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Theory of odd numbers
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Introduction : This new theory is used to define the structure of prime numbers. An organizational
scheme of primes so emerges.
I- Theory of odd numbers
1- New theoretical approach
The idea comes from science. It is accepted that the material is organized in a number of well
above three dimensions (see superstring theory, quasicrystals ...). There are therefore hidden
dimensions, that are not perceptible but they are presents in the materials.
To understand the organization of primes, we must seek a hidden setting that would allow us
to highlight the structure of odd numbers.
The "j" parameter matches a new axis of analysis. This parameter is link to the index k of odd
numbers Ni, with
. So we have
The set of properties of odd numbers is related to the parameter "j". This parameter is the
origin of a new space W for understanding the organization of prime numbers, and simplify
the mathematical formulas for describing the properties of prime numbers.
This study helps to define the structural elements governing the organization of odd numbers.
2- Organization scheme of odd numbers
To define the organizational scheme of odd numbers, it is necessary to define the structural
elements of odd numbers.
- The first structural element can build a working space for odd numbers. It is
. This space allows to study only the odd numbers.
This element is described in trigonometric form. This makes the link between the primes
and wave mechanics.
-
The second structural element is used to define a measurement interval.
This element allows to understand the evolution of the properties of prime numbers
regardless of the magnitude of the number studied, such as oscillations in the evolution
of the density of primes.
-
The third structuring element used to define the organizational scheme of odd numbers.
This element helps explain the distribution and symmetry of prime numbers.
This theory provides the elements for understanding the organization of primes and
therefore the associated properties.
This theory provides mathematical tools that facilitate the study of prime numbers. In addition,
this study showed the practical aspects in cryptography.
Authors : François WOLF, Marc WOLF
mathscience.tsoftemail.com
Date : 28 december 2015
Page 1 of 6
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Theory of odd numbers
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II- Advantages and Uses
1- Advantages
The advantages of this theory are to define mathematical tools:
- Characterization of odd primes
- determining a formula to account for primes (Warning: this is not a sieve)
- Prime numbers generator
Cette théorie explique les propriétés des nombres premiers, telles que :
-
changing properties of the primes on the entire scale of odd integers.
linking prime numbers to complex numbers (wave mechanics)
Describe the properties in the form of simple mathematical formulas
Solve conjectures
Explain the Ulam Spiral
The fractal dimension in the distribution of prime numbers
…so on
It is possible thanks to the understanding of the organization of prime numbers to obtain new
results both in theory and practice.
2- Uses
The uses are:
- Theoretical
Solving equations with prime numbers.
We solved the equation of twin primes and the equation of Goldbach. Warning, we are
not talking about conjectures.
Solve conjectures: Legendre, twin primes,…so on
-
Practical
The main application of primes is cryptography. The science of the secret.
The primes are the raw material of cryptography.
Our theory was used to determine a formula, within the structure of the odd numbers,
having unique properties.
This formula can be used as a generator of primes.
The density of primes generated by this formula is very high relative to the natural
.
The determination of large prime number is made easier with this formula. It is faster to
find large prime numbers with this formula.
Authors : François WOLF, Marc WOLF
mathscience.tsoftemail.com
Date : 28 december 2015
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Theory of odd numbers
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The mathematical form of numbers generated by this formula, associated with the
properties of this formula, allows the use of the deterministic test of primality of LucasLehmer.
This allows to get in a deterministic way, large prime numbers up to 1000 digits.
Conclusion: The organizational scheme of primes brings a new way to study the properties of
prime numbers, as well as a convenience for number theory and for cryptography science.
Authors : François WOLF, Marc WOLF
mathscience.tsoftemail.com
Date : 28 december 2015
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Theory of odd numbers
T-Soft@mail
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ANNEX 1: Summary of the File slides
attached: NewPrimesTheoryEN.pdf
1- Graphical representation showing the organization of odd numbers
The first structuring element used to represent the indices odd primes on a two-dimensional graph.
All of the results is linked to the new dimension defined with the first structural element
("Dimension" in the physical sense: setting an axis providing relevant analysis).
2- A characterization of a prime number with sinusoidal functions
This characteristic is obtained using only a single parameter, hence simplicity to solve equations
involving prime numbers such the equation of Goldbach.
3- Relationship between primes and wave mechanics hence Riemann Hypothesis
The combination of sinusoidal functions help explain why in the space of complex numbers, the sum
of these functions provides the real-axis zero at a single point. Hence the connection with Riemann
Hypothesis.
4- Solving Equations with prime numbers such that the equation of twin primes or Goldbach's
equation
Characterizing a prime number in a simple way allows to solve equations with several primes.
5- Explanation of oscillations in measures of properties
•
Definition of a natural period
•
Stabilization of the measures with virtual primes
6- Count the primes
•
Exact Formula
•
Approximate Formula
7- Generator of prime numbers
Authors : François WOLF, Marc WOLF
mathscience.tsoftemail.com
Date : 28 december 2015
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Theory of odd numbers
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•
Determine prime numbers quickly by testing less numbers
•
Identify large prime numbers with the deterministic test of Lucas-Lehmer
8- Ulam’s Spiral
The spiral of Ulam is explained by the structure of prime numbers in the second structural element.
9- Determination of the organization of odd numbers (Pattern and combined pattern)
Combination of sinusoidal functions allows the conservation of two properties:
•
Periodicity: natural period
•
Symmetry (Conjecture of Goldbach)
Obtaining a fractal-like structure: patterns combination; each pattern has the same structure, only
the scale increases.
10- Demonstration and explanation of the conjecture of Legendre
Combined patterns are used to obtain the combination of composite numbers giving the largest
sequence of contiguous composite numbers . This distance corresponds to the conjecture legendre.
11- Distance between two primes Gap(Pn)
The maximum distance between two prime numbers is related to the occurrence of twin primes
hence the link with the conjecture of Cramér.
ANNEX 2: 15 formulas connecting all
pairs of prime numbers
15_formula_for_twin primes.pdf
1- solve equation of twin primes
2- 15 formulas connecting all pairs of prime numbers
Authors : François WOLF, Marc WOLF
mathscience.tsoftemail.com
Date : 28 december 2015
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Theory of odd numbers
T-Soft@mail
Softwre EasyPrimes
ANNEX 3: Demonstration of the pairs of
twin primes - Twin Primes Theory.pdf
I- 3 items (A), (B) and (C) in space W
II- Number of pairs of twin primes and prime numbers
III- Illustration of the infinity of prime numbers and pairs of twin primes
IV- Demonstration of the conjecture
◦
1- infinity of (A)
◦
2- infinity of (B) and infinity of prime numbers
◦
3- infinity of (C) and infinity of pairs of twin primes
◦
4- properties
Authors : François WOLF, Marc WOLF
mathscience.tsoftemail.com
Date : 28 december 2015
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