T-Soft@mail mathscience.tsoftemail.com Odd Primes Theory W: Structure and organization of odd primes (8th problem of Hilbert) WOLF François and WOLF Marc PhDs in materials science Ceramists engineers IT engineers 2015 Authors : WOLF François;WOLF Marc Page 1 sur 14 08/09/2015
Summary of the study of prime numbers Scientific / Mathematical study REF : TWPS-001 V1.1 Target All For information purposes Version - Date Issuer Status / Track Changes V1.1 30/07/2015 François WOLF Marc WOLF Document creation Purpose of the paper Summary of the study of a new theory on prime numbers. Keywords Space W, prime numbers, odd numbers, Riemann conjectures of Goldbach, Legendre, Brocard, twin primes, primes characterization, properties, resolution of equations with two prime numbers (binary solution). Summary This new theory provides the foundation for understanding the observed properties of prime numbers. This naturally leads to the mathematical description of these properties and validation of several conjectures. Authors : WOLF François;WOLF Marc Page 2 sur 14 08/09/2015
Table of Contents SUMMARY OF THE STUDY OF PRIME NUMBERS...2 SUMMARY...4 INTRODUCTION...5 I. THE WORKSPACE DEDICATED TO ODD NUMBERS ALLOWED TO DESCRIBE THE PROPERTIES OF PRIMES THANKS TO STRUCTURAL ELEMENTS...5 I.1 IN ORDER TO FOCUS THE STUDY ON THE ODD NUMBERS, IT IS NECESSARY TO BUILD A DEDICATED WORKSPACE ONLY TO ODD NUMBERS....5 I.2 THE STUDY OF PRIMES CONSIDERED AS A PHYSICAL SYSTEM INDUCES THE DEFINITION OF STRUCTURAL ELEMENTS SUCH AS A MEASUREMENT INTERVAL AND A MEASUREMENT UNIT IN ORDER TO DESCRIBE THE PROPERTIES OF PRIME NUMBERS REGARDLESS OF MAGNITUDE OF THE NUMBERS. 7 II. THIS NEW THEORY PROVIDES NEW MATHEMATICAL TOOLS AND A NEW INSIGHT INTO THE 8 TH PROBLEM OF HILBERT....9 II.1 THE PERFORMANCE OF THE NEW FORMULA FOR PRIMES HAVE ALLOWED THEIR USE WITH DESKTOP COMPUTERS....9 II.2 THE 8TH PROBLEM OF HILBERT IN RELATION TO THREE CONJECTURES INVOLVING THE DISTRIBUTION OF PRIMES, IS EXPLAINED THANKS TO STRUCTURING ELEMENTS OF PRIMES.10 CONCLUSION... 14 Authors : WOLF François;WOLF Marc Page 3 sur 14 08/09/2015
Summary Prime numbers have become increasingly important in our daily life since the advent of computer communication and electronic transactions such as financial transactions on the Web. Indeed, the communications require a means for encrypting the sensitive information to enable secure communication. This medium is based on a special property of odd numbers that is the difficulty of factoring large odd numbers. Indeed, the properties of prime numbers remain a mystery since Euclid. A new theory of odd numbers reveals the mysterious organization of primes. This theory is based on the study of indices of odd numbers. This study led to the definition of a working space dedicated to odd numbers. This space takes into account the two dimensions that allow to study the primality of an odd number: its value and the value of its square root. The properties of odd numbers are based on three structural elements. Each element allows us to understand the behavior of the properties of odd numbers including the odd primes. These elements led to associate a sinusoidal function to each odd number. As in science with the nature of the material, a wave / corpuscle duality is defined with the odd numbers. Odd numbers behave as a physical system. A natural frequency has been demonstrated. This sinusoidal formula or wave function provides a characterization of an odd prime number. This characterization tool is use to solve equations whose problems are related to odd primes such as twin primes conjecture and Goldbach's conjecture. This duality is at the origin of validating the Riemann's hypothesis. The definition of measuring elements helped to understand the distribution of prime numbers. This understanding has led to the obtaining of mathematical functions that describe the following properties: - The exact count of primes, - The maximum distance between two consecutive prime numbers that allows to validate Legendre's conjecture and Brocard's conjecture, - The prediction of a prime number Other properties observed have been explained such that the spiral Ulam, the fractal nature of the distribution of prime numbers, the oscillatory character properties These results help to understand the evolution of the properties of prime numbers regardless of the magnitude of the number. This theory opens a new way of research on odd numbers. This new approach allows a better understanding of the organization of these numbers thus becoming less mysterious. Authors : WOLF François;WOLF Marc Page 4 sur 14 08/09/2015
Introduction Prime numbers are the basis of all electronic transactions security systems. They are the basis of the science of cryptology. However, the prime numbers and their distribution are a mystery since Euclid. Neither their structure nor any organization scheme is known. A new approach to the study of primes reveals their mysterious organization. This approach is based on the study of indices of odd numbers. The creation of a dedicated workspace with odd numbers helped define structural elements thus providing a better understanding of the organization of primes. These structural elements lead to the definition of new mathematical tools, and bring new light on the main conjecture of prime numbers such as those defined by the 8 th problem of Hilbert. I. The workspace dedicated to odd numbers allowed to describe the properties of primes thanks to structural elements The primes are constituted of odd numbers with the exception of one pair prime number "2". The study of primes therefore concern odd numbers. I.1 In order to focus the study on the odd numbers, it is necessary to build a dedicated workspace only to odd numbers. Odd integers can be cut out into three sets: the primes, the composite numbers and the identity element "1". The space of odd numbers, called space W (WOLF) consists solely of two complementary sets within this space: the primes and the composite numbers. This space is defined from the formula of odd numbers,. In order to remove the identity element, the numbers begin with the value "3" hence. The parameter k is called "index of the odd number Ni." This index is described using two parameters "j" and "n" belonging to the set of natural numbers, with the following formula: ( ). This formula is regarded as the first structuring element that are three in number This formula allows to construct a two-dimensional graph that we call space W. We have on the ordinate axis the parameter "j", and on the abscissa the index k. In fact, for each value of the parameter "j", we get the odd multiples of odd numbers described by ( ). We were able to define a work space in which each point obtained by the formula ( ) is the index of an odd number. The set of composite numbers is obtained when the "n" parameter is strictly greater than zero. The complementary set of composite numbers in the space W corresponds to prime numbers. This formula involves the characterization of a prime number using sinusoidal formulas. Authors : WOLF François;WOLF Marc Page 5 sur 14 08/09/2015
Determining the primality of an odd number is to test if this number is divisible by at least one of odd numbers less than or equal to the square root of that number. Using this formula, we note that for each value of the parameter "j", we can define a sinusoidal function : ( ( ) ). The indices k correspond to the intersection of this function with the abscissa axis. So if the index k of the odd number studied does not match any of the values generated by the formula ( ) for values of "j" included in the zero range and square root of the odd number, then this number is a prime number, hence the following theorem: Theorem : The following equation characterizes the odd prime numbers Ni except for the odd number «3» : ( ) ( ( ) ) With and * + With and In the case of an odd prime number, all sinusoidal functions are different from zero. Note: only the prime numbers 2 and 3 can not be characterized by the formula. We have associated with each odd number a wave function hence the concept of duality wave / corpuscle, the corpuscle representing the value of an odd number. This characterization is a mathematical tool to solve problems involving two primes (binary problem) or more. Indeed, X odd numbers of index, with odd numbers whose value is greater than 3, are prime numbers if and only if the multiplication of the following function is nonzero: ( ) We will apply this characterization for twin primes and the equation of Goldbach. - The twin primes correspond to two consecutive odd numbers. In the space W, the distance between the index of these two numbers is 1 unit. We consider the index of the first number equal to and the index of the next number equal to. Two consecutive odd numbers are prime if and only if ( ) ( ). Solving this inequality, provided in this study corresponds to the resolution of a trigonometric equation. - Similarly, the Goldbach equation involves two primes. We are only interested in the odd primes. The equation of Goldbach has a parameter corresponding to an even number that must be decomposed into two prime numbers, and two unknowns representing the two primes to search. The formula for characterizing a prime number allow to add an equation with two unknowns corresponding to the two prime numbers to be determined. Thus we have a system of two equations and two unknowns. This study has solved this system and understand why the number of decomposition of an even number increases when the value of the even number increases. Authors : WOLF François;WOLF Marc Page 6 sur 14 08/09/2015
Note : The resolution of the equations does not solve the associated conjectures. I.2 The definition of the space W dedicated to the odd numbers has enabled the characterization of odd primes. The definition of the structural elements is used to describe the behavior of the properties of prime numbers. The study of primes considered as a physical system induces the definition of structural elements such as a measurement interval and a measurement unit in order to describe the properties of prime numbers regardless of magnitude of the numbers. To study the properties of prime numbers, it is necessary to define the underlying structure to the organization of odd numbers. To do this, as in the case of the study of a physical phenomenon, we must define a measurement interval, and a measuring unit. The measuring interval, which corresponds to the second structural element, is defined between two successive remarkable points. Each remarkable point corresponds to a perfect square that is an odd number squared. Between two consecutive odd numbers squared such as this interval, all composite numbers in this interval are multiples of a prime number that is lower or equal to the lower bound of this range which is 37 in this example. This structural element allows us to understand the properties of prime numbers and mathematically it allows to describe the evolution of these properties to any order of magnitude of the number. Indeed, the measurements made within this range are stable because the composite numbers are multiples of a defined and fixed set of primes For each value of the parameter "j", we have a remarkable point corresponding to the number( ) squared. So, for each value of "j" we have an interval equal to ( ) ( ( ) ). The parameter "j" is the index of the square root of the perfect number ( ), and therefore the index of the number ( ). The parameter "j" sets the link between an odd number and its square root. This is the fundamental element of this new theory The set of properties of prime numbers, such as the density of prime numbers, the characterization of a prime number,... is defined with the parameter "j". The evolution of these properties depends on this parameter. The value of the measurement interval increases as the value of the parameter "j" increases. In addition, whenever the value ( ) corresponds to a prime number, new composite numbers which are multiple of the prime number, appear. The study of the properties of the primes in these intervals show that measured values per unit "j" vary from one interval to another even when the next value of "j" does not give a value ( ) corresponding to a prime number. The reason for the variability of the measurements is the fact that the measurement interval is too small to hold all the possible combinations* between the prime numbers. Considering the primes as a physical system and associating a wave function to each prime number, then the number of possible combinations between these wave functions will be located in an interval named natural period of the system equal to the multiplication of prime numbers between zero and ( ) for a "j" value Authors : WOLF François;WOLF Marc Page 7 sur 14 08/09/2015
given. The inverse of this value is the eigenfrequency of the system. The realization of measures in these specific periods shows an evolution of the properties of prime numbers without oscillation. The measures are stable. * The combinations of multiple of prime numbers correspond to the set of possible distances between n sets of multiple numbers of a prime number. Example: all combinations between the multiple numbers of the number 3 and multiple numbers of the number 5 are contained in an interval equal to 3*5=15. Note : The primes in these natural periods are considered as virtual primes. Indeed, beyond the measurement interval defined by the remarkable points for a given value "j", it is necessary to take into account the multiple numbers of prime numbers whose index "jp" is greater than the provided value "j", in order to be able to determine the primality of numbers located in this natural period. Not taking into account the multiple numbers of prime numbers whose index is greater than the given value "j", the numbers determined as prime numbers are either composites or prime numbers hence the label "prime virtual number". The evolution of the properties of prime numbers can be described by mathematical formulas based on the parameter "j". These formulas are used to extrapolate the evolution of these properties regardless of the magnitude of the number studied thanks to the stability of the properties within the measurement interval and the natural period. The study of the maximum distance between two primes requires the definition of the third structuring element, the combined pattern. A pattern matching three consecutive measurement intervals and therefore three consecutive value of "j". The combined pattern is associated with all the preceding patterns. The patterns are accumulated. The accumulation of patterns does not change the internal structure of patterns. The basic pattern is beginning to j = 0. It is the basis schema that organizes the primes. It helps to understand the symmetry of prime numbers. The accumulation of the patterns confers on primes a randomness in their occurrence. This is due to the fact that the combination of these structures forms a fractal structure. The combined pattern is used to define within its natural period a maximum distance ( ) between two consecutive primes virtual. This theoretical maximum distance, depending on "j", is described mathematically by the following formula : ( ) ( * +). It is equivalent to the following formula which depends on the number "n" corresponding to a natural number: ( ). We have shown that this distance is related to the occurrence of twin primes. The study of this distance within natural periods showed that the number of pair of virtual twin prime numbers are increasing without oscillations. Within the measurement intervals, we observe an increase of the maximum distance value which oscillates around a value that follows a logarithmic squared function, hence the maximum distance tends to this given value ( ( ( )) ) when tends toward infinity. This explains the Cramér's conjecture. This theoretical maximum distance demonstrated the Legendre's conjecture conjecture. and Brocard's Authors : WOLF François;WOLF Marc Page 8 sur 14 08/09/2015
Note : This maximum distance is the unit of measurement for the density of prime numbers. This space W dedicated to odd numbers helped define the structural elements governing the organization of prime numbers. These items shed new light on the nature of prime numbers, especially with the wave function associated with odd numbers. The properties of prime numbers should be described with the parameter "j", which allows to obtain the changes in these properties at any scale. The knowledge of the organization of prime numbers leads to the definition of new mathematical tools to study these numbers. It also leads to explain and/or demonstrate conjectures such as these of the 8 th problem of Hilbert. II. This new theory provides new mathematical tools and a new insight into the 8 th problem of Hilbert. The characterization of an odd prime is a new tool to solve mathematical equations involving two or more prime numbers. Other tools have been defined for determining primes. II.1 The performance of the new formula for primes have allowed their use with desktop computers. Successful methods count of primes today use an algorithmic method called sieve. Indeed, existing formulas are not efficient. However, these sieve methods have the disadvantage of having to take a lot of computer memory resource which limits their use to appropriate computers. This study has identified an exact formula for counting primes. This formula requires little memory resources allowing its use with desktop computers. The limit is related only to the computing power of the computer. Indeed, the computing time follows a proportional power : ( ) (. An approximate formula has improved performance: ). The generation of large prime numbers is a need for encryption of information called cryptography. This study has yielded a formula equivalent to ( ) ( ) which generates a percentage of prime number more important than the conventional percentage ( ). The numbers generated by this formula have two features : - The first characteristic is that the primes generated with the formula can not belong to a pair of twin primes, with the exception of the first number "7". - The second characteristic is fundamental to use the deterministic test Lucas-Lehmer. In fact, the numbers generated by the formula are divisible neither by the prime number 3, nor by the prime number 5. This allows to set the test parameter. This parameter is the same for all the numbers generated by the formula. The test is therefore more efficient. As with Mersenne primes, it is possible to use this test with this formula to determine deterministically very large prime numbers. Authors : WOLF François;WOLF Marc Page 9 sur 14 08/09/2015
Note : Within each measurement interval is a singular point. This formula is used to describe all of these singular points. Moreover, this formula, which has a single parameter, revealed two other features: - It is possible to predict which values of the parameter will enable to generate a prime number. When the formula generates a composite number, you can break it down into prime numbers through knowledge of numbers generated with previous parameter values. This formula generates an infinite number of primes. We obtain a generator of prime numbers. - This formula of the second degree represents an alignment of the Ulam s spiral. It generates composite numbers which can be decomposed into prime numbers. These prime numbers are all interconnected by formulas of the second degree. This explains at least partly the alignments of the spiral Ulam. In seeking the origin of twin primes, we determined within the structure of the odd numbers a particular formula linking a subset of prime numbers. This formula is called natural formula of primes:. We have developed an algorithm to determine the values of the parameter "m" that generate primes with the formula. This parameter "m" is related to arithmetic progressions:. The parameters and are linked to parameter "j", but only a part of "j" values allows to obtain values for and. With this formula, we have developed an algorithm to solve the following Diophantine equation:. All of the formulas is bonded to the parameter "j". This is central to understanding the structure of primes and mathematically describe the relationships between primes. Using this setting, structuring elements allow to shed new light on the conjectures of prime numbers and in particular those concerning the 8 th problem of Hilbert. II.2 The 8th problem of Hilbert in relation to three conjectures involving the distribution of primes, is explained thanks to structuring elements of primes. The mathematical demonstrations of conjectures are based on counting of composites in our study. Indeed, the direct count of primes, through the fractal nature of their structure, can be made neither theoretically via usable algebraic formulas, nor conveniently using efficient algorithms. But all composite numbers can be described using the algebraic formula ( ( ) ). This provides a formula to count the composite numbers and the primes. This study proposes a mathematical proof of the conjecture of twin primes. This demonstration is based on counting of pairs of composite numbers which are either two composite numbers or a composite number and a prime number. This count is done within each measuring interval which is related to the parameter "j". We show that the pair of twin primes increases according to the following mathematical formula: with. Note : the origin of twin primes comes from the parameter "j" of indices of the odd numbers ( ). The term "j" corresponds to a shift of the value «( )». This offset is at the origin of the set of pairs of twin primes except for these pairs (3,5) and (5,7). Authors : WOLF François;WOLF Marc Page 10 sur 14 08/09/2015
The structure of the patterns explains the reasons of the origin of the symmetry of prime numbers. The resolution of the Goldbach's equation allows us to understand the increasing number of decompositions of an even number into two prime numbers when the value of even number increases. These elements, linked to structural elements, helped define an equation with a single parameter corresponding to the even number. If this equation has a solution then the conjecture is invalidated. Otherwise, the conjecture is validated mathematically. We have not solved this equation. The question remains open. However, a mathematical proof of Goldbach's conjecture is defined. The space W can be extended to rational numbers. This allows to observe a single point where all wave functions associated with odd numbers are in phase. This means that the sum of these waves is zero and is zero only on the real axis at the abscissa point regardless of the value of parameter "j". This can be extrapolated in the Riemann s space thanks to two elements: - The wave functions This extract comes from this reference : Gilles LACHAUD «L hypothèse de Riemann» http://iml.univ-mrs.fr/editions/biblio/files/lachaud/2001b.pdf. «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 ( ) ( ( )) Authors : WOLF François;WOLF Marc Page 11 sur 14 08/09/2015
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? 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: ( ) ( ) ( Axe réel ( ) ) And 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 : the imaginary axis with abscissa. All solutions are therefore on. 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 constant corresponds to constant corresponds to the value on the real axis.. In space of Riemann, this - The eigenfrequency of the system prime numbers A link between the Riemann hypothesis was done with quantum physics by Hugh Montgomery and Freeman Dyson. Authors : WOLF François;WOLF Marc Page 12 sur 14 08/09/2015
The wave functions are used to represent primes as a duality wave / corpuscle (The corpuscle is the value of odd number). Odd numbers can be regarded as a physical system with an eigenfrequency which changes with each occurrence of a prime number. This is the sinusoidal component which establishes the link between the fundamental elements of mathematics (prime numbers) and the fundamental elements of physics (atoms). 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. These elements thus validate the Riemann's hypothesis. Authors : WOLF François;WOLF Marc Page 13 sur 14 08/09/2015
Conclusion The setting a new workspace dedicated to the odd numbers has allowed to identify the structural elements governing the organization of primes. The properties of the primes have been explained through these elements. They are all related to the basic parameter "j" which is involved in all the mathematical formulas and tools primes. This parameter is important because it makes the connection between the value of a number and the value of the square root. This relation corresponds to the test of primality of a natural number.. The structural elements have allowed a better understanding of the properties of prime numbers which yielded successful formulas in the search for prime numbers. In addition, it has provided new insights into the conjectures of primes, including the 8 th problem of Hilbert.. The duality wave / corpuscle of odd numbers helped to link the basic elements of mathematics that are prime numbers, and the basic elements of matter that are atoms. This theory of primes strengthens the technical choice for encrypting information using the RSA algorithm. Indeed, the determination of prime numbers requires a significant number of operations that requires a lot of computation time and so a high cost of calculation that result of the fractal structure of primes. The discovery of the space W opens a new area of knowledge about odd numbers and mainly on the odd primes. This knowledge will undoubtedly contribute in the near future to demonstrate many other conjectures and discover other properties on prime numbers thus becoming less mysterious. Authors : WOLF François;WOLF Marc Page 14 sur 14 08/09/2015