Polynomial Formula at the origin of the Prime Numbers Generator

Transcription

1 Polynomial Formula at the origin of the Prime Numbers Generator WOLF François; WOLF Marc F r a n c e 2 3 a v e n u e o f W a i l l y C r o i s s y 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 / A space W dedicated to odd numbers was created to study the properties of odd numbers. From this structure emerges a measurement interval for studying the properties of odd numbers. A singular point emerges from this interval. It is the cause of the polynomial formula which provides a generator of primes. Prime numbers are predictable using arithmetic sequences that connect the values of the parameter of the formula. This formula generates reliably large prime numbers with the primality test Lucas-Lehmer. Prime numbers obtained using this formula are all connected by formulas of the second degree which explains at least part of the Ulam Spiral.

2 Generator of prime numbers Chapter IV Polynomial Formula and prediction of prime numbers... 3 Introduction Context Subject of the study Definition of the Space W Determination of a basic unit in the space W Remarkable points Interval of measurement Characteristic of a base unit Singular points inside each base unit linked by a polynomial formula Study of composed numbers generated by the polynomial formula Numerical results Relation between the composed numbers and the parameter of the polynomial formula Characteristics of the arithmetical progression of the parameter of the formula Determination of formulas of decomposition of composed numbers UNNP Characteristic of the polynomial formula Generator of prime numbers Control of the divisibility of the generated numbers Specific conditions to test the primality of a number Test of Lucas-Lehmer Other possible tests The formula allows to generate an infinity of prime numbers Generator of prime numbers Conclusion References Index of graphics Index of tables Authors: François and Marc WOLF mathscience.tsoft .com Page 2

3 Chapter IV Polynomial Formula and prediction of prime numbers Introduction The prime numbers are unpredictable; knowing the «to predict the next prime number.» first prime numbers, it is impossible We are going to show that there is a polynomial formula of the second degree which allows to predict which values of the parameter of the formula allow to generate a prime number. Moreover, a prime number obtained by this formula can be associated by a unique way to a value of the parameter of the formula. Some large prime numbers can be obtained with the help of the test of primality of Lucas- Lehmer and of prime numbers of Mersenne. Note: The graphical representations, and the verification of theoretical and numerical results are made with the software MAPLE v A specific program in C/C++ has been made to evaluate the performance of new formulas and new algorithms allowing to obtain some prime numbers. The definitions following are used in this study. Formula/ abbreviations Representation/Explications 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 uneven number or odd number which is non-prime. This is a composed number. PUN is the abbreviation of an odd number which is prime. This is a prime number. Table 1: Definitions of some terms used in this study (Lexicon) 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. Authors: François and Marc WOLF mathscience.tsoft .com Page 3

4 1- Context 1.1- Subject of the study The subject of the study is to determine a polynomial formula having a single parameter and which allows to generate some prime numbers. The study of this parameter allows to predict the values which generate a prime number with this polynomial formula. This formula is linked to the structure and organization of indexes of odd numbers. This structure is represented by the space of work W 1. In this structure, the study of prime numbers is made inside an interval of measurement which is called «base unit». Indeed, the evolution of the number of prime numbers is stable inside this base unit. In each base unit, a singular point is present. These singular points are linked by a formula to the origin of the generator of prime numbers. These elements are described in the following paragraphs Definition of the Space W. The space of work, named space W, allows to work only with numbers corresponding to the odd numbers in the space N, and to distinguish two types of odd numbers: - the Uneven Numbers which are Not Prime (UNNP) - the Uneven Numbers which are Prime (PUN). An odd number Ni is described by the formula: with k representing the index or rank of the odd number. This formula is amended as follows: - We are going to begin the number «3». The number «1» is therefore excluded of the space W. - The parameter «k» corresponds to an affined parametric function described with the help of parameters and : With Each value of the parameter «j» generates a affined function which described a series of points. The space W corresponds to the numbers generated by the formula and. This formula ( ) allows to obtain all the odd numbers in the space N, but the number «1» which does not belong to the space W. Hence the following relation corresponding to the set of odd numbers (Ni) strictly superior to «1» in the space N: 1 The article «New theory of odd numbers 8th problem of Hilbert» explains the properties of this space [3]. Authors: François and Marc WOLF mathscience.tsoft .com Page 4

5 If, then we obtain all the odd numbers non-prime (UNNP), in other words, all the odd multiples of prime odd numbers. The formula allows to obtain for each value of the parameter an infinite sequence of numbers generated with the parameter, including into the set of natural integers N. The formula is constituted of two factors: - The factor corresponds to the «period» of the sequence. The numbers of the series of points are distant between them of a value equal to the «period» of the sequence. - The factor corresponds to a «shift». The initial point of the series of points is shifted of a value equal to relative to the origin of the coordinates of the space W. A sequence corresponds to a series of points whose the period is equal to and the shift is equal to. The sequence of index, or sequence, corresponds to a sequence points generated by the sequence is named.. The series of The space W is represented Figure 1. The parameters of the Figure 1 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 «linked with dotted line.», are Description of the graphic Figure 1: 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 and Marc WOLF mathscience.tsoft .com Page 5

6 The set of points in red obtained for the set of odd multiples of prime odd numbers (UNNP). They are the composed 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 1: 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 composed 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 composed 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 [8]. 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 and Marc WOLF mathscience.tsoft .com Page 6

7 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 * +. To understand the demonstration of the polynomial formula, the concept of basic unit is introduced below. 2- Determination of a basic unit in the space W 2.1- Remarkable points For each sequence, a remarkable point described by the following formula: is defined. The remarkable points are Hence with j 0. Note: These points correspond to the odd numbers to the square in the space N. The distance between two consecutive remarkable points minus one unit will be called base unit:. This distance corresponds to the following formula:. Hence:. Each base unit is composed of a number of points equal to the distance of a base unit plus «1», because the number of points which compose a number of interval is always superior of one unit relative to the number of intervals. Hence the following formula:. Note: The remarkable points correspond to the odd numbers to the square. So an odd number Ni corresponding to a remarkable point, we have the following relation: The following formula described a remarkable point mathematically: Hence: hence After factorization, the result following is obtained: It is an odd number to the square. Authors: François and Marc WOLF mathscience.tsoft .com Page 7

8 The graphic below represents some remarkable points and some base units. Space W Figure 2: Representation of the basic schema «base unit Ugw(j).» and of the The base unit is an interval which allows to determine the properties of prime numbers Interval of measurement A base unit corresponds to the distance between two consecutive points minus 1. This distance corresponds to ; hence. This unit allows to measure, when the value represents a prime number, an augmentation of the density of points UNNP (odd number non-prime). In other words, the filling rate of the axis k increases when corresponds to a prime number. These base units allow to analyze the evolution of the density of prime numbers when tends to the infinite. These base units are used as interval of measurement Characteristic of a base unit The density of the axis, on a given interval, corresponds to the number of points UNNP relative to the total number of existing points UNNP +PUN. If the base unit does not possess any more prime numbers from a point, this means that the density of the axis is equal to «1» from the point. Indeed, a sequence comes and densifies the axis to the condition that the period of the sequence is a prime number. If the density of the odd numbers non prime for a given sequence is equal to «1», then the following values of «j» will not increase this density. There will not be any more prime numbers beyond this sequence. Authors: François and Marc WOLF mathscience.tsoft .com Page 8

9 However, it was proved by Euclide [2] that there is an infinity of prime number. There is then inevitably one or several prime numbers in each of base units. Moreover the number of prime numbers increases when the base unit increases with. The subject of this study concerns a particular point which is unique in each base unit. This point is called singular point. It allows to define a polynomial function of the second degree with one unknown. 3- Singular points inside each base unit linked by a polynomial formula 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. 3.1 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 Authors: François and Marc WOLF mathscience.tsoft .com Page 9

10 An other mathematical formulation is possible with and. This allows to obtain the following formula: 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:. Authors: François and Marc WOLF mathscience.tsoft .com Page 10

11 In the space N, we must again multiply by 2 and add 3 because. 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». 3.2 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 and Marc WOLF mathscience.tsoft .com Page 11

12 Figure 3: 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, except for the first number 7. 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. The singular points are described by a polynomial formula: This formula allows to explain an alignment of the spiral of Ulam. It is this one: This alignment is at the origin of other alignments. This formula allows to predict some prime numbers according to some prime numbers obtained previously, either directly, either indirectly by decomposing the composed number obtained by the formula. The decomposition of a number obtained by the formula is made by only with the prime numbers obtained previously. The following paragraphs explain in detail the use of the formula for the prediction of prime numbers. Note: The formula does not generate any common number with the formula of prime numbers 2, with an exception for the number 7. This number corresponds to 2 Chapter III: prime numbers Formula [5] Authors: François and Marc WOLF mathscience.tsoft .com Page 12

13 the unique solution of the following equation:. The two formulas generate some different prime numbers except for the prime number «7». We have therefore two distinct sets of prime numbers generated by two formulas of different nature: an exponential formula and a polynomial formula of the second degree. This shows the diversity of possible relations between the prime numbers. 4- Study of composed numbers generated by the polynomial formula We have created a space W which allows to work only with some odd numbers excluding the number 1, source of an unmanageable shift. We begin therefore to study the odd numbers from the number 3. This has allowed us to obtain a regular stucture of odd numbers with a basic schema. The results are the followings: 1- separating of points UNNP (odd numbers non-prime) and of points PUN (prime odd numbers). 2- determination of remarkable points which correspond to the odd numbers to the square and represent the point from which the density of UNNP increases for a value «j» given, when the value corresponds to a prime number. 3- determination of a base unit corresponding to the interval between two consecutive remarkable points GW(j). This is the interval of measurement used to study the properties of prime numbers. A singular point defined in the base unit has allowed the determination of a polynomial formula. We are going to determine the composed numbers (UNNP) generated by the polynomial formula according to the parameter «j». The composed numbers UNNP obtained by the polynomial formula are divisible by one or several prime odd numbers (PUN) obtained also by the polynomial formula. As an odd number prime described by the formula the index of this number. The value generated by the polynomial formula, for a value «j» given, is divisible by a prime number if this equation is satisfied: With ( ) and We check if the value is divisible by an odd number prime included in the interval [0, ]. We are going to test numerically this formula in order to to study the factorization of composed numbers obtained with this polynomial formula. Authors: François and Marc WOLF mathscience.tsoft .com Page 13

14 4.1- Numerical results The graphic below shows the values of, between 0 and 100, for which we obtain a prime number; the points of abscissa with a value of ordinate to 1 correspond to a prime number. When the value gives, with the formula, a prime number, the value of this number is given in the list below. But when the value of do not gives a prime number then the value in the list corresponds to zero. The values are given in the order for from zero to 100. Note : Each prime number is equal to the formula. Therefore for, we obtain the prime number 7 which is equal to hence. The first composed number is obtained for,. This number is decomposed in two factors: 7 and 17. When the number, given by the formula, is not a prime number, we decompose this number in prime factors. The first list gives the first prime number which divides the studied number: we have in bold the first divisor of composed numbers (UNNP). Indeed, the others numbers correspond to the prime numbers. This second list gives the second divisor of composed numbers. The underlined values have an other divisor given in the third list. This third list gives the third divisor of composed numbers if it exists. Result: the first composed number is obtained for,. This number is decomposed in two factors: 7 and 17. How are decomposed the numbers UNNP generated by the formula? Is there a link between «j» and the prime numbers which decompose the number generated by? Authors: François and Marc WOLF mathscience.tsoft .com Page 14

15 4.2- Relation between the composed numbers and the parameter of the polynomial formula What can we observe on the prime numbers which divide the composed numbers generated by the formula Fgwm1(j)? - All the prime numbers which divide a number non-prime (UNNP) generated by the formula, with the exception of the last divisor, can be obtained either by the prime numbers previous generated by the formula, either by the decomposition of composed numbers generated by the formula. By example, we take the first composed number generated which corresponds to the fifth value for. This value corresponds to. We have two divisors and therefore two prime numbers which are «7» and «17». The first divisor «7» corresponds to the first value given by the formula with, hence. The second divisor "17" is a prime number new which is not generated by the formula. We consider that this prime number is associated to the value of «j». Therefore, the prime number «17» is associated to j=4. We obtain also a composed number for. The value is equal to:. We do not obtain a new prime number. Indeed, "17" is a prime number which has been obtained by decomposing the number generated by the formula with j=4. The prime numbers obtained by decomposition of a number generated by Fgwm1(j) find themselves in others decompositions of numbers Fgwm1(j). Some values «j» are therefore not associated to a prime number. We obtain also a composed number for. The value is equal to:. We obtain again a new prime number "313" non generated by the formula. However, we have also the number «17» which comes from the decomposition of and therefore. The prime number «313» is associated to the value j=95. We note therefore than the prime numbers, which divide the composed number, correspond either to the prime numbers given directly by the formula, either to the prime numbers linked to the decomposition of a composed number obtained previously by the formula. If all the factors of a composed number have been obtained previously, then the value of the parameter «j» is not associated to a prime number. This case is not frequent. Otherwise, we associate the new prime number to the value of the parameter «j» which has generated the composed number. Advantage: To determine if a number obtained by the formula is a prime number or a composed number, it is sufficient to divide it by the previous prime numbers. The determination of the primality of a number, given by this formula, is therefore faster because we avoid to search the prime numbers. Moreover, it is not required of divide the number by all the prime numbers which exist between 0 and the square root of this number. Authors: François and Marc WOLF mathscience.tsoft .com Page 15

16 - For each prime number obtained either directly by the formula, either by the decomposition of a composed number obtained with the formula, we obtain two arithmetical sequences which are linked to the value associated «j». We have a trace on the graphic below for the values of included in the interval [0, 35], the prime numbers which compose the value generated by the formula. The axis of ordinates corresponds to the value Ni of divisors, and therefore of prime numbers which compose The point which corresponds the first divisor, is of color black. If the value generated is a prime number, then only this point is present. The point corresponding the second divisor is of color rouge. The third point corresponding the third divisor, if it exists, is of color blue. The fourth point corresponding the fourth divisor, if it exists, is of color green. Remarque: This graphic is represented in the space N. We are not in the space W. (Note: ) Figure 4: Graphic representation of arithmetical sequences which links the values of the parameter «j» of the formula Fgwm1(j). Authors: François and Marc WOLF mathscience.tsoft .com Page 16

17 Some series of points appear on the graphic. By example, for, we have. This is the odd number prime (PUN) which is associated to. We observe than we can determine two arithmetical sequences which give all the values of «j» for which the formula gives a number divisible by 7. The first sequence is: The second sequence is: Can these sequences be generalized to all the prime numbers associated to the values of «j»? 4.3- Characteristics of the arithmetical progression of the parameter of the formula Number of divisors The graphic below shows the regularity of arithmetical sequences obtained for each prime number generated by the formula. Moreover, we observe some intervals of values in which no one divisor exists. prime numbers Figure 5: Representation of factors of composed numbers generated by the formula Fgwm1(j) The formula generates prime numbers and composed numbers. The composed numbers are divisible only by the prime numbers obtained previously, either directly with the formula, either by the decomposition of composed numbers obtained previously with the formula. Authors: François and Marc WOLF mathscience.tsoft .com Page 17

18 By example, no one of the composed numbers studied is not divisible by the following values: 3, 5, 11, 13, 19, 29...and so on. We observe on the graphic above some intervals of values which are without divisors. By example, no one of the prime number which are located between 50 and 70, does not divide a number obtained with the formula The results show two arithmetical sequences for each prime number depends on the prime number. First sequence: prime number associated to the value. Second sequence: prime number associated to the value. associated to «j». We observe two sequences whose the period The graphic below shows than the number of divisor increases with the value «j», but it increases slightly. Indeed, we do not exceed 5 divisors for of values of «j» up to 15087, which corresponds to a value equal to. We obtain 6 divisors from up to more than The value is not exceeded, because the time of calculation become excessive. Figure 6: Numbers of factors of numbers generated by the formula Fgwm1(j) according to the parameter «j» We have defined two arithmetical sequences. Is there a link between these two sequences? Determination of two arithmetical sequences For each prime number obtained, we have the following arithmetical suites with : Authors: François and Marc WOLF mathscience.tsoft .com Page 18

19 First sequence: prime number associated to the value. With Second sequence: prime number associated to the value. With Hence We have two possibilities: 1- The formula generates a prime number for a value «j» given. This number corresponds the prime number associated to «j». Hence and, as: And Hence 2- The formula generates a composed number for a value «j» given. The decomposition of this number gives the factors which correspond to the prime numbers. If all the prime numbers have been associated to a previous value «j», then this case is not taken into account. Indeed, whatever is the prime number which would be associated to this value of «j», we would only obtain some arithmetical sequences corresponding to the subset of sequences which exist. It is not useful nor required of take into account this case. If a prime number is associated to this value «j», then we obtain with : With Hence Presentation of 3 examples. Authors: François and Marc WOLF mathscience.tsoft .com Page 19

20 1- For j=0, we obtain the prime number «7». The formula generates a prime number. For the prime number " ", we have:, with hence. We have because the prime number " " appears the first time for. The prime number «7» is therefore associated to. Hence the following sequences: First arithmetical sequence: Second arithmetical sequence: These two sequences generate some values of which give with the formula Fgwm1() of numbers divisible by : ( ) 2- For, we obtain the prime number. The formula generates a prime number. For the prime number "23", we have: and hence First arithmetical sequence: Second arithmetical sequence: 3- For, we obtain the prime number associated «17». The formula generates a composed number corresponding to the prime numbers «7» and «17». The prime number «7» is associated to. The prime number «17» is therefore associated to. For the prime number "17", we have and hence Indeed, for, we obtain the value. The value «17» appears therefore the first time for hence. First arithmetical sequence: Second arithmetical sequence:...and so on All these arithmetical sequences give the values of for which the formula number associated to j, as. generates some composed numbers divisible by the prime Can we determine of formulas which allow to describe mathematically the prime numbers which decompose the UNNP obtained by the formula? Authors: François and Marc WOLF mathscience.tsoft .com Page 20

21 4.4- Determination of formulas of decomposition of composed numbers UNNP Decomposition mathematical of the formula Fgwm1(j) for the two arithmetical sequences Each arithmetical sequence gives a series of value for, hence and. We are going to demonstrate that these values of allow to generate, with the help of the formula, some composed numbers divisible by the prime number associated to «j», as. We have two possible cases: 1- Case 1: the formula generates a prime number with the parameter. As the following formula: *The first arithmetical sequence Hence ( ) corresponds to the following formula: With The formula ( ) is therefore divisible by. Therefore, we can calculate the factors of composed numbers: For j=0 and n=1, we have:, with, Fgwm1(0)=7. Hence PUN(7)=41 For j=0 and n=2, we have:, with, Fgwm1(0)=7. Hence PUN(14)=137 For j=1 and n=1, we have:, with, Fgwm1(1)=23. Hence PUN(24)=113 *The suite arithmetical ( ) corresponds to the following formula: Authors: François and Marc WOLF mathscience.tsoft .com Page 21

22 With The formula ( ) is therefore divisible by. Therefore, we can calculate the factors of composed numbers: For j=0 and n=0, we have:, with. Hence PUN(4)=17 For j=0 and n=1, we have:, with. Hence PUN(11)=89 For j=1 and n=0, we have:, with. Hence PUN(19)=73 2- Case 2: the formula generates a composed number with the parameter. As a composed number which possesses of divisors or factors. corresponds to the greater divisor. This divisor is the prime number associated to the value «j». We consider in this example that there is only two factors: a prime number obtained with a value previously named and the factor. The factor corresponds to a value obtained previously with the formula hence. Note: by considering several factors, we obtain the same conclusions. Hence: Hence The factor depends on the following values of :, hence the following notation: We would have the same conclusions if the divisor was linked to the divisors obtained with of the previous values of such as *As the arithmetical sequence corresponding to the following formula: then we obtain the formula hereafter: ( ) Authors: François and Marc WOLF mathscience.tsoft .com Page 22

23 = The formula ( ) is therefore divisible by ( ). Therefore, we can calculate the factors of composed numbers as showed by the following examples: For, and, we have: and hence. Hence No new prime number. Indeed, the number 7 is associated to and the number 17 is associated to. For, and, we have and. Hence PUN(38)=367 * As the arithmetical sequence corresponding to the following formula: then we obtain the formula hereafter: ( ) = The formula ( ) is therefore divisible by ( ). Therefore, we can calculate the factors of composed numbers as showed by the following examples : For, and, we have:,. Hence PUN(10)=31 For, and, we have:,. Hence PUN(27)=191 We are going to considerer the case of a factor obtained by the decomposition of a composed number, in order to show than the results obtained are identical. Indeed, the composed numbers obtained by the formula are always divisible by some prime numbers obtained previously, either directly with the formula, either by decomposition of the composed number. Authors: François and Marc WOLF mathscience.tsoft .com Page 23

24 As the formula which generates a prime number with the parameter. As the two following factors: and As the arithmetical sequence corresponding to the following formula: ( ) With We obtain the following formula: ( ) The formula is therefore divisible by. Therefore, we can calculate the factors of composed numbers as by example: For,,, and, we have:,, hence PUN(41)=233 These results led to obtain a set of prime numbers linked to the formula connecting some singular points. This formula corresponds to a polynomial formula of the second degree. The spiral of Ulam highlights some alignments of prime numbers. These alignments correspond to the polynomial formulas of the second degree of type. Is there a link between the alignments of the spiral of Ulam and the formula? Spiral of Ulam We have represented the prime numbers obtained with the formula on the graphic below. The axis of abscissa corresponds the parameter. The axis of ordinates corresponds to the value of divisors and therefore of prime numbers which compose Authors: François and Marc WOLF mathscience.tsoft .com Page 24

25 The point which corresponds the first divisor, is of color black. If the value generated is a prime number, then only this point is present. The point corresponding to the second divisor is of color red. The third point corresponding to the third divisor, if it exists, is of color blue. The fourth point corresponding to the fourth divisor, if it exists, is of color green. By example, as the following number: ; 7 is the first factor, 79 is the second, 103 is the third and 3313 is the fourth factor. When there is only one factor, this means than the number generated by the formula is a prime number. Prime Numbers Prime numbers obtained with the formula Fgwm1(n) Figure 7: This graphic shows the evolution of factors of composed numbers obtained with the formula according to the parameter. These results allow to explain at least partly the spiral of Ulam. Indeed, the prime numbers are all linked by of formulas of the second degree. These relations correspond to the formulas previously defined. By example, when the decomposition is limited to 2 factors with, the second factor respects the following formulas: Authors: François and Marc WOLF mathscience.tsoft .com Page 25

26 For the sequence, we have : For the sequence, we have: When the decomposition is limited to two factors with as first factor following formulas: For the sequence, we have : with, the second factor respects the. with For the sequence, we have : with. When the number of factors increases, the formula becomes more complex; however all the formulas are linked to the parameter to the power two. These are only quadratic functions. The value corresponds to the value of the parameter associated the prime number. If the formula is constituted by three factors ( ), then the first factor an a value associated equal to. This explains why the formulas are not traced on the graphic. Indeed, the value «abscissa.» does not correspond to the value on the axis of These formulas complexes explain the link with the spiral of Ulam. What is the origin of the Ulam Spiral? The basic units Ugw(j) help to understand this property The diagram below shows the change in the size, or range, of base units according to "j". Authors: François and Marc WOLF mathscience.tsoft .com Page 26

27 Space N Interval of a base unit: Distance between two singular points S(j) Singular points : The base units correspond to an interval in which the distance increases in an arithmetic progression following a reason point equal to 14 The interval between two singular points also follows an arithmetic progression with and. With Indeed, we have : The table below provided examples. j Suite with an initial The formula Fgwm1(j) used to connect one and only one point of each basic unit. We can define a family of functions for linking other prime numbers using a formula of the second degree such as that Fgwm1(j) For example, we can start with the prime number 11 that follows the prime number 7, and connect it with the prime number 29 that follows the prime number of 23. And so on. The family of functions that we have defined is as follows: Authors: François and Marc WOLF mathscience.tsoft .com Page 27

28 With even number et prime number. The arithmetic sequence that connects the dots matches. Note : it is possible to connect points that are separated by several base units. The formulas are of the second degree but reason always correspond to a multiple of 8. We have shown in Chapter I [3] that the number of primes increases in basic units when "j" increases. This means that the number of possibilities for connecting primes using arithmetic sequences increases. Demonstration : All composite numbers in the space W are represented by the formula with. We have shown in [3] in paragraph 3.2, that new composite numbers appear in a basic unit from the remarkable point. These points match for each sequence "j" to the value. To describe the points corresponding to composites from that point, we define the parameter belonging to natural integers, hence. For obtaining the index of prime numbers, we must shift the value of Y units because otherwise we will get only composite numbers. We obtain a formula allowing us to get indices of primes and indices of composites : To get the values of the numbers Ni, we apply the formula, hence : Hence With even number and odd composite number or odd prime number. Dirichlet's theorem on primes in arithmetic progression (Gustav Lejeune Dirichlet) : «For all natural integers g >0 and m prime to one another, there is an infinite number of primes of the form, where is a positive integer.» We will describe the arithmetic progression that connects the numbers generated by the family of functions. As the parameter is even, we have. Authors: François and Marc WOLF mathscience.tsoft .com Page 28

29 The numbers generated by the functions correspond to the following formula (cf. the table above for an example with X = 12 and P = 7 hence Y = 6) : With ( ) ( ) So we get the case of an arithmetical progression : which corresponds to an odd number and. We have therefore g>0 and m prime to one another. However, the variable is equal to. The variable belongs to the set of natural integers N, but corresponds to only a subset of de N. We have therefore an incomplete arithmetic progression. is this incomplete arithmetic progression contains an infinity of prime numbers? We conjecture that YES it is. Paragraph 5.6 shows that there are an infinity of prime numbers in arithmetic progression defined by X = 12 and P = 7. This is the case of the function. Special case:? We know that the terminals of base units consist of two consecutive odd numbers squared. So we have an odd number corresponding to the even number squared least one unit in the middle of that range. The index of the odd number is equal to : with Hence The value of this number is equal to: ( ). Hence A unit is added to provide the even number squared, hence: Hence ( ) Authors: François and Marc WOLF mathscience.tsoft .com Page 29

30 with X=16 and P=17, we get and and. This case should also contain an infinity of prime numbers. Note : We have the following relationship: Hence With and. Conclusion : The set of prime numbers can be connected by formulas of the second degree. This property is linked to the organization of the numbers described using base units. The interval of these base units increases following an arithmetic progression. This property allows to connect the points between base units using the formula of the second degree. The origin of the spiral Ulam therefore arises from the nature of these base units which corresponds to the difference of two consecutive odd numbers squared. Within this interval, the position of a number corresponds to the first odd number squared (minimum limit of the range which is a function of the parameter ) at which is added a shift according to the parameter "j" and a constant C. Hence the formulas of the second degree type. The numbers obtained using these formulas correspond to an incomplete arithmetical progression. Indeed, only a subset of an arithmetic progression is represented by those numbers. However, we conjecture that these sets should contain an infinity of prime numbers. 5- Characteristic of the polynomial formula 5.1- Generator of prime numbers The formula generates of prime numbers and of composed numbers. These composed numbers can be decomposed owing to the prime numbers previously associated to the values of the parameter «j». - For each prime number (PUN) generated with the formula, we obtain, as showed in the previous paragraph, two arithmetical sequences ( and ). These sequences correspond to the values of the parameter of the formula. The formula generates some composed numbers (UNNP) with the values «j» obtained with the arithmetical sequences. Authors: François and Marc WOLF mathscience.tsoft .com Page 30

31 j prime numbers Sequences arithmetical obtained with the prime numbers of the formula We obtain therefore the 2 following arithmetical sequences: with with if and only if is a prime number Therefore and will give some UNNP. Table of arithmetical sequences of prime numbers obtained with the formula - The composed numbers obtained with the formula can be defined and decomposed owing to the previous prime numbers. For j=4, we obtain a composed number. Indeed, the value Fgwm1(4)=119 is decomposed in two prime numbers: "7" and "17". It is possible to predict that the value gives a composed number. Indeed, for j=0, we obtain two arithmetical sequences which give the values of for which the formula generates some multiples of "7". These two arithmetical sequences are: and. The sequence, for n=0, gives a value equal to "4", hence. This means that for, the number generated by the formula ( ) is a composed number divisible by the prime number "7". The two arithmetical sequences are linked by the following formula: : NEW prime numbers Sequences arithmetical obtained with the prime numbers PUN. These numbers PUN are obtained by decomposing the number UNNP generated with the formula 4 119=17* We obtain therefore the two arithmetical sequences following: for 7 287=41* for Authors: François and Marc WOLF mathscience.tsoft .com Page 31

32 10 527=31* =89* is the prime number obtained by decomposing the number corresponds the new prime number which appears during the decomposition of the number. By example with, we obtain : If no one prime number new appears, then the value is not =73* associated to a prime number; this is a case without prime number. The value j=18 is in this case of figure. We can not have 2 new prime numbers for a value given =71*7* Therefore, if we have a prime number PUN(j) associated to «j», the sequences and give some UNNP with the formula, as and divisible by PUN(j). Table of arithmetical sequences of prime numbers linked to the composed numbers obtained with the formula For each prime number associated to a value, we determine two arithmetical sequences which allow to predict which are the following values of which give some composed numbers UNNP with the formula. The values of «j» which do not belong to these sequences generate some PUN. It is therefore possible to predict the following prime number generated by the formula owing to the previous values generated by this same formula. This new prime number is obtained either directly by the formula, either by decomposing the composed number into prime numbers. Be careful, all the composed numbers do not allow to obtain a new prime number. In other words, all the values of «j» do not have an associated prime number. The disadvantage to have to determine the following prime number is that we must use the formula for each value of in the increasing order from zero. We obtain thus a generator of prime numbers. Authors: François and Marc WOLF mathscience.tsoft .com Page 32

33 5.2- Control of the divisibility of the generated numbers We have showed than the numbers generated by the formula are divisible neither by 3 nor by 5. We have showed that we can predict, with the help of arithmetical sequences associated to a prime number, the values of the parameter which allow to obtain with the formula a value divisible by this prime number. We have showed than the prime numbers which are not associated to a value can not divide the numbers. By example, the numbers following are not associated to any value such: { }. - It is therefore possible to search some values of for which we are certain that the number generated will not be divisible by one or several prime numbers selected in the list of prime numbers associated to «j». By example: we observe on the Figure 5 that the prime number 7 is the number which divides the most often a composed number. The composed numbers generated by the formula, and which have as divisor 7, are obtained for some values of which are represented by two arithmetical sequences: and. If the we take then, we are certain to obtain some numbers non divisible by 7 with the following formula:. If we wish to obtain some numbers non divisible by 7 and by 17, then we must resolve a Diophantine equation of the first order with two unknown by the method of Bezout. We search the numbers of identical value generated by the sequences Hence and. Solution: and We obtain then. We added the number 1 in order to the number generated by the formula is divisible neither by 7 nor by 17 hence : For the prime numbers 7 and 23, we have the following equation to resolve to obtain some numbers divisible by 7 and by 23: Hence the solution We added «1» to obtain of values non divisible by 7 and by 23, hence, and we obtain the formula: Authors: François and Marc WOLF mathscience.tsoft .com Page 33