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

**Authors:** Philip Gibbs

**Comments:** 13 Pages.

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

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

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

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

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

**Authors:** Bing He, Long Li

**Comments:** 16 Pages.

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

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

**Authors:** A. A. Frempong

**Comments:** 6 Pages. Copyright © by 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

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

**Authors:** Brekouk

**Comments:** 12 Pages.

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

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

**Authors:** Wei Ren

**Comments:** 17 Pages.

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

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

**Authors:** Wei Ren

**Comments:** 22 Pages.

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 $x \in[0]_2$, $x\in[1]_4$, 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, $CODE(x \in [1]_4)=100,$
$CODE((x-3)/4 \in [0]_4)=101000,$ $CODE((x-3)/4 \in
[2]_8)=10100100,$ $CODE((x-3)/4 \in [5]_8)=10101000,$ $CODE((x-3)/4
\in [1]_{32})=10101001000,$ $CODE((x-3)/4\in [3]_{32})=10101010000,$
$CODE((x-3)/4\in [14]_{32})=10100101000.$ The experiment results
again confirm above discoveries. We also give a conjecture on $x \in
[3]_4$ 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

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

**Authors:** N.Prosh

**Comments:** 6 Pages.

About prime numbers and new way of find prime numbers

**Category:** Number Theory

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

**Authors:** Brekouk

**Comments:** 12 Pages.

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

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

**Authors:** A. A. Frempong

**Comments:** 6 Pages. Copyright © by 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

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

**Authors:** Junnichi Fujii

**Comments:** 2 Pages.

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

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

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by 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

[10] **viXra:1609.0123 [pdf]**
*replaced on 2017-09-12 06:14:32*

**Authors:** T.Nakashima

**Comments:** 5 Pages.

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

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

**Authors:** Bijoy Rahman Arif

**Comments:** 5 Pages.

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

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

**Authors:** Bijoy Rahman Arif

**Comments:** 4 Pages.

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

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

**Authors:** Bijoy Rahman Arif

**Comments:** 3 Pages.

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

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

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by 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

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

**Authors:** Aleksandr Tsybin

**Comments:** 3 Pages.

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

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

**Authors:** Predrag Terzic

**Comments:** 5 Pages.

Polynomial time compositeness tests for generalized Fermat numbers are introduced .

**Category:** Number Theory

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

**Authors:** Aleksandr Tsybin

**Comments:** 14 Pages.

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

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

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

The even Goldbach conjecture states that any even integer greater than four may be expressed as the sum of two odd primes. The odd Goldbach conjecture states that any odd integer greater than seven must be expressible as a sum of three odd primes. These conjectures remain unverified. In this paper we explore the possible constraints that exist on the smallest possible counterexample of the even Goldbach conjecture. We prove that the odd numbers immediately flanking the smallest counterexample of the even Goldbach conjecture are themselves expressible as the sum of three odd primes and are therefore consistent with the odd Golbach conjecture.

**Category:** Number Theory

[1] **viXra:1609.0012 [pdf]**
*submitted on 2016-09-01 00:00:52*

**Authors:** D. D. Somashekara, S. L. Shalini, K. N. Vidya

**Comments:** 15 Pages.

In this paper, we give an alternate and simple proofs for Sear’s three term 3 φ 2 transformation formula, Jackson’s 3 φ 2 transformation formula and for a nonterminating form of the q-Saalschütz sum by using q exponential operator techniques. We also give an alternate proof for a nonterminating form of the q-Vandermonde sum. We also obtain some interesting special cases of all the three identities, some of which are analogous to the identities stated by Ramanujan in his lost notebook.

**Category:** Number Theory