# Number Theory

## 1609 Submissions

 viXra:1609.0425 [pdf] replaced on 2016-10-27 10:44:46

### Regular Rational Diophantine Sextuples

Authors: Philip Gibbs

A polynomial equation in six variables is given that generalises the definition of regular rational Diophantine triples, quadruples and quintuples to regular rational Diophantine sextuples. The definition can be used to extend a rational Diophantine quintuple to a weak rational Diophantine sextuple. In some cases a regular sextuple is a full rational Diophantine sextuple. Ten examples of this are provided.
Category: Number Theory

 viXra:1609.0398 [pdf] replaced on 2016-10-26 15:46:30

### Inéquation de Goldbach Démonstration

Authors: BERKOUK Mohamed

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentale des nombres premiers , ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture à savoir la Sommation et la primalité des ses éléments, ...et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .
Category: Number Theory

 viXra:1609.0384 [pdf] submitted on 2016-09-26 21:46:39

### An Appell Series Over Finite Fields

Authors: Bing He, Long Li

In this paper we give a finite field analogue of one of the Appell series and obtain some transformation and reduction formulae and the generating functions for the Appell series over finite fields.
Category: Number Theory

 viXra:1609.0383 [pdf] replaced on 2016-10-01 23:01:09

### Beal Conjecture Proved & Specialized to Prove Fermat's Last Theorem

Authors: A. A. Frempong

Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2, identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y. One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. Two proof versions are covered. The first version begins with only the terms in the given equation, but the second version begins with the introduction of ratio terms which are subsequently and "miraculously" eliminated to allow the introduction of a much needed term for the necessary condition for c^z = a^x + b^y to have solutions or to be true. Each proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.
Category: Number Theory

 viXra:1609.0377 [pdf] submitted on 2016-09-26 11:05:09

### Conjectures de C.goldbach-Démonstration

Authors: Brekouk

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .
Category: Number Theory

 viXra:1609.0374 [pdf] submitted on 2016-09-26 10:09:55

### Collatz Conjecture for $2^{100000}-1$ is True - Algorithms for Verifying Extremely Large Numbers

Authors: Wei Ren

Collatz conjecture (or 3x+1 problem) is out for about 80 years. The verification of Collatz conjecture has reached to the number about 60bits until now. In this paper, we propose new algorithms that can verify whether the number that is about 100000bits (30000 digits) can return 1 after 3*x+1 and x/2 computations. This is the largest number that has been verified currently. The proposed algorithm changes numerical computation to bit computation, so that extremely large numbers (without upper bound) becomes possible to be verified. We discovered that $2^{100000}-1$ can return to 1 after 481603 times of 3*x+1 computation, and 863323 times of x/2 computation.
Category: Number Theory

 viXra:1609.0373 [pdf] submitted on 2016-09-26 10:14:45

### Induction and Code for Collatz Conjecture or 3x+1 Problem

Authors: Wei Ren

Collatz conjecture (or 3x+1 problem) has not been proved to be true or false for about 80 years. The exploration on this problem seems to ask for introducing a totally new method. In this paper, a mathematical induction method is proposed, whose proof can lead to the proof of the conjecture. According to the induction, a new representation (for dynamics) called code'' is introduced, to represent the occurred $3*x+1$ and $x/2$ computations during the process from starting number to the first transformed number that is less than the starting number. In a code $3*x+1$ is represented by 1 and $x/2$ is represented by 0. We find that code is a building block of the original dynamics from starting number to 1, and thus is more primitive for modeling quantitative properties. Some properties only exist in dynamics represented by code, but not in original dynamics. We discover and prove some inherent laws of code formally. Code as a whole is prefix-free, and has a unified form. Every code can be divided into code segments and each segment has a form $\{10\}^{p \geq 0}0^{q \geq 1}$. Besides, $p$ can be computed by judging whether $x \in_2$, $x\in_4$, or computed by $t=(x-3)/4$, without any concrete computation of $3*x+1$ or $x/2$. Especially, starting numbers in certain residue class have the same code, and their code has a short length. That is, $CODE(x \in _4)=100,$ $CODE((x-3)/4 \in _4)=101000,$ $CODE((x-3)/4 \in _8)=10100100,$ $CODE((x-3)/4 \in _8)=10101000,$ $CODE((x-3)/4 \in _{32})=10101001000,$ $CODE((x-3)/4\in _{32})=10101010000,$ $CODE((x-3)/4\in _{32})=10100101000.$ The experiment results again confirm above discoveries. We also give a conjecture on $x \in _4$ and an approach to the proof of Collatz conjecture. Those discoveries support the proposed induction and are helpful to the final proof of Collatz conjecture.
Category: Number Theory

 viXra:1609.0358 [pdf] submitted on 2016-09-25 11:28:38

### About Prime Numbers

Authors: N.Prosh

About prime numbers and new way of find prime numbers
Category: Number Theory

 viXra:1609.0353 [pdf] submitted on 2016-09-25 09:09:01

### Démonstration Des Deux Conjectures de C.goldbach

Authors: Brekouk

ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .
Category: Number Theory

 viXra:1609.0263 [pdf] replaced on 2016-10-10 20:15:27

### Fermat's Last Theorem Proved on a Single Page Revisited

Authors: A. A. Frempong

Honorable Pierre de Fermat could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Three similar versions of the proof are presented, using a single page for each version. The approach used in each proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary conditions in the solution for n = 2. The necessary conditions in the solutions for n = 2. will guide one to determine if there are solutions when n > 2.. For the first two versions, the proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1; and for the third version, on (a^2 + b^2)/c^2 = 1, with n = 2, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary conditions in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. For the first version of the proof, the proof began with reference to a right triangle. The second version of the proof began with ratio terms without any reference to a geometric figure. The third version occupies about half of a page. The third version of the proof began without any reference to a geometric figure or ratio terms. The second and third versions confirmed the proof in the first version. Each proof version is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a single page proof (considering the advantages) using inverse proportion, would be$107,250,000.
Category: Number Theory

 viXra:1609.0258 [pdf] submitted on 2016-09-17 09:37:47

### A Definition for 1 Second

Authors: Junnichi Fujii

The definition in time in the present-day physics is insufficient. Several problems which are to reconsider a definition in time and concern in time can be settled.
Category: Number Theory

 viXra:1609.0157 [pdf] replaced on 2016-09-21 20:45:45

### Beal Conjecture Proved on Half of a Page

Authors: A. A. Frempong

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple, (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system
Category: Number Theory

 viXra:1609.0123 [pdf] replaced on 2018-02-25 02:41:46

### Aﬃrmative Resolve of Riemann Hypothesis

Authors: T.Nakashima

First, we prove the relation of the sum of the mobius function and Riemann Hypothesis. This relationship is well known. I prove next section, without no tool we prove Riemann Hypothesis about mobius function. This is very chalenging attempt.
Category: Number Theory

 viXra:1609.0121 [pdf] submitted on 2016-09-09 13:54:25

### Oppermann’s Conjecture and the Growth of Primes Between Pronic Numbers

Authors: Bijoy Rahman Arif

In this paper, we are going to prove Oppermann’s conjecture which states there are at least one prime presents between first and second halves of two consecutive pronic numbers greater than one. Subsequently, we are going to prove the logarithmic sum of primes between two pronic numbers increase highest magnitude of log(4).
Category: Number Theory

 viXra:1609.0115 [pdf] submitted on 2016-09-09 08:08:37

### The Number of Primes Between Consecutive Squares: A Proof of Brocard’s Conjecture

Authors: Bijoy Rahman Arif

In this paper, we are going to find the number of primes between consecutive squares. We are going to prove a special case: Brocard’s conjecture which states between the square of two consecutive primes greater than 2 at least four primes will present. Subsequently, we will approximate the number of primes between consecutive square
Category: Number Theory

 viXra:1609.0112 [pdf] submitted on 2016-09-09 06:28:05

### Attacking Legendre’s Conjecture Using Moivre-Stirling Approximation

Authors: Bijoy Rahman Arif

In this paper, we are going to prove a famous problem concerning prime numbers. Legendre’s conjecture states that there is always a prime p with n^2 < p < (n+1)^2, if n > 0. In 1912, Landau called this problem along with other three problems “unattackable at the presesnt state of mathematics.” Our approach to solve this problem is very simple. We will find a lower bound of the difference of second Chebyshev functions using a better Moiver-Stirling approximation and finally, we transfer it to the difference of first Chebyshev functions. The final difference is always greater than zero will prove Legendre’s conjecture.
Category: Number Theory

 viXra:1609.0080 [pdf] replaced on 2016-09-16 01:53:45

### Fermat's Last Theorem Proved on Half of a Page

Authors: A. A. Frempong

Fermat's last theorem has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary condition in the solution for n = 2. The necessary condition in the solutions for n = 2 will guide one to determine if there are solutions when n > 2. The proof in this paper is based on the identity (a^2 + b^2)/c^2 = 1 for a Pythagorean triple, a, b, c, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary condition in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. The proof began without reference to any geometric figure or ratio terms. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a half-page proof (considering the advantages) using inverse proportion, would be$214,500,000.
Category: Number Theory

 viXra:1609.0052 [pdf] submitted on 2016-09-04 16:05:23

### Perhaps This is the Proof?

Authors: Aleksandr Tsybin

This problem is devoted a huge number of articles and books. So it does not make sense to list them. I wrote this note 10 years ago and since then a lot of time I tried to find the error in the reasoning and I can not this to do. I’ll be glad if someone will be finds a mistake and even more will be happy if an error will be not found.
Category: Number Theory

 viXra:1609.0048 [pdf] submitted on 2016-09-05 06:28:40

### Probable Prime Tests for Generalized Fermat Numbers

Authors: Predrag Terzic

Polynomial time compositeness tests for generalized Fermat numbers are introduced .
Category: Number Theory

 viXra:1609.0046 [pdf] submitted on 2016-09-04 16:01:51

### Algorithm of Representation of Prime Numbers Determinants of the Special Kind

Authors: Aleksandr Tsybin

For a positive integer n I construct an n × n matrix of special shape, whose determinant equals the n-th prime number, and whose entries are equal to 1,-1 or 0. Specific calculations which I have carried out so far, allowed me to construct such matrices for all n up to 63. These calculations are based on my own method for quick calculations of determinants of special matrices along with a variation on the Sieve of Eratosthenes.
Category: Number Theory

 viXra:1609.0025 [pdf] submitted on 2016-09-02 07:36:57

### The Smallest Possible Counter-Example of the Even Goldbach Conjecture if Any, Can Lie Only Between Two Odd Numbers that Themselves Obey the Odd Goldbach Conjecture

Authors: Prashanth R. Rao