime factor in the form 3m+1. For instance, when Nmax=5 if the number of pairs is NC=3 and i=2, the value of z1 can take the value 1 and 2. But z1 cannot take the value 3 because z2 must be greater than or equal to 1. There is a minimum value and a maximum value of Z denoted respectively Zmin and Zmax. These values depend on the 3 following parameters: i, t and NC. An algorithm is performed using this table to get the main enumeration. The combinations of pairs of prime factors are counted in a unique way. The combinations for which the inverse Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 185

186 combination is a main combination already included are removed from the main enumeration. The algorithm is given in appendix 6. The table below gives the number of combinations composed of NC pairs of prime factors for the first seven primes greater than the prime number 3. The primes are in the interval [5 ; 23]. This number of combinations is given depending on the value of i that is the number of prime factors in the form 3m+1. The value of i is between 1 and 3 for Nmax=7. These data are also represented graphically. NC i=1 i=2 i=3 Nmax=7 Sum of i i=1 i=2 i= Nmax= From Nmax=2, a new series of combinations of pairs of prime factors appears each 2 primes added to Nmax. The value of i max increases then of 1. The first combinations of this series exist for a number of pairs of prime factors equal to NCmin=. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 186

187 The distribution of combinations of NC pairs of prime factors is represented by a bell curve close to a normal distribution. This bell curve has a increasing phase and then a decreasing phase. This distribution is observed both for each value of i as well as the overall curve that sums the number of combinations of NC pairs for each value of i. An axis of symmetry is present at NCmax/2. This symmetry is not perfect because the values of both side of the axis are not exactly the same. We have previously shown that when the value of NC is in the range [NCmax-i max + 1 ; NCmax] each combination of NC pairs of prime factors is composed of all prime factors. The number of prime factors is Nmax. The elements U NC of the series a C can be written using the following formula: The value of U nc+1 is then less than the value of U nc if and only if the number of combinations is lower. - for i>1 and NC=NCmax, the number of combinations is - for i>1 and NC=NCmax-1, the number of combinations is ( ) - for i>1 and NC=NCmax-2, the number of combinations is (( ) ( ) ) ( ) For the two last elements of the series, we get: U NCmax U NCmax-1 because. For NCmax-2 and NCmax-1, we get the inequality: U NCmax-1 U NCmax-2 because with Nmax > 5 and i > 1. For the last values of U NC of the series a C, the inequality U NC+1 U NC is true. This inequality is true for all elements U NC with the value of NC in the interval [NCmax-imax+1 ; NCmax]. In this interval, when the value of NC increases the number of combinations decreases. The inequality is then true. For i=1, the evolution of the number of main combinations of NC pairs of prime factors is similar to this one of the number of combinations of n prime factors which allows to calculate the series a B and the number of composite numbers (B). The values are respectively: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 187

188 and. For Nmax+2 and, the evolution of the number of combinations of NC pairs of prime factors is similar to the evolution of the number of combinations of n prime factors. The factor is present in only one of the products. For the values are respectively: and. This similarity is found in the next paragraph in the estimation of the value of a C. The convergence of the series a C is mathematically demonstrated by the method of the absurd. We have demonstrated that the number of pairs of prime numbers is written Ncnpj = (A) - (B) + (C). The number of pairs of composite numbers (C) cannot exceed the number of pairs of odd numbers (A). If (C) = (A) then there is no more primes. But Euclid's theorem says that there are infinitely many primes. This means that the value of (C) is always less than the value of (A): (C) < (A). We can write: So Therefore the value of a C converges to a value strictly less than 1/3. This means that the alternating series converges. The inequality would be then true whatever the value of NC. This inequality is discussed in the following paragraph with the calculation of a C c Alternating series: estimation of the value of ac This alternating series has a particularity. The value of each element U NC of the series increases when Nmax tends to infinity. This behavior is identical to this one of the alternating series a B. The value of the element U 1 increases when the number of primes Nmax increases because the number of combinations of pairs of prime factors also becomes greater. This behavior is similar for U NC with element of the series a C is:. The value of the last The product of the prime factors Fn is composed of Nmax prime factors. Thus for Nmax=25, U 156 is the last element of the series: These numbers are in the range [3 ; 103]. The value of this product is about. If we want to calculate the value of the alternating series with a great value of Nmax, it is necessary to use a mathematic library for the great numbers and a supercomputer. The value of the series a C is calculated as follows: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 188

189 ( ) There are as many main combinations as inverse combinations. The product is the same for a main combination and an inverse combination because it contains the same prime factors. The calculation can then be written: ( ) For Nmax between 1 and 40, the numerical values of a part of the elements Un are given in the table below. The table in appendix 7 and section A contains more elements. In addition, a graphical representation of the evolution of the elements U NC depending on the value of Nmax is given in this appendix section B. The algorithm to compute a C and the enumeration of the pairs of composite numbers (C) is available in section C and D of the appendix 7. Nmax U1 U2 U3 U4 U5 U6 U7 2 0, ,98E-02 5,19E ,21E-02 1,52E-02 2,13E-03 4,00E ,21E-02 2,78E-02 7,11E-03 2,23E-03 4,70E-04 7,84E , ,35E-02 1,56E-02 6,56E-03 2,35E-03 7,67E-04 1,98E , ,01E-02 2,68E-02 1,38E-02 6,54E-03 2,97E-03 1,24E , , ,93E-02 2,33E-02 1,32E-02 7,35E-03 3,95E , , ,02E-02 5,07E , , , , , , , , , , , , , , , , , We observe that is inferior to whatever the value of NC for the first values of Nmax. When the value of NC increases, the ratio between and becomes lower. The accuracy of the value a C then depends on the number of elements calculated. For Nmax=7, if a precision ε with 2 decimals (10-2 ) is searched, the calculation of a C is stopped to U5, i.e. when <. However when the value of Nmax increases, the values of increase up to that the same precision is got for a higher index NC of, i.e. with NC>5. For a value of Nmax near 20, the value of U1 becomes lower than the value of U2. The inequality is then no more valid. The coefficient a C is written:: a C =U1-U2+U3-U4+U5-U6+U7-U8+U9-U10... Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 189

190 but also a C =U1-(U2-U3)-(U4-U5)-(U6-U7)-(U8-U9)-(U10... In a convergent alternating series, the first element U1 has always the highest value. This first element is called U1Max. When the number of prime numbers Nmax increases the numeric values of U NC show that the value of U2 becomes lower than this one of U1 but that the value of U1-(U2-U3) is positive. If we write U1Max= U1-(U2-U3) and the following conditions are true U1Max > U4 > U5>... then the alternating series is written a C = U1Max-(U4-U5)-(U6- U7)-(U8-U9)-(U10... and then a C = (U1Max-U6-U7)-(U8-U9)-(U10... and so on. Thus the first element U1max is modified by an addition with the 2 next elements in order to always have a positive value which is higher than the value of the fourth element in absolute value. The theory is: when the value of Nmax increases, as the first element U1Max plus the 2 next elements is greater than the fourth element and so on, the inequality is true. This theory is identical to that found in the calculation of the series a B. The table and graphic below show the evolution of the value of a C depending on the value of Nmax. The coefficient a C is computed with all elements (U1 to U16) and also respectively up to the 6th and the 7th elements of U NC : U1 to U6 and U1 to U7. The real value of a C is between these 2 last values. Nmax a C (U1-U16) 0, , , , , , , U1-U6 0, , , , , , , U1-U7 0, , , , , , , ,09 0,08 0,07 0,06 0,05 0,04 0,03 0,02 ac (U1-U16) U1-U6 U1-U7 0, Nmax The evolution of the value of coefficient depending on the value of Nmax converges to a value greater than However, the use of a personal computer is insufficient to get the value to which the coefficient converges when Nmax tends to infinity. It is necessary to use a supercomputer. Another method is therefore needed to get an estimation of the value of this coefficient. We know that the numerical value of (C) evolves according to a second degree polynomial depending on the value of the index of the sequence of points j. We have then calculated the number of composite numbers for sequences of points j up to a value of j equal to The Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 190

191 New theory of odd numbers 8th problem of Hilbert 2013 formula which computes the value of (C) determined to the paragraph " Enumeration of the pairs of prime factors" is: (C) = ( ) This formula is not used for big numbers because it requires too many calculating with numbers too big. We then used a primality function to make this calculation. The result is shown in the graphic below C=f(j) C= 0,5009 j 2-423j R² = C Poly. (C) The value of (C) evolves depending of the value of j according to the formula: (C) = = 0,5009 j 2-423j When the value of j tends to infinity, the formula (C) evolves with the monomial of the highest degree of j: The estimated value of a C is then equal to 0,5009 divided by 2 and so a C = Infinity of the pairs of twin primes We will demonstrate in which way the number of primes Ncnpj evolves towards infinity. The number of pairs of twin primes is calculated by the formula Ncnpj = (A) - (B) + (C) that is also written Ncnpj=(A) - ( (B) - (C) ). (A) is the number of pairs of odd numbers (NCI) Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 191

192 New theory of odd numbers 8th problem of Hilbert 2013 which are not multiple of the value 3. The graphs below show the evolution of values of (A), (B) - (C) and Ncnpj when the index of sequence of points j increases up to j= The two first graphs represent the values of (A) and (B)-(C) for the values of j respectively in the range [ ] and [ ]. We can observe a divergence between the 2 curves representing (B)-(C) and (A). The number of pairs of composite numbers and mixed numbers (1 prime number and 1 composite number) that is equal to "(B)-(C)" increases more slowly than the number of pairs of odd numbers "(A)". The number of pairs of prime numbers increases when j increases. The last graph shows the evolution of the number of pairs of twin primes Ncnpj for the values of j between 0 and (A) = NCI (B)-( C ) Poly. ((A) = NCI) (A) = 0,6667 j2 + 2 j + 1,2223 R² = Poly. ((B)-( C )) (B)-(C) = 0,6566 j2-60,867 j R² = Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 192

193 New theory of odd numbers 8th problem of Hilbert (A) = NCI (B)-( C ) Poly. ((A) = NCI) Poly. ((B)-( C )) (A) = 0,6667 j2 + 2 j + 1,2223 R² = 1 (B)-(C) = 0,6566 j2-60,867 j R² = Ncnpj = (A)-(B-C) Nncpj = (A) - (B-C) = 0,0101x ,867x R² = 0, We have demonstrated that the number of pairs of odd numbers (A) which are not multiple of 3, evolves according to a second degree polynomial depending on the value of j. Similar results were also found for the formula (C) representing the evolution of the number of pairs of composite numbers and for the formula (B) representing the evolution of the number of composite numbers which are not multiple of 3. The formula Ncnpj = (A) - (B) + (C) evolves Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 193

194 according to a second degree polynomial depending on the value of j if and only if the coefficient of the monomial which has a degree of 2 is greater than zero. When the value of j tends to infinity, we get: Hence The graphic shows that the number of pairs of prime numbers Ncnpj increases according to a second degree polynomial: Ncnpj = 0,0101 j ,867 j When j tends to infinity, the number of pairs of prime numbers evolves according to the following formula: There is then an infinity many twin primes Conclusion It has been shown that the number of pairs of twin primes Ncnpj is written depending on 3 parameters (A), (B) and (C) according to the formula: Ncnpj =(A)-(B)+(C) or Ncnpj=(A) - ((B)-(C)). It has also been shown that the number of primes Nnp is written: Nnp=2*(A)-(B). - the parameter (A) is the number of pairs of odd numbers which are not multiple of 3. - the parameter (B) is the number of composite numbers which are not multiple of 3 - the parameter (C) is the number of pairs of composite numbers which are not multiple of 3. It has been shown that these 3 parameters are written depending on the value of j according to a second degree polynomial. In the space W, a value of j is the index of a sequence of points which are multiple of an odd number. When the value of j increases, the value of (A) evolves as the integer part of a second degree polynomial. Lorsque j croit, la valeur de (A) évolue selon la partie entière d'un polynôme du second degré. The values of (B) and (C) evolve according to infinity of addition and subtraction of integer part of second degree polynomials. A polynomial evolves towards infinity as its monomial of highest degree. The evolution of the integer part of a polynomial is as the evolution of the integer part of the polynomial. The values of the parameters (A), (B) and (C) evolves towards infinity according to a monomial of second degree with respectively a coefficient 2*a, 2*a B and 2*a C. The coefficients a B and a C are dependent on the infinity of primes. They are written as an alternating series. It has been shown that these alternating series are convergent. The calculation of the value of the coefficients a B and a C requires a large number of operations with a large number of prime numbers. Using a supercomputer is needed. A primality function was used to calculate the number of composite numbers (B) and the number of pairs of composite numbers (C) according to the parameter j. The values of coefficients a B and a C were estimated from the plot Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 194

195 of these values according to j up to a value equal to j= A second degree polynomial function corresponds to each curve with a regression coefficient of 1. The values of coefficients are: a, a B et a C. The number of primes tends to infinity according to the following formula: The number of pairs of twin primes tends to infinity according to the following formula: The number of pairs of twin primes is then infinite. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 195

196 6- Method for determining the couples of prime numbers that compose an even number (Goldbach's Conjecture). 6.1 Determination of Goldbach's equation Definition and context Goldbach's conjecture is a mathematical unproven assertion that states as follows: «Any even integer greater than 3 can be written as the sum of two primes». Therefore an even number «N» is equal to the sum of two primes. This formulation takes the following mathematical form : With N : Even number represented by the following formula:, with. and are prime numbers. The mathematical tool defined in paragraph 5 (Theorem 5.1) uses the base units. The value of the index of the first point of the first base unit is equal to "3" because. This value corresponds to the odd number "9". The use of the mathematical tool that characterizes an odd prime is only possible with numbers greater than "3". So the study of the Goldbach's conjecture is only possible with even numbers greater than "9". The decomposition of an even number greater than "10" or equal to "10" in two primes implies that these two prime numbers are odd because the difference between the two even numbers provides another even number greater or equal to the first even number. Only the even number "4" can be decomposed into two first even numbers with the even prime number "2". All even numbers less than the value "10" and above "3" check the Goldbach's formula. The first odd numbers constitute the set of prime numbers except the even prime number "2". The first odd prime number has the value "3". This study takes into account only the odd primes. The following formulas are used to describe an odd number : The parameters and represent the indices of odd numbers. The use of the characterization of an odd prime number implies that the first odd prime number "3" is not taken into account to solve the equation. Indeed, the decomposition of an even number with the first odd prime number "3" can not be resolved using this characterization Definition of the equations The Goldbach's equation to solve is as follows: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 196

197 Équation 8 : Goldbach's equation with odd primes (4=2+2 is not taken into account) In the context of the study of odd primes, the solution is to demonstrate that there is for all values of the parameter greater than or equal to "5", two indexes corresponding to odd primes which respect the Goldbach's equation. The use of the characterization of an odd prime number allows to jointly determine the primality of two odd numbers. The inequality to solve is the following : ( ( ** ( ) With With.. The index «k1» corresponds to a prime number if and only if : With. The index «k2» corresponds to a prime number if and only if : With. We showed in section 3.4- Singular points inside each base unit, that to be able to prove the primality of an odd number, we could use sequences «j» greater than the value to solve the inequation. However the condition must be applied. Reminder : The value is the index of the integer part of the value of the square root of studied number.. La valeur utilisée dans cette étude correspond à celle du nombre pair. En effet, cette valeur est supérieure à celle des nombres impairs qui composent le nombre pair. Hence [ ] Hence the following inequality: ( ( * ) Inequation 9 : inequation which allows obtaining two indexes each representing a prime number. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 197

198 6.2 Resolution of Goldbach's equation Constraints The previous paragraph gave an equation and an inequation (Equation 5 and inequation 6) with two unknowns and and a parameter. To solve the inequation 6, we have two conditions : Condition 1- [ Condition 2- Another condition is defined from the equation 5. The unknown variable ] is defined in relation to k1 through the equation 5 as follows: Équation 10 : formula which connects the two indices using the parameter «k» The variables and has a symmetrical role in the equation 5. The maximum value of the indices to study leads to the following equality:. Hence : * + The characterization of a Prime number does not take into account the prime number "3". Thus the indices studied in the equations are strictly greater than zero. Hence the following condition : Condition 3- The values of the variable k1 are within the interval: * + Note : it is possible to extend the interval until * prime numbers. +, so we find twice the same couples of Solution We have a system of two equations, Equation 6 and inequation 7 with two unknowns,, to solve. The resolution of the inequation 6 allows the production of both indexes and each corresponding to an odd prime number. The resolution of an equation is simpler. We will first solve the following equation: and ( ( * ) Équation 11 : equation for obtaining all the values of indexes k1 and k2 which are not jointly primes The values k1 that solve this equation are therefore excluded from the solutions which correspond aux elements k1 in the range ]0 ; ]. Indeed, the solutions sought correspond to the values k1 and k2 that give nonzero values for the function. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 198

199 The resolution of the equation 8 gives values for indexes and which does not jointly correspond to prime numbers. In other words, at least one of the two indices represents a composite number. Equation 7 is used to write the equation 8 as follows: ( ( *) As So the equation can be written : ( ( ( + ( +)) This equation is true if at least one of the values of between 0 and the following relation:, allows to obtain ( + ( + Hence ( + ( + With with The condition is written as follows : ( + ( + Knowing the following relationship:, we get : ( + ( + Solutions : The solutions obtained are as follows for the case : Solution 1 : the solutions for the variable k1 correspond to values that do not respect the equality hence : Solution 1 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 199

200 With and [ Hence ] and. The set of values of k1 is given by the following relationship: With [ ] The term is odd. The term must be odd so that the sum with the previous element is an even number. This allows the numerator to be divisible by two. It is therefore necessary that the number «n» is even so that is odd. Hence the following relation : with. Hence With By substituting m with m+1, we get : With Note : The term refers to all odd multiples of the odd primes. This term generates all points UNNP except the number "1". The relationship between variables k1 and k2 is as follows : Hence the solution 1 can be written as : Explanation: This solution makes the link between variables and. The values of variable corresponding to values for which the variable corresponds to a prime number. Solution 2 : Hence the following solution : Solution 2 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 200

201 with The term must be odd so that equality is respected. It is therefore necessary that the number «n» is even so that is odd. Hence the following relation: with. Hence the following solution: Hence Explanation: This solution requires that the variable k1 does not correspond to composites (UNNP). Conclusion : We have. The solutions 1 and 2 provide the set of values of the variable k1 for which Goldbach's equation is validated. The values of the variable k1 are located in the following interval: * +. The values of the parameter «j» are in the range [0 ; [ ]]. possible values of k1 * + k : indices of odd numbers * + Values of k for each value of «j» and «m» A B If the list of possible values for the variable k1 is {A, B}, thus if A is the index of a prime number then A is a solution and if B is the index of a prime number then B is a solution. Figure 29: Synthesis of the method for solving Goldbach's equation The solutions for the variable k1 are given by the following formula : Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 201

202 The values of the index of the variable k1 must be different from values provided by the formula above. Explanation of the formula k1 : We have two terms: - The first is the even number minus the number «3» : - The second corresponds to the set of odd composites (UNNP) : We subtract these two terms to divide them by two. Each term corresponds to an odd number. The subtraction of two odd numbers is always divisible by two. We have three possible forms of an even number (See paragraph 6.3 mathematical forms of primes that correspond to solutions of Goldbach's equation) : Form 1- If, thus we get : Hence The first term corresponds to a multiple of three. This means that one removes at a multiple number of three, the odd multiples of odd primes (UNNP). As there are an infinity of prime numbers, this means that the set of values k1 is never empty. In addition, there is always numbers in the list of possible values for k1 corresponding to the prime numbers of the form. Indeed, we can write the inequation in this mathematical form : We know that corresponds to an odd number of the form in this case. Indeed, we are looking for primes of the form. Hence With Regardless of the value, there is always a value to obtain a prime number of the form and a prime number of the form (See paragraph The foundations of a mathematical proof page 205). In addition, the number of values that generate a prime number increases as the value of parameter increases. Form 2- If, thus we get : Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 202

203 Hence The first term corresponds to a number with the form. This means that one removes at a number of this form, the odd multiples of odd primes (UNNP). All multiple of three are removed. This means that all numbers of the form are not solution of the equation of Goldbach. Only the primes of the form are solutions of the equation Goldbach if the even number is of the form. As there are an infinity of prime numbers of the form, this means that the set of values k1 is never empty. In addition, there is always numbers in the list of possible values for k1* corresponding to the prime numbers of the form. Indeed, we can write the inequation in this mathematical form : We know that corresponds to an odd number of the form in this case. Hence Hence : With Regardless of the value, there is always a value to obtain a prime number of the form and a prime number of the form (See paragraph The foundations of a mathematical proof page 205). In addition, the number of values that generate a prime number increases as the value of parameter increases. Form 3- If, thus we get : Hence The first term corresponds to a number with the form. This means that one removes at a number of this form, the odd multiples of odd primes (UNNP). All multiple of three are removed. This means that all numbers of the form are not solution of the equation of Goldbach. Only the primes of the form are solutions of the equation Goldbach if the even number is of the form. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 203

204 As there are an infinity of prime numbers of the form, this means that the set of values k1 is never empty. In addition, there is always numbers in the list of possible values for k1* corresponding to the prime numbers of the form. Indeed, we can write the inequation in this mathematical form : We know that corresponds to an odd number of the form in this case. Hence Hence : With Regardless of the value, there is always a value to obtain a prime number of the form and a prime number of the form (See paragraph The foundations of a mathematical proof page 205). In addition, the number of values that generate a prime number increases as the value of parameter increases. * We have demonstrated that there is an infinity of twin primes. So this allows us to affirm that to shift the UNNP numbers of unit keeps the fact of always having primes in the set of possible values of k1. When the value of the even number N increases, the value 'increases. This means that the interval of the values k1 increases because the maximum value of k1 that is [ increases with. The number of prime numbers within this interval increases. This explains why the number of decomposition increases with the value of the even number. Moreover, the mechanisms of running of the conjecture has been explained in paragraph Combined patterns and Goldbach s conjecture page 58. The resolution of the equation confirms the validity of the symmetry of Goldbach. The set of prime numbers, as well as the set of composite numbers have an internal symmetry. The conjecture has been verified for values of up to The resolution of the equation of Goldbach: [9] which reinforces our theory. - explains the increase of the number of decompositions of an even number in two primes, - valid the symmetry of Goldbach. Note : There is another solution: does not provide positive solutions for Solution excluded:. It is therefore excluded. ]. But this solution Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 204

205 With Hence No positive solutions for the variable is possible with. The following paragraph gives the mathematical basis for a method that allows to get a mathematical proof of Goldbach s conjecture The foundations of a mathematical proof We will determine the mathematical condition for solving mathematically the Goldbach's conjecture. According to Goldbach any even number is the sum of two prime numbers, hence: with and : primes. ( is an even number) Determination of the mathematical condition 1- Definition We define four sets of odd numbers in the interval integer greater than or equal to 10 : with Npair a positive even - Set of odd numbers Ei with nei the number of elements into this set. - Set of odd multiples of the odd primes E1 with ne1 the number of elements into this set. with corresponds to the odd multiples of the odd primes. - Set of values that correspond to the even number minus the odd multiples of the odd primes E2 with ne2 the number of elements into this set. with corresponds to the odd multiples of the odd primes. - Set of common odd numbers Ecm between sets E1 and E2 with necm the number of elements into this set. To get the number of couples of primes that decompose an even number Npair, we will determine a formula. To get the number of primes that decompose an even number Npair, we will determine a formula. The relationship between the two formulas is as follows : Note : the number «4» is added to in order to take into account the two odd numbers which is removed «1» and «3» and the two even numbers associated. The formula consists of the following elements: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 205

206 Note : the odd number "3" should not be taken into account because the formula for the characterization of prime numbers does not run with this prime number. The odd number "1" is not considered because this is not an odd multiple of odd primes. We will determine the sets E1, E2, and Ecm concerning this even mathematical form :. Note : The two other mathematical even forms ( and ) have smaller sets for E1, E2, Ecm. However the same type of calculation is to be done. Formula in space N We will consider two cases corresponding to two different intervals: a- We will search for the couples of primes whose the sum corresponds to an even number in the interval* +. Description of the graph : The chart below shows the three sets of numbers E1, E2 and Ecm. - The set E1 corresponds to the odd multiples of the odd primes. Hence. The first number in this set is. - The set E2 corresponds the numbers obtained by subtracting the odd multiples of the odd primes to the even number studied (Npair). Hence. - The set Ecm corresponds to common numbers between the sets E1 and E2. They are represented on the graph by black circles. The function to count this points is. (commun=common) For example with Npair = 36, we have the following common odd number: Hence the number «9» belongs to the set of Ecm. Note : If there is a couple of prime numbers with the number "3", this couple can not be determined with this method. The number of couples of prime numbers given by the formula corresponds to a minimum. Indeed, an extra couple of prime numbers with the number "3" is possible but not systematic. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 206

207 0 Determination of the sets, and in the interval, with The origin of numbers from the set E1 is the number Numbers Numbers -1 Prime number Possible values of the variable. Ecm : Common number between E1 and E2 Only numbers in the interval are taken into account. Sense of the runs of the composites origin of the counting of the set E2 This figure shows the method for determining the sets E1, E2 and Ecm in the interval. Determination of the formula These numbers are excluded «1» and «3» hence : - Both numbers are removed from the odd numbers hence. - The number of composites is calculated in the interval + + and in the Hence interval + + in order not to take into account the numbers «1» and «3». ( ( * * ( ( *) Équation 12: Formula calculating the number of couples of prime numbers that decompose an even number Hence ([ ] + We have four couples of prime numbers that are solutions of this equation system. These couples are :. b- We will search for the prime numbers whose the sum is equal to the even number in the interval [ ]. Determination of the formula Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 207

208 The interval for determining the prime numbers corresponding to the variable, can be extended to this interval ]. We excluded the numbers "1" and "3" hence : - Both numbers are removed from the odd numbers hence. - The number of composites is calculated in the interval and in the interval in order not to take into account the numbers «1» and «3». Hence the following formula : After simplification, we obtain the following formula with : With. Equation 13: Formula calculating the number of prime numbers that decompose an even number All terms of this formula are known except the term. We have eight prime numbers which are solutions of the equation system. These numbers are:. Determination of the sets, and in the interval The origin of numbers from the set E1 is the number Nombre Prime number Possible values of the variable:. Ecm : Common number between E1 and E2 Only numbers in the interval are taken into account. Nombre Sense of the runs of the composites : -1 origin of the counting of the set E2 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 208

209 This figure shows the method for determining the sets E1, E2 and Ecm in the interval. Formula in space W We will define the method for determining the number of decompositions. Decomposition of the even number «Even number hence Determination of : * + *» with the form + Determination of the maximum limit of the interval of values of the variable : * + [ ] Hence the interval of possible solutions. possible values of k k : indices m = 0, 1, 2,3 9 4 m = 0, 1 k1=2 k1=1 k1=7 k1=5 Solution 1 : List of possible values of k1 : [ ] a- Solution 1 : List of possible values of k1 b- Solution 2 : Among the values of the list, only the values corresponding to prime numbers are kept. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 209

210 k1 Prime number? 1 5 yes 2 7 yes 5 13 yes 7 17 yes The solutions for the variable Np1 are : The solutions for the second prime number correspond to: Hence : Hence the solutions are : With * + No other solution exists with the number "3". Result: the number of couples of prime numbers is 4. This corresponds to 8 primes. Determination of the formula in space W. We will consider two cases corresponding to two different intervals: a- We will search for the couples of primes whose the sum corresponds to an even number in the interval* +. We excluded the numbers "1" and "3" hence : - One takes into account only the indices ; hence the number of odd number is equal to. - The number of composites is calculated in the interval and in the interval in order not to take into account the numbers «1» and «3». D où D où We have four couples of prime numbers that are solutions of the equation system. These couples are :.. b- We will search for the prime numbers whose the sum is equal to the even number in the interval [ ]. The interval for determining the prime numbers corresponding to the variable, can be expanded to the interval. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 210

211 On exclue le nombre «1» et «3» d où : - One takes into account only the indices ; hence the number of odd number is equal to.. - The number of composites is calculated in the interval and in the interval in order not to take into account the numbers «1» and «3». Hence the following formula : With * + Hence We have eight prime numbers that are solutions of the equation system. These primes are:. 2- Determination of the condition to search Le problème initial était de résoudre une équation avec un paramètre (le nombre pair), et deux variables inconnues (deux nombres premiers). Solving the equation system has reduced the problem to a parameter and an unknown variable for each form of even number. The formula to be determined is composed of the following terms: The terms are function of the parameter. The function was determined in the reference [5] Distribution of prime numbers. The term to be defined is the number of common elements between sets of E1 and E2. The formula must be defined in order to mathematically validate the Goldbach conjecture. Indeed, the formula is used to set the formula as a function of the only parameter. If the formula is always greater than zero, regardless of the value of the parameter with, thus the Goldbach's conjecture is validated. The function provides the common number of composite numbers between the two sets of composite numbers corresponding to Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 211

212 and with is the odd multiples of odd primes. The function is calculated as follow : We want to know the number of composite numbers that respects the following equation : Hence With et. If one wishes to know the total number of composite numbers, the search interval corresponds to the set of composite numbers up to. The last five numbers are removed in order to eliminate the odd numbers "1" and "3". The first odd multiple number is 9, hence the following interval:. If one wishes to reduce the computation, the interval then corresponds to half the even number. Indeed, the two parameters and is as follows :. are symmetrical relative to the point. The calculation interval Determination of the formula. The set of odd numbers to be tested is the following. 1- The number is it a prime number? We have a function named that allows to know whether the number N is a composite number. The function returns a Boolean value. If the number is a prime number, then the function returns the value "0". If the number is a composite number, the function returns the value "1". 2- The number is it a prime number? The function is used to answer. 3- The number is accounted if and only if 4- The formula is as follows : Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 212

213 With Or With : the number of composed odd numbers less than or equal to X, et which is the number of primes less than or equal to X. These functions have been defined in the study of the distribution of prime numbers [5]. Hence the following formula from Equation 13: Formula calculating the number of prime numbers that decompose an even number page 208 : With CASE 1 CASE 2 How to simplify the formula? We know that we have three cases: - Case 1 : : no multiple numbers of prime numbers in the interval. - Case 1 : : one number that is an odd multiple of prime numbers in the interval. - Case 1 :. : two numbers that are an odd multiple of prime numbers in the interval. Indeed, in the interval, there are 0, 1 or 2 numbers that are odd multiples of prime numbers. The number of multiple is a function of the mathematical form of the even number. - If then. It is an odd multiple of the prime number «3». Therefore, the number of odd multiples of primes is equal to 1 or 2 in the interval. - If then. It is an odd multiple of the prime number «3». Therefore, the number of odd multiples of primes is equal to 1 or 2 in the interval. - If The number of odd multiples of primes is equal to 0, 1 or 2 in the interval. CASE 1 : ou The number of odd multiples of primes is equal to 1 or 2 in the interval. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 213

214 This difference of one unit does not change the validity of the equation to solve. Indeed, the number of primes which decomposes an even number is necessarily even. Hence the following formula with : With We have the following elements : Hence : - - Number of odd primes =. One unit "1" is removed from because it corresponds to the first even number «2». With The number of composite numbers exceeds very quickly the number of prime numbers. Hence. The number of common numbers between the sets E1 and E2 must grow faster than the difference between and if the Goldbach's conjecture is valid, hence : We know [5] : [ ] With N an even number, we get : Hence the following condition to prove Goldbach's conjecture : With Équation 14: Condition de Goldbach CASE 2 : Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 214

215 We consider the following formula with : With The numerical computations show that if then the number of couples of primes is greater than three. But the number of primes in a couple of primes is twice that value, hence. This means that if then hence. If one of the three couples prime numbers corresponds to a couple with the number three, then we must remove those two primes numbers hence. This explains the condition.so if the above inequality is satisfied, then the Goldbach s conjecture is valid. We have the following elements : Hence : - - Number of odd primes =. One unit "1" is removed from because it corresponds to the first even number «2». With The number of composite numbers exceeds very quickly the number of prime numbers. Hence. The number of common numbers between the sets E1 and E2 must grow faster than the difference between and if the Goldbach's conjecture is valid, hence : We know [5] : With N an even number, we get : [ ] Hence the following condition to prove Goldbach's conjecture : With Équation 15 : Condition of Goldbach Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 215

216 Condition of Goldbach : We will consider the most strict condition, the equation 15. We know that the term is the number of odd number. It must be shown that regardless of the value of the parameter, the number of odd number is always greater than twice the number of odd multiples of the odd primes minus the number of common odd multiple of the sets E1 and E2. Note: we have to solve this inequation with the three mathematical forms of an even number. Summary : The problem has been reduced to a parameter Npair (even number) and a condition to resolve. This condition is a function only of the parameter r. The resolution of the equation is out of reach. We will check the result numerically according to the value of the even number and the mathematical form of even number Numerical computations The «comet Goldbach» gives a representation of the number of possible decompositions of an even number as sum of two primes. The graph Figure 30 shows that the line is empty. Indeed, all even numbers numerically tested have at least one decomposition as sum of two prime numbers. The conjecture was checked numerically to a value equal to [9]. The scatter plot in the form of comet tail is organized into bands more or less dense (cf. book of Jean-Paul Delahaye, Merveilleux nombres premiers: voyage au cœur de l arithmétique, Belin-Pour la science. (2012), page 127). The graph below shows that these bands depend on the form of even number. We have three possible forms as shown in paragraph 6.3 mathematical forms of primes that correspond to solutions of Goldbach's equation page 226. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 216

217 Zoom Figure 30: Goldbach's comet depending on the form of the even number The even form has a number of decomposition greater than the two other forms. Indeed, this mathematical form can be decomposed in the sum of two prime numbers of the form and. The two other mathematical even forms use either the form or the form. The graph shows that whatever the form of the even number, the number of decompositions in the sum of two primes increases with the value of the even number as predicted by the solution of the equation of Goldbach Conclusion The numerical results show that regardless of the form of the even number, the number of decomposition is always greater than zero. In addition, the number of decomposition increases when the value of the even number increases. The theoretical calculation of the number of decomposition of an even number must confirm that whatever the value of the even number, the number of decomposition is strictly greater than zero. The work goes on The following paragraph gives the calculation method to determine the decomposition of even numbers from examples with the three forms of an even number. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 217

218 6.2.4 Examples The even numbers that are studied: 10, 14, 36, Decomposition of the even number «10» Even number Hence Determination of : * + Determination of the maximum value of the interval of values of the variable : [ ] Hence the interval of possible solutions. So one value is available :. The solution 1 of the equation gives us all the values of which are to be eliminated from the interval of possible solutions. The sequences «j» to be investigated are within the range [0 ; ]. So only the sequence «j=0» will be studied. The following table gives the values of to remove. Sequence «j» 1 2 No solution exists. There is no values to remove. - - The solution 2 of the equation requires the elimination, in the interval of possible solutions, indices corresponding to composites UNNP. The index «k1=1» does not correspond to a composite number UNNP. The final solution is therefore k1=1. This index is the prime number «5». The index k2 is equal to the value «1» which corresponds to the prime number «5». So the even number is divided into two prime numbers:. The verification of the decomposition with the prime odd number "3" shows that a second possibility exists:. This possibility can not be determined by solving the equation 8 as already mentioned previously. 2- Decomposition of the even number «14» Even number hence Determination of : * + Determination of the maximum value of the interval of values of the variable : [ ] Hence the interval of possible solutions. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 218

219 The solution 1 of the equation gives us all the values of which are to be eliminated from the interval of possible solutions. The sequences «j» to be investigated are within the range [0 ; ]. So only the sequence «j=0» will be studied. The following table gives the values of to remove. Sequence «j» Only one value exists. It must be removed from the interval of possible solutions. This leaves us the value «2». The solution 2 of the equation requires the elimination, in the interval of possible solutions, indices corresponding to composites UNNP. The index «k1=2» does not correspond to a composite number UNNP. The final solution is therefore k1=2. This index is the prime number «7». The index k2 is equal to the value «2» which corresponds to the prime number «7». So the even number is divided into two prime numbers:. The verification of the decomposition with the prime odd number "3" shows that a second possibility exists:. 3- Decomposition of the even number «36» Nombre pair D où Détermination de : * + Determination of the maximum value of the interval of values of the variable : [ ] Hence the interval of possible solutions. The solution 1 of the equation gives us all the values of which are to be eliminated from the interval of possible solutions. The sequences «j» to be investigated are within the range [0 ; ]. The following table gives the values of Sequence «j» to remove Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 219

220 The values in the interval are taken into account. We must therefore remove the following values: 3,4,6. This leaves the following possible values:. The solution 2 of the equation requires the elimination, in the interval of possible solutions, indices corresponding to composites UNNP. None of the remaining values in the interval corresponds to a composite number UNNP. Solutions for the index are as follows:. The solutions for the second prime number correspond to : The final solution provides several couples of prime numbers to decompose the even number. These couples are given in the following table: Note : the prime odd number "3" does not provide a couple of primes to decompose the even number. 4- Decomposition of the even number «102» Nombre pair D où Détermination de : * + Determination of the maximum value of the interval of values of the variable : [ ] Hence the interval of possible solutions. Sequence «j» The solution 1 of the equation gives us all the values of which are to be eliminated from the interval of possible solutions. The sequences «j» to be investigated are within the range [0 ; ]. La valeur de la séquence j=3 n est pas étudiée car elle correspond à un sous-ensemble de la séquence j=0. The following table gives the values of to remove Useless because it is a subset of the sequence j = 0. The values in the interval are taken into account. We must therefore remove the following values :. This leaves the following possible values :. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 220

221 The solution 2 of the equation requires the elimination, in the interval of possible solutions, indices corresponding to composites UNNP. The following values correspond to composites UNNP :. Solutions for the index are as follows:. The solutions for the second prime number correspond to : The final solution provides several couples of prime numbers to decompose the even number. These couples are given in the following table: Note : the prime odd number "3" does not provide a couple of primes to decompose the even number Decomposition of the even numbers according to the mathematical form of the even number 1- Decomposition of the even number «44» with the form Even number hence Determination of : * + Determination of the maximum value of the interval of values of the variable : [ ] Hence the interval of possible solutions. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 221

222 Possible values of k k : indices m = 0, 1, 2, 3, 4, m = 0, 1, 2 k1=2 k1=5 k1=6 k1=9 Solution 1 : List of the possible values of k1 : [ ] a- Solution 1 : List of possible values of k1 b- Solution 2 : Among the values of the list, only the values corresponding to prime numbers are kept. k1 Prime number? 2 7 yes 5 13 yes 6 15 no 9 21 no The solutions for the variable Np1 are : The solutions for the second prime number correspond to: Hence : Hence couples which are solutions are: There is also another solution with the number "3" : Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 222

223 Result : Prime numbers that are solutions correspond to prime numbers with this form. 2- Decomposition of the even number «40» with the form Even number hence Determination of : * + Determination of the maximum value of the interval of values of the variable : [ ] Hence the interval of possible solutions. Possible values of k k : indices m = 0, 1, 2, 3, m = 0, 1, 2 k1=4 k1=7 k1=3 Solution 1 : List of possible values of k1 : [ ] a- Solution 1 : List of possible values of k1 b- Solution 2 : Among the values of the list, only the values corresponding to prime numbers are kept. k1 Prime number? 3 9 no 4 11 yes 7 17 yes The solutions for the variable Np1 are : Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 223

224 The solutions for the second prime number correspond to: Hence : Hence couples which are solutions are: There is also another solution with the number «3» : Result : Prime numbers that are solutions correspond to prime numbers with this form. 3- Decomposition of the even number «42» with the form Even number hence Determination of : * + Determination of the maximum value of the interval of values of the variable : [ ] Hence the interval of possible solutions. Possible values of k k : indices m = 0, 1, 2,3, m = 0, 1, 2 k1=1 k1=5 k1=4 k1=8 Solution 1 : List of possible values of k1 : [ ] a- Solution 1 : List of possible values of k1 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 224

225 b- Solution 2 : Among the values of the list, only the values corresponding to prime numbers are kept. k1 Prime number? 1 5 yes 4 11 yes 5 13 yes 8 19 yes The solutions for the variable Np1 are : The solutions for the second prime number correspond to: Hence : Hence couples which are solutions are: No other solution exists with the number «3». Result : Prime numbers that are solutions correspond to prime numbers with these form and. The even numbers studied in the examples are all decomposable into two primes. They therefore respect the conjecture of Goldbach. As the number increases, the possible number of decomposition in two primes increases. Indeed, the limit value increases with the value of the even number. The number of indexes of prime numbers among the list of possible values constituted by the range [0 ; ] increases with the value of the even number N. Conclusion : The resolution of the equation of Goldbach explains why the number of decomposition into two primes increases when the value of the even number increases. However, the number of decompositions, for an even number of the form is greater than for the other two forms. Indeed, with this form, the prime numbers can be either of the form or of the form. For the other two forms of the even number, the prime numbers are either of the form or of the form. This reduces the possibilities of decomposition. We explained in paragraph Combined patterns and Goldbach s conjecture page 58, the origin of decomposition of even numbers in two primes. This decomposition is due to an internal symmetry of the basic patterns. This symmetry is true regardless of the scale of magnitude of the even number studied. We demonstrated that the maximum distance between two Prime numbers is less than. We have shown that this distance can be approximated by. The number of prime numbers in the interval increases almost linearly with. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 225

226 We explained the origin of twin primes, and we showed that there is an infinity of twin primes by using the base units. We also proposed a mathematical proof of the conjecture of twin primes. The twin primes are involved in the conjecture of Cramer to demonstrate the approximation of the maximum distance between two primes by. In addition, the twin primes are also involved in the Goldbach conjecture. Indeed, the existence of twin primes ensures that there are always primes of the form 6m + 1 and form 6m-1 in the base units and therefore in the interval. The conjecture has been verified for numerical values of N up to decomposition increases with N which confirms our theory. and the number of All of these elements helps to understand the conjecture of Goldbach. However, the mathematical resolution of this conjecture is coming in particular by solving the equation 14. The following paragraph establish the forms of primes which decomposes an even number according to the form of this even number. 6.3 mathematical forms of primes that correspond to solutions of Goldbach's equation We will show in the first part that the two prime numbers that make up an even number, have a mathematical form which depends on the Euclidean division of the even number by 3. The even number N is defined by the formula. We will study the divisibility of the number «k» by 3 in order to show the influence of this parameter on the mathematical form of values of the solution 1: With The Euclidean division of by 3 may only take one of three forms:, and equivalent to with q corresponding to a natural number. As demonstrated in paragraph Erreur! Source du renvoi introuvable., every prime number other than 2 and 3 can be written in the space N as with. We will study what forms have the prime numbers that solve the inequation 6, paragraph 6.1.2, depending on the shape of the number. 1- Consider the first forms to, thus a- Let us take the following form of prime numbers: The solution for is: hence Hence the following solution: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 226

227 For a given even number N, the value of the parameter is fixed. The value «g» is in the interval because the value is greater than zero. The term with corresponds to the set of points UNNP, with the exception of the number "1". This means that if solutions exist, they may take the form. Indeed, there is an infinity of prime numbers with the form. b- Let us take the following form of prime numbers: Hence the following solution: In the same manner as previously, if solutions exist, they may take the form. Indeed, there is an infinity of prime numbers with the form. The couple of primes (k1, k2) must be made of each of the two possible forms of a prime number. The formula of Goldbach is then written as follows: Hence Solutions are possible. If the prime numbers are of the same form then no solution is possible. Conclusion : The solutions of the form et can exist if the parameter «k» has the form. 2- Consider the following form : a- Let us take the following form of prime numbers : Hence the following solution: The term with corresponds to the set of points UNNP, with the exception of the number "1". This means that if solutions exist, they may take the form. Indeed, there is an infinity of prime numbers with the form. b- Let us take the following form of prime numbers : Hence the following solution: Hence the solution: Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 227

228 If, then the above equation is satisfied regardless of the value «g» in the interval This means that no prime number of the form solves the inequation 6 if «k» has the form. Conclusion : No solutions of the form the form. can exist if the parameter «k» has 3- The last form is the following : a- Let us take the following form of prime numbers : Hence the following solution: The term with corresponds to the set of points UNNP, with the exception of the number "1". This means that if solutions exist, they may take the form. Indeed, there is an infinity of prime numbers with the form. b- Let us take the following form of prime numbers : Hence the following solution: Hence the solution: If, then the above equation is satisfied regardless of the value «g» in the interval This means that no prime number of the form solves the inequation 6 if «k» has the form. Conclusion : No solutions of the form can exist if the parameter «k» has the form. Summary: We have the even number N. This number is represented as follows: with. The mathematical form of the two prime numbers, that breaks down an even number, is related to the form of the Euclidean division of the index «k» of the even number. - If, then the solutions of couples of twin primes (k1, k2) take respectively the form and and vice-versa, - If, thus the primes which correspond to solutions will have only the form, - If, thus the primes which correspond to solutions will have only the form. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 228

229 Conclusion : The formula that characterize an odd prime number has allowed to get a method for determining the set of twin primes which decomposes an even number, with the exception of the couple of primes containing the prime number "3". In addition, we explained how the decomposition of an even number is related to the divisibility of an even number per "3". The resolution of the equation of Goldbach has helped explain why the number of possible decompositions into two primes increases with the value of the even number. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 229

230 Conclusion This theory W is used to describe the basic rules governing the organization of primes. These rules are defined through three structural elements : - The basic schema This element allows to construct the workspace W. Only odd numbers are represented through their index in this space. In addition, the prime numbers are mathematically characterized what distinguishes the set of prime odd numbers and the set of odd composites. - The base unit We demonstrated that the proper interval for measuring the properties of prime numbers, corresponds to the base unit. This unit is the second structural element of the space W. Within these base units, we have defined a unit of measurement. This unit is the maximum distance between two primes in a base unit. Moreover, measurements within natural periods guarantee obtaining stable measures. Indeed, these measures will not oscillate around a mean value. These results help to understand the evolution of the properties of prime numbers regardless of the order of magnitude of numbers studied. - The basic patterns and combined patterns These patterns offer an internal structure that explains the distribution and the symmetry of primes. These elements define a regular structure of prime numbers. Prime numbers do not appear randomly. They are built around a repetitive and symmetrical structure which remains the same regardless of the scale: when the value «j» of the base schema increases, then the base unit and the natural period increase together. But rules defining the structure remain unchanged which allows the study of the properties of primes by the base units. The underlying structure is linked to the base pattern defined by the prime numbers 3, 5, 7. The superposition of the patterns corresponds to combined patterns which explains the fractal structure of primes. These combined patterns retain the properties of patterns including their internal structure. This structure and this organization of prime numbers explain : the need for using discrete mathematics to measure accurately the properties of odd numbers, the difficulty of finding simple relationships between these numbers, the high cost in terms of computing time, to determine the properties of these numbers, starting with finding them. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 230

231 The understanding of the structure of prime numbers has led us to the following results: a characterization of odd primes, the understanding and the determination of the exact distribution of odd primes, the validation of conjectures such as the Riemann hypothesis, the twin primes, the conjecture of Legendre, Goldbach's conjecture, The explanation of the conjectures such as Goldbach's conjecture and Cramer's conjecture. This theory allows us to understand the properties of prime numbers. In addition, it also validates the evolution of these properties at any scale. This theory makes the connection between the primes and the following mathematical elements: - The constants - The trigonometry - The complex numbers - The fractal dimension This theory confirms the technical choice of prime numbers for encryption of information using the RSA algorithm. Indeed, the determination of prime numbers requires a significant number of operations that requires a high computational time due to the fractal structure of primes. The discovery of the space W opens a new area of knowledge of the odd numbers and mainly on the odd primes. This knowledge will undoubtedly contribute in the near future to demonstrate many other conjectures as well as discover other properties on prime numbers thus becoming less mysterious Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 231

232 References [1] Jean-Paul Delahaye, Merveilleux nombres premiers: voyage au cœur de l arithmétique, Belin-Pour la science. (2012). [2] EUCLIDE, Les œuvres d Euclide traduites littéralement par F. Peynard, 1819, nouveau tirage avec une introduction de Jean Itard, Librairie scientifique et technique Albert Blanchard, Paris,1993 [1,2,4]. [3] M. R. Murty, Primes in certain arithmetic progressions, J. Madras Univ. (1988), [4] I. Schur, Uber die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, Sitzungber. Berliner Math. Ges. 11 (1912), [5] François Wolf et Marc Wolf, Distribution of prime numbers, site web mathsciences.tsoft .com. [6] P. Erdős. Beweis eines Satzes von Tschebyschef. Acta Szeged 5 (1932), Lire en ligne: [1] [archive]. [7] Joseph Bertrand. Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme. Journal de l'ecole Royale Polytechnique, Cahier 30, Vol. 18 (1845), [8] P. Tchebychev. Mémoire sur les nombres premiers. Journal de mathématiques pures et appliquées, Sér. 1(1852), (Preuve du postulat: p ). [9] Tomás Oliveira e Silva, Siegfried Herzog et Silvio Pardi, «Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18», Math. Comp., vol. 83, 2014, p [10] Gilles LACHAUD «L hypothèse de Riemann» [11] NOVA ACTA ACADEMIAE SCIENTIARUM IMPERIALIS PETROPOLITANAE TOMUS XII, «Essai sur les nombres premiers par M. KRAFFT» présenté à l académie le 12 avril 1798 p [12] G.H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, (1979) [13] G.H. Hardy and J.E. Littlewood, Some problems of 'Partitio Numerorum' III : On the expression of a number as a sum of primes, Acta Mathematica, (1922), vol. 44, p [14] Gustavo Funes, Damián Gulich, Leopoldo Garavaglia, Mario Garavaglia, HIDDEN SYMMETRIES AMONG PRIMES, Form and Symmetry: Art and Science, Buenos Aires Congress, 2007 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 232

233 Indexes of graphics Figure 1 : Graphical representation of odd integers in the space N... 9 Figure 2 : Graphical representation of the odd integers with their index in the space W... 9 Figure 3 : The graphic corresponds to the space W and represents a part of the first five sequences «j». Some numbers PUN are also represented Figure 4 : Representation of the basic schema and the base unit Ugw(j), and the equivalence of the points k(j) and k(m) connected by arrows with a green color Figure 5 : Representation of an odd number in the space W Figure 6 : Representation of singular points Figure 7 : Chart representing the base unit ) and the natural period of the system per sequence «j» Figure 8: Determination of virtual primes PUNv for the sequence "jmax=1" Figure 9 : Distribution of points UNNP within the natural period of the basic pattern Figure 10: Evolution of the density of the points within base units Figure 11: Schema of the structure of the two forms PUNv+ and PUNv- in the basic pattern 41 Figure 12: Graph showing the maximum distance between two prime numbers for the first base unit j= Figure 13: This figure shows how to combine two series of consecutive composite numbers 69 Figure 14: evolution of the maximum distance between two primes within a natural period of a combined pattern Figure 15: Evolution of the maximum distance between two primes PUN per base unit «j» 72 Figure 16: This figure shows that a minimal configuration can not exist within a base unit beyond the value Figure 17 : Curve representing the evolution of the ratio UNNP / PUN based on the base unit j Figure 18: Representation of the evolution of the number of points PUN per base unit Ugw(j) Figure 19: Trigonometric representation of the indices of the odd numbers Figure 20 : Graphical representation of primes determined with the formula Sf(k) Figure 21: Representation of the basic schema without the shifting «j», i.e. k'(j,n) Figure 22: Representation of 2 components of the basic schema: the offset "j" and the period Figure 23: Chart depicting the values of and the values corresponding to «k+2» Figure 24 : Chart depicting the values of and the values corresponding to «k+2» Figure 25 : Chart depicting the values of and the values corresponding to «k+2» Figure 26 : Graphic representing the number of twin primes in each base unit Figure 27: Evolution of the density in the base units Ugw(j) of the seqences «j» Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 233

234 Figure 28: Comparison of the approximate formulas of the enumeration of the pairs of twin primes with Figure 29: Synthesis of the method for solving Goldbach's equation Figure 30: Goldbach's comet depending on the form of the even number Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 234

235 APPENDIX 1 : Distribution of points PUNv within the natural period of the basic pattern Mt(1) We are going to observe the distribution of numbers of the form 6m+1 and 6m-1 respectively corresponding to the forms 3m+1 and 3m+2 in space W. We are going to count the number of each of these forms corresponding to the points NIPv, NIPv+ et NIPv-. The sequences taken into account for the calculation of composites are :. These sequences correspond to the following multiples. The studied natural period is equal to :. This period starts with the number. Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 235

236 Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 236

237 Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 237

238 Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 238

239 k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter Ni k Y=k (+ ) ( - ) coun ter This table shows that within the natural period, the number of points NIPv+ is equivalent to the number of points NIPv-. The number of points NIPv+ runs after the number of points NIPv-. This is because the numbers NIPv- starts racing. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 239

240 APPENDIX 2: representation of the points connecting to the 15 formulas giving birth to the first prime number of a pair of twin primes The first prime number of a pair of twin primes is highlighted in yellow with its index k and the k+2 value associated. In the tables below, the values Ni, k and k+2 between k=0 and k=738 are shown. This value 738 corresponds to the result of the following calculation k=3+7*105=738. We represented 7 series of 105 points by superposing the 6 last series to the first series. The presence of the first prime number a pair of twin primes is then highlighted to only 15 specific locations. That is, where a value "0" is present for j = 0 to j = 2 and so DPej=3 to Dpej=7. These odd numbers cannot be factorized into prime factors with the prime numbers 3, 5 and 7. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 240

241 1st formula Ni=11+210*x (1) (2) (3) (4) (5) (6) (7) (8) Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 241

242 (9) (10) (11) 11th formula: Ni= *x (12) (13) (14) (15) 15th formula: Ni= *x Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 242

243 APPENDIX 3: results of the study of < A) Determination of the factor We will determine of the partial sum of the products made of the highest Nmax product. ANP is dependent of x, i.e. the number of factors which composes the product. When the product is composed of two factors, one of them is Nmax, we get: For x factors, the partial sum is written: The general formula is: so ( ) B) Tables of the ratios for Nmax=9, 21 and 26 and x between 1 and 5. Nmax=9 Sum Ratio R Partial sum Ratio Rp x R = ANx/AN(x-1) Rp = ANPx/ANP(x-1) , , , , , , , , Nmax=21 Sum Ratio R Partial sum Ratio Rp x R = ANx/AN(x-1) Rp = ANPx/ANP(x-1) 5 2,792E+09 35, , , , , , , Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 243

244 Nmax=26 Sum Ratio R Partial sum Ratio Rp x R = ANx/AN(x-1) Rp = ANPx/ANP(x-1) 5 2,592E+10 56, , , , , , , , C) Values of a minimum for n=1 and n>1 according to X for Nmax equal to 11. The 2 tables below give the values of a respectively for the ratios R=ANx/AN(x-1) and Rp=ANPx/ANP(x-1). The ratios and the values of a are in red. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 244

245 Nmax=11 Sum X Ratio R R = ANx/AN(x-1) Partial sum Ratio Rp For n=1 and Nmax tends to infinity, U1 > U2 For n>1 and Nmax tends to infinity, Un > U(n+1) Rp = ANPx/ANP(x-1) 1/R a must be greater than : a > a must be greater than: , ,5 1, a>(9/2)*1/r 5, a>1*1/r 1, , U2>U3 a > for n=2 U3>U4 a > for n= , ,4 0, , , , , , , , ,25 0, , , , , ,2 0, , , , , , , a>(nmax+2-x)*(nmax+1-x) / (2*(Nmax-x)*1/R 2, a>(nmax+2-x)/(n+1)*1/r 0, , a>(nmax+2-x)*(nmax+1-x) / (2*(Nmax-x)*1/R 1, a>(nmax+2-x)/(n+1)*1/r 0, , a>(nmax+2-x)*(nmax+1-x) / (2*(Nmax-x)*1/R 1, a>(nmax+2-x)/(n+1)*1/r 0, , a>(nmax+2-x)*(nmax+1-x) / (2*(Nmax-x)*1/R 0, a>(nmax+2-x)/(n+1)*1/r 0, , a>(nmax+2-x)*(nmax+1-x) / (2*(Nmax-x)*1/R 0, a>(nmax+2-x)/(n+1)*1/r 0, , a>(nmax+2-x)*(nmax+1-x) / (2*(Nmax-x)*1/R 0, a>(nmax+2-x)/(n+1)*1/r 0, , a>(nmax+2-x)*(nmax+1-x) / (2*(Nmax-x)*1/R 0, a>(nmax+2-x)/(n+1)*1/r 0, , a>(nmax+2-x)*(nmax+1-x) / (2*(Nmax-x)*1/R 0, a>(nmax+2-x)/(n+1)*1/r 0, , Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 245

246 Nmax=11 Sum X Ratio R R = ANx/AN(x-1) Partial sum Ratio Rp For n=1 and Nmax tends to infinity, U1 > U2 For n>1 and Nmax tends to infinity, Un > U(n+1) Rp = ANPx/ANP(x-1) 1/Rp a must be greater than : a > a must be greater than : , ,5 0, a>(9/2)*1/r 3 a>1*1/r 0, , U2>U3 a > for n=2 U3>U4 a > for n= , ,4 0, , , , , , , , ,25 0, , , , , ,2 0, , , , , , , a>(nmax+2-x)*(nmax+1- X)/(2*(Nmax-x)*1/R 1,25 a>(nmax+2-x)/(n+1)*1/rp 0, , a>(nmax+2-x)*(nmax+1- X)/(2*(Nmax-x)*1/R 1 a>(nmax+2-x)/(n+1)*1/rp 0,5 0,375 a>(nmax+2-x)*(nmax+1- X)/(2*(Nmax-x)*1/R 0,875 a>(nmax+2-x)/(n+1)*1/rp 0, ,35 a>(nmax+2-x)*(nmax+1- X)/(2*(Nmax-x)*1/R 0,8 a>(nmax+2-x)/(n+1)*1/rp 0, , a>(nmax+2-x)*(nmax+1- X)/(2*(Nmax-x)*1/R 0,75 a>(nmax+2-x)/(n+1)*1/rp 0, , a>(nmax+2-x)*(nmax+1- X)/(2*(Nmax-x)*1/R 0, a>(nmax+2-x)/(n+1)*1/rp 0, ,3125 a>(nmax+2-x)*(nmax+1- X)/(2*(Nmax-x)*1/R 0,6875 a>(nmax+2-x)/(n+1)*1/rp 0, , a>(nmax+2-x)*(nmax+1- X)/(2*(Nmax-x)*1/R 0, a>(nmax+2-x)/(n+1)*1/rp 0,4 0,3 Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 246

247 D) Graph 1: Evolution of the coefficient a according to N a= *ln(1+n) N Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 247

248 E) Graph 2: Evolution of the coefficient b according to N b = The coefficient b tends to 1 when the value of N increases. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 248

249 APPENDIX 4: result of the calulation of the elements Un of the aleternating series ab according to the parameter Nmax A) Value of the elements Un of the alternating series The set of elements of the series are present up to Nmax = 16. Then, related to the duration of the calculation of the elements Un, only a part of the elements have been computed. For the 4 first elements with Nmax=60, the duration of the calculation was 5 h : 37 m : 45 s. For the 5 first elements with Nmax Nmax=40, the duration of the calculation was 17 h : 42 m : 48 s. For the 3 first elements with Nmax=200, the duration of the calculation was 1 day 4 h : 0 m : 3 s. The computer used owns a processor Intel core i7-4700hq 2.40 GHz with 8 Go of memory. Nmax +U1 -U2 +U3 -U4 +U5 -U6 +U7 -U8 +U9 -U10 +U11 -U12 +U13 -U14 +U15 -U16 +U17 1 0,2 2 0, , ,52E , , ,82E-02 2,60E-03 6,66E , , ,02E-02 7,67E-03 4,74E-04 1,55E-05 2,06E , , ,97E-02 0, ,18E-03 6,40E-05 2,12E-06 3,93E-08 3,09E , , , ,63E-02 3,10E-03 2,56E-04 1,51E-05 6,34E-07 1,89E-08 3,92E-10 5,32E-12 4,27E-14 1,53E , , , ,85E-02 5,35E-03 5,41E-04 4,08E-05 2,34E-06 1,03E-07 3,46E-09 8,96E-11 1,76E-12 2,57E-14 2,72E-16 1,95E-18 8,51E-21 1,71E , , , ,00E-02 7,71E-03 8,83E , , , ,31E-02 1,06E-02 1,35E , , , ,52E-02 1,35E , , , ,59E-02 1,89E , , , , , , , , , , , , Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 249

250 B) Graphic representation of the evolution of the elements Un according to Nmax. 1,8 1,6 1,4 1,2 1 0,8 0,6 +U1 -U2 +U3 -U4 0,4 0, Nmax Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 250

251 C) Algorithm to compute the value of a B and its elements Un according to Nmax. 'Compute the coefficient a for (B) 'Select the number of primes Nmax used to compute the value of a. Nmax 'Build the array of primes numbers (Nmax) => ArrayPrimes(i) and i belongs to [0 ; Nmax] 'The primes numbers are sorted from lowest to highest value: ArrayPrimes(0)=3 ArrayPrimes(i) 'Compute U1 For i=1 To Nmax afirst=afirst+1/arrayprimes(i) Next Call FunctionRecordInTextFile(1, asecond) 'Record the value for U1 'Initialise the array to record the combinations C(j,Nmax). 'The value recorded is the result of the multiplication between primes numbers into ArrayPrimes(i) ArrayCombination() 'Compute the sum of Un with n>=2 For j=1 to Nmax 'Initialisation asecond=0 n=j+1 'n is the reference of the element Un 'Compute the sign of the element Un SignOsum=(-1)^j 'Recursive function to compute the combination MaxComb=C(j,Nmax) and return an array ' ArrayCombination() with the result of the multiplication done between primes numbers ArrayCombination()=RecusiveFunction-ReturnArrayCombination() for EachComb=0 to MaxComb asecond=asecond+1/arraycombination(eachcomb) next 'Compute the sum of Un with n>=2 SumOfaSecond=SumOfaSecond+SignOfSum*aSecond 'Record the result of asecond into a text file for the value n=j+1 (Un) Call FunctionRecordInTextFile(j, asecond) 'Option: Exit function is depending of the number of element n (Un) wanted if OptionElementWanted=True then if n > N-wanted then Exit For end if end if Next a=afirst+sumofasecond Return a Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 251

252 D) Algorithm to compute (B)=(B1)+(B2) according to Nmax 'Compute the number of composed numbers (B), not multiple of 3 for a value of J 'Select the value of Nmax = number of primes numbers Nmax 'Build the array of primes numbers (Nmax) => ArrayPrimes(i) and i belongs to [0 ; Nmax] 'The primes numbers are sorted from lowest to highest value: ArrayPrimes(0)=3 ArrayPrimes(i) 'Compute the value of the odd number NiMax. (B) is computing for Ni = NiMax Jmax=(ArrayPrimes(Nmax)-3)/ KMax=2*Jmax*Jmax+6*Jmax-1+3 NiMax=2*KMax+3 'Compute for the first element of the serie (B1) for the combinations C(1,Nmax) For i=1 To Nmax 'Fix = function which returns the integer part of the value BFirst=aFirst+Fix((NiMax-ArrayPrimes(i))/(2*ArrayPrimes(i))) Next 'Initialise the array to record the combinations C(j,Nmax). 'The value recorded is the result of the multiplication between primes numbers into ArrayPrimes(i) ArrayCombination() 'Compute the sum of next element of the serie = (B2) for the combinations C(j,Nmax) For j=1 to Nmax 'Initialisation BSecond=0 'Compute the sign of the element SignOsum=(-1)^j 'Recursive function to compute the combination MaxComb=C(j,Nmax) and return an array 'ArrayCombination() with the result of the multiplication done between primes numbers ArrayCombination()=RecusiveFunction-ReturnArrayCombination() for EachComb=0 to MaxComb 'Fix = function which returns the integer part of the value BSecond=BSecond+Fix((NiMax+ArrayCombination(EachComb))/(2*ArrayCombination(EachComb))) next 'Compute the sum of the elements of the serie (B2) SumOfBSecond=SumOfBSecond+SignOfSum*BSecond Next 'Exit function if the value of the element BSecond is 0 because of the next value is less than this 'BSecond value if BSecond = 0 then Exit For end if 'Return the result (B)=(B1)+(B2) B=aFirst+SumOfaSecond Return B Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 252

253 APPENDIX 5: number of main combinations of pairs of prime factors according to the number of the pairs of primes NC for Nmax=5 The table below give the number of main combinations of pairs of prime factors: - for Nmax=5 - for un number NC pairs of prime factors with NC between 2 and 6 - for i=2 which corresponds to the number of primes in the form 3m+1 NC => i Nmax 5 2 ( * ( * ( * ( * ( * ( * ( * ( * ( ( * ( * * ( * ( * ( * ( ( * ( * * ( * ( ( * ( *) ( * ( * ( * ( * ( * The enumeration of the main combinations is done from 10 ways to combine 5 primes with 2 prime factors in the form 3m+1 and 3 prime factors in the form 3m+2. There are = 10 ways to take 2 prime factors in the form 3m+1 among 5 prime factors. Prime factors in the form Cell 1 Cell 2 Cell 3 Cell 4 3m+1 3m+2 Cell 5 Cell 6 Cell 7 Cell 8 3m+1 3m+2 Cell 9 Cell 10 3m+1 3m+2 To form 2 pairs of prime factors with 2 factors in the form 3m+1, it is needed to take 2 prime factors among 3 in the form 3m+2. For that, 1 prime factor among 3 is selected and then another prime factors among the 2 remaining prime factors is selected. The number of combinations is then. This enumeration is done for the 4 first ways to combine 5 primes (cells 1 to 4). For the 3 next ways, the first prime factors in the form 3m+2 is no more take into account to avoid to take inverse combinations with some combinations got for the cell 3. So 1 factor among 2 is selected and then the second factor is selected. The number of combinations is then. For the 3 last ways to combine 5 primes (Cells 8 to 10), there is only 1 prime factor in the form 3m+2 that is not enough to form 2 pairs with 2 factors in the form 3m+2. The combinations of 2 pairs which can be formed with 2 factors in the form 3m+1 and 1 factor in the form 3m+1 are the inverse combinations of the main combinations obtained for i=1. The number of main combinations of 2 pairs is then :. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 253

254 To form 3 pairs of prime factors with 2 factors in the form 3m+1, it is needed to take 2 prime factors among 3 in the form 3m+2 for the first factor in the form 3m+1 and 1 prime factor among 3 in the form 3m+2 for the second factor in the form 3m+1. The inverse case exists with 2 prime factors among 3 in the form 3m+2 for the second factor in the form 3m+1. This enumeration must be done for the 4 first ways to combine 5 primes (Cells 1 to 4). The number of combinations is then:. For the 3 next ways, the first prime factors in the form 3m+2 is no more take into account to avoid to take inverse combinations. So 2 prime factors among 2 remaining factors in the form 3m+2 are selected for the first factor in the form 3m+1 and then 1 factor among 2 remaining in the form 3m+2 are selected for the second factor in the form 3m+1. The inverse case exists. The number of combinations is then:. For the 6 last ways to combine 5 primes (Cells 5 to 10), it is needed to take the combinations of 3 pairs formed with the 3 primes factors in the form 3m+2. These combinations have no inverse combinations because only 2 prime factors are present in the form 3m+1. It is needed to take prime factors in the form 3m+2 among 3 for the first prime factor in the form 3m+1 and then the remaining factor for the second prime factor in the form 3m+1. The inverse case also exists. The number of combinations is then:. The number of main combinations of 3 pairs is then:. To form 4 pairs of prime factors with 2 factors in the form 3m+1, it is needed to take: - either 3 prime factors among 3 in the form 3m+2 for the first factor in the form 3m+1 and 1 prime factor among 3 in the form 3m+2 for the second factor in the form 3m+1. The inverse case also exists with 3 prime factors among 3 in the form 3m+2 for the second factor in the form 3m+1. This enumeration must be done for the ways to combine 5 primes (Cells 1 to 10). The number of combinations is then:. - or 2 prime factors among 3 in the form 3m+2 for the first factor in the form 3m+1 and 2 prime factors among 3 in the form 3m+2 for the second factor in the form 3m+1. The inverse case does not exist because there is as many prime factors in the form 3m+2 for the first factor in the form 3m+1 as for the second factor in the form 3m+1. All combinations already exist. This enumeration must be done for the 4 first ways to combine 5 primes (Cells 1 to 4). The number of combinations is then:. For the 6 last ways to combine 5 primes (Cells 5 to 10), it is also needed to take the combinations of 4 pairs formed with 2 prime factors in the form 3m+2 for the first factor in the form 3m+1 and 2 prime factors in the form 3m+2 for the second factor in the form 3m+1. However for the second factor in the form 3m+1, one of the prime factors in the form 3m+2 must contain the third prime factor not used for the first factor in the form 3m+1. The inverse case does not exist because there is as many prime factors in the form 3m+2 for the first factor in the form 3m+1 as for the second factor in the form 3m+1. All combinations already exist. The number of combinations is then:. For the 3 ways to combine 5 primes, cells 5 to 8, it is needed to take into account the combinations of 4 pairs formed with 2 prime factors in the form 3m+2. The first prime factor in the form 3m+2 is not take into account to avoid the inverse combinations. It is needed to take for each prime factor in the form 3m+1, the 2 prime factors in the form 3m+2. The number of combinations is then:. The number of main combinations of 4 pairs is then:. To form 5 pairs of prime factors with the factors in the form 3m+1, it is needed to take 3 prime factors among 3 in the form 3m+2 for the first prime factor in the form 3m+1 and then 2 prime factors among 3 in the form 3m+2 for the second factor in the form 3m+1. The inverse case also exists with 3 prime factors among 3 in the form 3m+2 for the second factor in the form 3m+1. This enumeration must be done for the ways to combine 5 primes (Cells 1 to 10). The number of combinations is then:. To form 6 pairs of prime factors with 2 factors in the form 3m+1, it is needed to take 3 prime factors among 3 in the form 3m+2 for the first factor in the form 3m+1 and then 3 prime factors among 3 in the form 3m+2 for the second factor in the form 3m+1. The inverse case does not exist because there is as many prime factors in the form 3m+2 for the first factor in the form 3m+1 as for the second factor in the form 3m+1. All combinations already exist. This enumeration must be done for the ways to combine 5 primes (Cells 1 to 10) The number of combinations is then:. Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 254

255 APPENDIX 6: Algorithm to enumerate the combinations of pairs of prime factors 'Compute the number of combinations of couples of primes factors depending on NC and Nmax 'Select the value of Nmax = number of primes numbers Nmax 'Build the array of primes numbers (Nmax) => ArrayPrimes(i) and i belongs to [0 ; Nmax] 'The primes numbers are sorted from lowest to highest value: ArrayPrimes(0)=3. After selecting Nmax, the compute is done on ArrayPrimes(1) to ArrayPrimes(Nmax) ArrayPrimes(i) '1- NC=1 ; Compute the number of pair of Primes (First part of the enumeration of the pair of primes factors) ArrayCombination()=RecusiveFunction-ReturnArrayCombination() for EachComb=0 to MaxComb 'Record the enumeration of the combinations as text file call RecordFirstPartInTextFile() 'Count the number of combination NumberOfCombination= NumberOfCombination+1 next 'Record the number of combination for NC=1 call RecordNumberOfCombination() '2- NC >= 2 ; Compute the number of combinations of NC pairs of primes factors (Second part) Tmin = fix(nmax/2) 'fix is a function which returns the integer part of a number 'Loop on i values For i=1 To Tmin 'Primes factors in 3m+1 t=nmax-i NCmax=(i*t) NCmin=2 If i>2 Then NCmin=i 'Loop on NC values For NC= NCmin To Nmax 'Compute Zmin and Zmax If (NC-(i-1))>t Then 'Numeric value to dispatch on other combinations for z ZValueToDispatch=(NC-(i-1))-t End If 'Compute Zmin and Zmax Zmin=1 DeltaValueNC=NC-i If DeltaValueNC <t Then Zmax=1+ DeltaValueNC Else Zmax=t DeltaMaxVal=(i-1)*(t-1) If DeltaValueNC >DeltaMaxVal Then Zmin= DeltaValueNC -DeltaMaxVal+1 End If End If 'Exception for i=1, only 1 combination If i=1 Then Zmin=fix(NC/i) Zmax=(j-(i-1)) If Zmax>t Then Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 255

256 'Numeric value to dispatch on other combinations for z ZValueToDispatch =Zmax-t Zmax=t End If End If 'Loop on each value of Z For z=zmin To Zmax 'When i>1 => Sum of multiplication of elements C(z,t) If i>1 then ZValueToDispatch = NC-z-(i-1) If i=1 Then 'Only 1 combination C(z,t) 'Compute the combinations of 1 pair of primes factors ArrayCombination()=RecusiveFunction-ReturnArrayCombination2() Else 'Compute the combinations of NC pairs of primes factors ' => C(z2,t) * C(z3,t)... 'There is ZValueToDispatch to dispatch on the Zx values 'Get combinations for Zx and number of combinations = NumOfSumOfProd1 Call GetAllCombinationsForZx(NumOfSumOfProd1,ZCombination()) For EachCombination=0 To NumOfSumOfProd1 For iposition=1 To (i-1) 'Do the product of 2 combinations C(z1,t)*C(z2,t) ArrayCombProduct ()=ReturnArrayProductOfComb() NEXT 'Do the product of 2 combinations C(z1,t)*C(z2,t)+ C(z3,t)*C(z4,t) ArraySumProduct ()=ReturnArraySumOfProductComb() NEXT End if 'Loop on Z values NEXT 'Do the permutation of the primes factors in 3m+1 to get i values among Nmax C(i,Nmax) 'A filter is applied to avoid the inverse combinations Call GetAllCombinationsWithPermutation() 'Computer the inverse combination of the main combinations Call GetAllInverseCombinations() 'Record the enumeration of the inverse and main combinations as text file Call RecordSecondPartInTextFile() 'Record the number of combination for NC > 1 Call RecordNumberOfCombination() 'Loop on NC values NEXT 'Loop on i values NEXT 'Record the number of inverse and main combinations depending on i et NC values Call RecordInverseAndMainCombinations(i, NC) Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 256

257 APPENDIX 7: result of the calculation of the elements Unc of the alternating series ac according to the parameter Nmax A) Value of the elements Unc of the alternating series The set of elements of the series are present up to Nmax = 8. Then, related to the duration of the calculation of the elements U NC, only a part of the elements have been computed. For Nmax=8, the duration of the calculation was 3 days 5 h : 26 m : 53 s. For the 2 first elements with Nmax Nmax=40, the duration of the calculation was 23 h : 57 m : 6 s. The computer used owns a processor Intel core i7-4700hq 2.40 GHz with 8 Go of memory Nmax +U1 -U2 +U3 -U4 +U5 -U6 +U7 -U8 +U9 -U10 +U11 -U12 +U13 -U14 +U15 -U16 2 0, ,98E-02 5,19E ,21E-02 1,52E-02 2,13E-03 4,00E ,21E-02 2,78E-02 7,11E-03 2,23E-03 4,70E-04 7,84E , ,35E-02 1,56E-02 6,56E-03 2,35E-03 7,67E-04 1,98E-04 4,33E-05 4,12E , ,01E-02 2,68E-02 1,38E-02 6,54E-03 2,97E-03 1,24E-03 4,80E-04 1,56E-04 4,18E-05 7,53E-06 6,28E , , ,93E-02 2,33E-02 1,32E-02 7,35E-03 3,95E-03 2,05E-03 9,83E-04 4,34E-04 1,70E-04 5,75E-05 1,58E-05 3,12E-06 3,81E-07 2,16E , , ,02E-02 5,07E , , , , , , , , , , , , , , , , , Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 257

258 B) Graphic representation of the evolution of the elements U NC according to Nmax. 0,6 0,5 0,4 U1 U2 U3 U4 U5 0,3 0,2 0, Nmax Auteurs : François WOLF - Marc WOLF mathscience.tsoft .com Page 258

259 C) Algorithm to compute a C The algorithm in appendix 6 determines the set of combinations of NC pairs of prime factors. The algorithm to compute the value of a C uses this last algorithm and adds at the end the calculation of the value of a C. 'Loop on each value of NC from 1 to NCmax For NC=1 to NCmax 'Sign of the element of the serie a NC SignF=(-1) (NC+1) 'Only main combinations. For each number of NC pair of primes factors, ' there are a MaxVal of combinations. For EachCombination=1 to MaxVal 'Get each unique primes factor and compute Fn =product of primes factor Fn= product of each unique primes factor 'Compute the value a NC a NC = a NC + 1/(Fn) Next 'Compute a NC for the main and inverse combinations a NC = 2*a NC / 3 'Compute a C a C = a C + SignF *a NC Next 'Record the values a NC and a C in a text file Call RecordInTextFile(a NC, a C ) Authors : François WOLF - Marc WOLF mathscience.tsoft .com Page 259

260 D) Enumeration of the pairs of composite numbers (C) and calculation of the value of (C). The algorithm in appendix 6 determines the set of combinations of NC pairs of prime factors. The algorithm to compute the value of a C uses this last algorithm and adds at the end the calculation of the coefficients a and b of the formulas which allows to compute the value of (C). The coefficients a and b exist for each combination of NC pairs of prime factors. The calculation is done for the main and inverse combinations. (C) = ( ) 'Select the value Ni matching with the value Nmax (Max number of primes) k=(ni-3)/2 'Loop on each value of NC from 1 to NCmax For NC=1 to NCmax 'Sign of the element of the serie a NC SignF=(-1) (NC+1) 'For each number of NC pair of primes factors, there are a MaxVal of combinations. For EachCombination=1 to MaxVal '2 cases: for NC=1, solve 2 Linear Diophantine Equations and only one for NC>1 if NC=1 then 'Only 1 pair of primes factors 'Get the primes factors: the first P1 at 3m+1 and the second P2 at 3m+2 'Get the interval for P1 and P2 IP1=(P1-3)/2 IP2=(P2-3)/2 ' Linear Diophantine Equation: A X + B Y = C a=p1*p2 A=P1 B=-1*P2 C= IP2- IP1-1 'Get the b value = x0 with a recursive function using 'Extended Euclide Algorithm Call SolveLinearDiophantineEq(A,B,C,a,IP1,IP2,x0,y0) 'Solve the second equation to get only the pair of primes not multiple of 3 A=a B=-3 IP1=x0 IP2=1 C=IP2-IP1 a=a * 3 'Get the b value = x0 Call SolveLinearEqDiophantine(A,B,C,a,IP1,IP2,x0,y0) 'Record the values a and b=x0 for each combination in an array Call RecordResult(a,x0,ArrayEachCombinationNC()) else For EachPairOfPrimesFactors=1 to NC 'Search the values a and b for the combination of the pairs Authors : François WOLF - Marc WOLF mathscience.tsoft .com Page 260

261 ' of primes factors in the array ArrayEachCombinationNC() If not SearchA&BInArray(a,b,ArrayEachCombinationNC()) then 'Search the value a2 and b2 for the last pair of primes factors Call SearchA&BInArray(a2,b2,ArrayEachCombinationNC()) A= TmpA B=a2 IP1= TmpB IP2=b2 C=IP2-IP1 'Return the value a = product of unique primes factors a=returnvalue(a,a,b) 'Get the b value = x0 Call SolveLinearEqDiophantine(A,B,C,a,IP1,IP2,x0,y0) 'Record the values a and b=x0 for the combination in an array Call RecordResult(a,x0,ArrayEachCombinationNC()) else TmpA=a TmpB=b end if Next End if Next 'Compute (C) for the value Ni selected (k=(ni-3)/2) For EachCombination=1 to MaxVal 'Return the value a and b from the array ArrayEachCombinationNC() 'Fix = function which returns the integer part of the value (C)=(C)+ SignF * Fix((k - b + a) / a) Next Next 'Display the result (C) Messagebox "(C) = "+(C) Authors : François WOLF - Marc WOLF mathscience.tsoft .com Page 261

262 APPENDIX 8: table with the pair of values (1, Ni) The table below includes the odd number 1 as a factor in the first column j = -1 with DPej=1. The odd number 1 is represented by a pair of binary values "11" on the line Ni=1. The pairs of binary values "11" overlap in the column j = -1. All odd numbers are written as the product of the two factors 1 and the odd number itself. These pairs of odd numbers are not taken into account in the study. The column j = -1 is not displayed in the others tables. Authors : François WOLF - Marc WOLF mathscience.tsoft .com Page 262