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2016 - 1601(14) - 1602(18) - 1603(77) - 1604(56) - 1605(28) - 1606(18) - 1607(23) - 1608(19) - 1609(21)

Any replacements are listed further down

[1325] **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

[1324] **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

[1323] **viXra:1609.0360 [pdf]**
*submitted on 2016-09-25 13:51:26*

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

The odd Goldbach conjecture states that every odd integer greater than seven may be expressed as the sum of three odd primes whereas the even Goldbach conjecture states that every even integer greater than four may be expressed as the sum of two odd primes. Harald Helfgott has provided a convincing proof of the Odd Goldbach conjecture. The even Goldbach conjecture, however remains unproven. We start with the assumption that first counterexamples of both the even and odd Goldbach conjectures exist. We identify a relationship between the first counterexample of the odd conjecture with the counterexample of the even. Invoking Helfgott’s proof of the Odd Goldbach conjecture, we study the value of the first possible even counter-example and it must be infinitely large so that the first odd counter-example may also be infinitely large thereby non-existent.

**Category:** Number Theory

[1322] **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

[1321] **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

[1320] **viXra:1609.0263 [pdf]**
*submitted on 2016-09-18 00:13:23*

**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

[1319] **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

[1318] **viXra:1609.0157 [pdf]**
*submitted on 2016-09-13 00:19:51*

**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 where a, b, and c are relatively prime positive integers. 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

[1317] **viXra:1609.0123 [pdf]**
*submitted on 2016-09-09 13:09:01*

**Authors:** T.Nakashima

**Comments:** 6 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

[1316] **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

[1315] **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

[1314] **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

[1313] **viXra:1609.0080 [pdf]**
*submitted on 2016-09-06 23:20:51*

**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, 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

[1312] **viXra:1609.0059 [pdf]**
*submitted on 2016-09-05 11:45:39*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

Near m, the destance of primes is lower order than logm. This is the Legendre’s conjecture.

**Category:** Number Theory

[1311] **viXra:1609.0058 [pdf]**
*submitted on 2016-09-05 11:49:34*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

This is the positive answer of Gilbreath's conjecture

**Category:** Number Theory

[1310] **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

[1309] **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

[1308] **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

[1307] **viXra:1609.0030 [pdf]**
*submitted on 2016-09-03 03:04:18*

**Authors:** Alexander K

**Comments:** 2 Pages.

Fermat last theorem, fermat-catalan conjecture, beal conjecture proof

**Category:** Number Theory

[1306] **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

[1305] **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

[1304] **viXra:1608.0449 [pdf]**
*submitted on 2016-08-31 17:53:09*

**Authors:** Joe Chizmarik

**Comments:** 2 Pages. This is a proof by contradiction.

We first prove a weak form of Fermat's Last Theorem; this unique lemma is key to the entire proof. A corollary and lemma follow inter-relating Pythagorean and Fermat solutions. Finally, we prove Fermat's Last Theorem.

**Category:** Number Theory

[1303] **viXra:1608.0439 [pdf]**
*submitted on 2016-08-30 21:46:42*

**Authors:** Watcharakiete Wongcharoenbhorn

**Comments:** 4 Pages. English

We study on the cycle in the Collatz conjecture and there is something surprise us. Our goal is to show that there is no Collatz cycle

**Category:** Number Theory

[1302] **viXra:1608.0429 [pdf]**
*submitted on 2016-08-31 09:51:31*

**Authors:** Gyeongmin Yang

**Comments:** 5 Pages.

This article is based on how to look for a closed-form expression related to the odd zeta function values and explained what meaning of the expansion of the Euler zigzag numbers is.

**Category:** Number Theory

[1301] **viXra:1608.0390 [pdf]**
*submitted on 2016-08-28 19:28:48*

**Authors:** Lucas Allen

**Comments:** English, 4 pages, ideas and examples

This paper presents a method of calculating powers and sums of powers using binomial coefficients. The method involves finding analogues of Pascal's triangle for each power and then showing that powers and sums of powers are the sums of binomial coefficients multiplied by constants. The constants are unique for each power. This paper presents a general idea and not a formal proof.

**Category:** Number Theory

[1300] **viXra:1608.0375 [pdf]**
*submitted on 2016-08-28 00:35:34*

**Authors:** Nathan Sponder

**Comments:** 10 Pages.

We discuss the asymptotics of the sum $\sum_{k=1}^{m} e^{ \frac{{\ln(k)}^n}{k} }-1$ for $n \geq 0 $. Our main interest is to show the asymptotics of this sum and show expressions for the constants tied to the asymptotics of the sum as well as in particular show the properties of the constants associated with the sum.

**Category:** Number Theory

[1299] **viXra:1608.0367 [pdf]**
*submitted on 2016-08-26 21:10:58*

**Authors:** Raymond Aschheim, Carlos Castro Perelman, Klee Irwin

**Comments:** 28 Pages. Submitted to IJGMMP

Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential,
and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the $asymptotic$
(very large $N$) region. The ordinates $\lambda_n$ are the positive imaginary parts of the nontrivial zeta zeros in the critical line : $ s_n = { 1\over 2} + i \lambda_n $.
The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches
not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula ($without$ any truncations) for the number $ N (E) $ of zeroes in the critical strip with imaginary part greater than $0$ and less than or equal to $E$.

**Category:** Number Theory

[1298] **viXra:1608.0356 [pdf]**
*submitted on 2016-08-25 20:25:15*

**Authors:** Zhang Tianshu

**Comments:** 15 Pages.

Positive integers which can operate to 1 by the set operational rule of the conjecture and positive integers got via contrary operations of the set operational rule are one-to-one correspondence unquestionably. In this article, we classify positive integers to prove the Collatz conjecture by the mathematical induction via operations of substep according to confirmed two theorems plus a lemma in advance.

**Category:** Number Theory

[1297] **viXra:1608.0144 [pdf]**
*submitted on 2016-08-12 21:14:51*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 06 Pages.

Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper the Multiplicative Coupled Fibonacci Sequence has been generalized for r t h order with some new interesting properties.

**Category:** Number Theory

[1296] **viXra:1608.0140 [pdf]**
*submitted on 2016-08-12 21:20:08*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

In this paper, some properties of k Fibonacci and k Lucas numbers are derived and proved by using matrices S and M. The identities we proved are not encountered in the k Fibonacci
and k Lucasnumber literature.

**Category:** Number Theory

[1295] **viXra:1608.0139 [pdf]**
*submitted on 2016-08-12 21:22:02*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 04 Pages.

In this paper we defined general matrices Mk(n,m),
Tk,n and Sk(n,m) for k-Fibonacci number. Using these matrices we find some new summation properties for k-Fibonacci and k-Lucas
numbers.

**Category:** Number Theory

[1294] **viXra:1608.0138 [pdf]**
*submitted on 2016-08-12 21:23:18*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 08 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. In this paper, I present some properties of multiplicative coupled Fibonacci sequences of fourth order under two specific schemes.

**Category:** Number Theory

[1293] **viXra:1608.0137 [pdf]**
*submitted on 2016-08-12 21:24:35*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. There are 32 different schemes of generalization for the Fibonacci sequences of fifth order in the case of two sequences [1]. I introduce their recurrent formulas below.

**Category:** Number Theory

[1292] **viXra:1608.0135 [pdf]**
*submitted on 2016-08-12 21:46:46*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their associated matrices, this approach is di®erent and never tried in k Fibonacci sequence
literature.

**Category:** Number Theory

[1291] **viXra:1608.0134 [pdf]**
*submitted on 2016-08-12 21:48:49*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 14 Pages.

In this paper, we defined new relationship between k Fibonacci and k Lucas sequences using continued fractions and series of fractions, this approach is different and never tried in k Fibonacci sequence literature.

**Category:** Number Theory

[1290] **viXra:1608.0133 [pdf]**
*submitted on 2016-08-12 21:50:43*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 08 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one
sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first
introduced coupled Fibonacci sequences of second order in additive form. There are 8 different schemes of generalization for the Tribonacci sequences in the case of two sequences & there are 16 different schemes of generalization for the Tetranacci sequences in the case of two sequences. I introduce their recurrent formulas below.

**Category:** Number Theory

[1289] **viXra:1608.0128 [pdf]**
*submitted on 2016-08-12 13:47:27*

**Authors:** Michael M. Ross

**Comments:** Pages.

By defining a function for a linear equation (of slope-intercept form) to be a composite generator, I am able to show that a subset of odd-value slopes must always have even solutions. I apply this function to generate a parity truth table for any perfect square interval that demonstrates the unequal cardinality of the subsets of odd and even composites. Using elementary set theory I demonstrate that this inequality is deterministic, eliminating the possibility of a prime-free interval. This method successfully attacks Legendre's conjecture, providing a logical-conceptual framework for a formal proof.

**Category:** Number Theory

[1288] **viXra:1608.0100 [pdf]**
*submitted on 2016-08-09 21:54:15*

**Authors:** Choe Ryong Gil

**Comments:** 20 pages, 4 tables

In this paper we consider the Riemann hypothesis (RH) by the Euler function and primorial numbers. The paper consists of two parts. In the first part, we find a new sufficient condition for the RH from well known Robin theorem and prove it under a certain condition, which would be called the condition (d). In the second one, we prove that the condition (d) holds.
Keywords; Riemann hypothesis, Euler function, Primorial number.

**Category:** Number Theory

[1287] **viXra:1608.0082 [pdf]**
*submitted on 2016-08-08 11:34:13*

**Authors:** Bengt Månsson

**Comments:** 10 Pages.

A formula giving the $n$:th number of a sequence defined by a recursion formula plus initial value is deduced using generating functions. Of particular interest is the possibility to get an exact expression for the n:th term by means a recursion formula of the same type as the original one. As for the sequence itself it is of some interest that the original recursion is non-linear and the fact that the sequence grows very fast, the number of digits increasing more or less exponentially. Other sequences with the same rekursion span can be treated similarly.

**Category:** Number Theory

[1286] **viXra:1608.0062 [pdf]**
*submitted on 2016-08-06 04:06:33*

**Authors:** Lucas Allen

**Comments:** English, 6 pages, equations and examples

This paper presents a “formula” (more or less) for prime numbers in a specific interval. This formula is then used to partially prove the Goldbach conjecture and the twin primes conjecture. The proofs are incomplete however and have not been reviewed by anyone.

**Category:** Number Theory

[1285] **viXra:1607.0569 [pdf]**
*submitted on 2016-07-31 17:30:32*

**Authors:** Hervé G.

**Comments:** 19 Pages.

It is presented an elementary proof that Beta(3)=Pi^3/32. Beta is the Dirichlet Beta function.

**Category:** Number Theory

[1284] **viXra:1607.0557 [pdf]**
*submitted on 2016-07-30 15:05:24*

**Authors:** Simon Plouffe

**Comments:** 3 Pages.

An extension of a known result of Ramanujan is used to produce sums with exponential terms that gives a representation of many prime numbers.

**Category:** Number Theory

[1283] **viXra:1607.0551 [pdf]**
*submitted on 2016-07-29 12:19:22*

**Authors:** Richard Broxley Omeston

**Comments:** 1 Page.

In this paper I show how the equivalence of the summation of the Móbius function with the Zeta function allows for proof of the Riemann Hypothesis.

**Category:** Number Theory

[1282] **viXra:1607.0536 [pdf]**
*submitted on 2016-07-28 15:35:29*

**Authors:** José de Jesús Camacho Medina

**Comments:** 1 Page.

The present article shows an unpublished formula to evaluate twin primes, the formula is based on the theorem of Wilson and contains mathematical functions such that greatest common divisor, factorial and floor function.

**Category:** Number Theory

[1281] **viXra:1607.0522 [pdf]**
*submitted on 2016-07-27 10:29:42*

**Authors:** Matilda Walter

**Comments:** 3 Pages.

Lemoine - Levy Conjecture, probably the least known of the 'Goldbach Conjectures', states that
every positive odd integer > 5 is a sum of a prime and double of a prime. We present a simple sieve
procedure for finding all existing solutions to the problem for any given odd number > 5.

**Category:** Number Theory

[1280] **viXra:1607.0468 [pdf]**
*submitted on 2016-07-25 01:02:22*

**Authors:** Dhananjay P. Mehendale

**Comments:** 3 Pages

This paper proposes a generalised ABC conjecture and assuming its validity settles a generalised version of Fermat’s last theorem.

**Category:** Number Theory

[1279] **viXra:1607.0437 [pdf]**
*submitted on 2016-07-23 12:06:52*

**Authors:** Kunle Adegoke

**Comments:** 15 Pages.

Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular cases of the more general formulas derived here.

**Category:** Number Theory

[1278] **viXra:1607.0434 [pdf]**
*submitted on 2016-07-23 12:24:01*

**Authors:** Terubumi Honjou

**Comments:** 7 Pages.

Chapter12. Challenge "proof of the Lehman expectation".
A mathematics difficult problem biggest in history.
[1] With the mathematics difficult problem "proof of the Lehman expectation" biggest in history.
[2] I challenge the difficult problem Lehman expectation that rejected the geniuses challenge for 150 years.
[3] It is challenged the mystery of the prime number, a mathematics difficult problem biggest in history, proof of the Lehman expectation.
[4] Neology of the Lehman expectation. A point of intersection that all 0 points are straight.
[5] An elementary particle pulsation principle founds a door of the Lehman expected proof.

**Category:** Number Theory

[1277] **viXra:1607.0400 [pdf]**
*submitted on 2016-07-21 22:44:27*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We give an illogical point in Dirichlet's proof, therefore the used infininite descent is not
powered in his proof

**Category:** Number Theory

[1276] **viXra:1607.0381 [pdf]**
*submitted on 2016-07-20 11:59:45*

**Authors:** Peter Bissonnet

**Comments:** 10 Pages.

Prime products are analyzed from various points of view, with an emphasis on graphical representation and analysis. A prime product N is determined to have two integer coordinates D and m. These coordinates are related to the solutions of a parabola, as well as to right triangles, in what the author calls a ‘backbone - rib’ representation. A prime number or a prime product fall on three dimensional helices, which can be represented in two dimensions as sets of parallel lines. If a prime or a prime product can be represented by 6s - 1, then helix 1 or H1 is designated; if a prime or a prime product can be represented by 6s + 1, then helix 2 or H2 is designated. The integer s is really a composite number, which can be represented as s = r + n, where r is the row number and n is the grouping number called the complex number, both determined from the two dimensional representation of the double helices.
It is also discovered that, due to the mathematical form relating N to D and m, that there must be Lorentz - like transformations between N, D, and m and a new set Nʹ, Dʹ and mʹ; however, the concept of velocity and the speed of light seem out of place in this instance. Nevertheless, the question is asked as to whether or not prime products can be considered to be away to unite relativity and quantum mechanics, which also depends upon integers in a large measure.

**Category:** Number Theory

[1275] **viXra:1607.0360 [pdf]**
*submitted on 2016-07-18 13:58:50*

**Authors:** Reuven Tint

**Comments:** 13 Pages. Original written in Russian

A variant of the solution with the help of Bill hypothesis direct evidence "Great" Fermat's theorem elementary methods rows. New are "invariant identity" (keyword) and obtained by us in the text, the identity of the work, which allowed directly to solve the FLT, and several others.

**Category:** Number Theory

[1274] **viXra:1607.0359 [pdf]**
*submitted on 2016-07-18 15:28:12*

**Authors:** Matilda Walter

**Comments:** 2 Pages.

We present a simple sieve algorithm for finding all existing solutions to the binary Goldbach
problem for a given even number 2N > 4.

**Category:** Number Theory

[1273] **viXra:1607.0178 [pdf]**
*submitted on 2016-07-15 06:08:08*

**Authors:** Zhang Tianshu

**Comments:** 25 Pages.

In this article, we first classify A, B and C according to their respective odevity, and thereby get rid of two kinds from AX+BY=CZ. Then, affirmed AX+BY=CZ in which case A, B and C have at least a common prime factor by several concrete equalities. After that, proved AX+BY≠CZ in which case A, B and C have not any common prime factor by mathematical induction with the aid of the symmetric relations of positive odd numbers concerned after divide the inequality in four. Finally, reached a conclusion that the Beal’s conjecture holds water via the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.

**Category:** Number Theory

[1272] **viXra:1607.0161 [pdf]**
*submitted on 2016-07-12 19:35:56*

**Authors:** Michael Pogorsky

**Comments:** 7 Pages.

The theorem is proved by means of general algebra. The proof is based on set of polynomials a=uwv+v^n; b=uwv+w^n; c=uwv+v^n+w^n and their modifications deduced as required for a,b,c to meet condition of being integers and to satisfy equation a^n+b^n=c^n. The equation also requires existence of positive integers u_p and c_p such that a+b is divisible by u_p^n and c is divisible by u_p c_p. Then the contradiction revealed between expressed by deduced polynomials a^n+b^n and c^n.

**Category:** Number Theory

[1271] **viXra:1607.0159 [pdf]**
*submitted on 2016-07-13 11:11:18*

**Authors:** Maaninou Youssef

**Comments:** 1 Page.

by this theory you can chek any number if it is a prime or not
also you can generat a new prime number

**Category:** Number Theory

[1270] **viXra:1607.0095 [pdf]**
*submitted on 2016-07-07 16:22:11*

**Authors:** Kolosov Petro

**Comments:** 2 Pages.

The basic and short description of proposed entry to Wolfram MathWorld, Finite Difference, starting from line ”An n-th power has a constant
nth finite difference. For example, take n=3 and make a difference table”.
The entry appears to to wide the properties of high order difference of
n-th power.

**Category:** Number Theory

[1269] **viXra:1607.0094 [pdf]**
*submitted on 2016-07-07 16:24:21*

**Authors:** Kolosov Petro

**Comments:** 2 Pages.

The basic and short description of proposed entry to Wolfram MathWorld, ”Power”, properties section. The entry appears to connect the
high order finite difference and derivative of Power function.

**Category:** Number Theory

[1268] **viXra:1607.0093 [pdf]**
*submitted on 2016-07-07 16:26:41*

**Authors:** Kolosov Petro

**Comments:** 12 Pages.

This paper presents the way to make expansion for the next form function: y =
x n , ∀(x,n) ∈ N to the numerical series. The most widely used methods to solve this
problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that
is, derivative and integral are inverse operators). The paper provides the other kind of
solution, except above described theorems.

**Category:** Number Theory

[1267] **viXra:1607.0087 [pdf]**
*submitted on 2016-07-07 12:02:36*

**Authors:** Ryan Zielinski

**Comments:** 81 Pages. This work is licensed under the CC BY 4.0, a Creative Commons Attribution License.

What is a general expression for the sum of the first n integers, each raised to the mth power, where m is a positive integer? Answering this question will be the aim of the paper....We will take the unorthodox approach of presenting the material from the point of view of someone who is trying to solve the problem himself. Keywords: analogy, Johann Faulhaber, finite sums, heuristics, inductive reasoning, number theory, George Polya, problem solving, teaching of mathematics.

**Category:** Number Theory

[1266] **viXra:1607.0072 [pdf]**
*submitted on 2016-07-07 03:56:39*

**Authors:** Korn Rakpradit

**Comments:** 47 Pages.

The opinions of this work are stalking and proving in details the derivation of Riemann Zeta Function and Riemann Hypothesis, which Riemann did roughly for more than 150 years ago without proof, and correcting all mistakes about the boundaries of the integrals and the undefined or divergent functional equations which caused very big problems to this Riemann Hypothesis.
Proof or disproof of Riemann Hypothesis’s derivation will be very useful for mathematicians and physicists because it is widespread in use though it is unproved officially for a long time.

**Category:** Number Theory

[1265] **viXra:1607.0058 [pdf]**
*submitted on 2016-07-06 02:42:43*

**Authors:** Tomer Shushi

**Comments:** 4 Pages.

In this paper we give a general proof for the irrationality property of numbers which take a certain form of infinite sums. As special cases we prove the irrationality of infinite sums involving pi and e constants. This proof is obtained in the spirit of the well known Fourier's proof for the irrationality of e.

**Category:** Number Theory

[1264] **viXra:1607.0029 [pdf]**
*submitted on 2016-07-03 05:30:52*

**Authors:** Andrea OSSICINI

**Comments:** 16 Pages.

It is shown that an appropriate use of so-called << double equations >> by Diophantus provides the origin of the Frey elliptic curve and from it we can deduce an elementary proof of Fermat’s Last Theorem

**Category:** Number Theory

[1263] **viXra:1607.0003 [pdf]**
*submitted on 2016-07-01 03:27:48*

**Authors:** Pingyuan Zhou

**Comments:** 13 Pages. If so-called strong Goldbach number sequence introduced in this paper is acceptable then the existence of strong Goldbach number sequence will imply both Goldbach and twin prime conjectures.

Abstract: In this note, we present a new and direct appraoch to prove the Goldbach conjecture that if the existence of the limit of

**Category:** Number Theory

[1262] **viXra:1606.0345 [pdf]**
*submitted on 2016-06-30 11:00:58*

**Authors:** Scott B. Guthery

**Comments:** 8 Pages.

An algorithm for recursively generating the sequence of solutions of a prime constellation is described. The algorithm is based on an polynomial equation formed from the first n elements of the constellation. A root of this equation is the next element of the sequence.

**Category:** Number Theory

[1261] **viXra:1606.0327 [pdf]**
*submitted on 2016-06-29 07:46:25*

**Authors:** Mohamed Ouiter

**Comments:** 7 Pages.

From the definition that a number is Prime if he admits that two dividers and any number not prime can it write as a product of prime factors, I was able to identify a simple law that governs the distribution of prime numbers.

**Category:** Number Theory

[1260] **viXra:1606.0310 [pdf]**
*submitted on 2016-06-28 09:09:00*

**Authors:** Peter Bissonnet

**Comments:** 14 Pages.

When a subset of sequential integers is arranged in a specific way, there appears a paired set of slanted straight lines along which prime numbers seem to naturally arrange themselves in a repeated fashion. This arrangement can further be observed to be a two dimensional surface applicable to the cylinder. If this arrangement is on a piece of paper, then one can fold the paper in the form of a cylinder, and the paired set of slanted straight lines unite at the page edges to form a double helix winding down the cylinder. Paired primes are thus seen to be an association between primes residing on these two paired helixes, and further analysis shows that there appears to be two types of paired primes. Is the prime number double helix nature’s secret for jump starting life (DNA) and thereby defeating entropy (at least in the initial stage) by creating order from order instead of order from disorder? The prime number trends examined exhibit right handed chirality; left handed chirality is obtained by symmetry considerations.

**Category:** Number Theory

[1259] **viXra:1606.0306 [pdf]**
*submitted on 2016-06-28 09:30:16*

**Authors:** Peter Bissonnet

**Comments:** 2 Pages. This is a Russian journal named "Almanac of Modern Science and Education" and can be found at the following link: http://www.gramota.net/eng/editions/5.html

This very short paper illustrates a mathematical method which illustrates the possibility of reducing a prime product (N = PQ) from two independent variables, namely the two original prime numbers P and Q, to only one independent variable which determines both prime numbers.

**Category:** Number Theory

[1258] **viXra:1606.0302 [pdf]**
*submitted on 2016-06-27 21:24:29*

**Authors:** T.Nakashima

**Comments:** 1 Page.

reration of recurring decimal and primitive root

**Category:** Number Theory

[1257] **viXra:1606.0271 [pdf]**
*submitted on 2016-06-25 20:21:56*

**Authors:** T.Nakashima

**Comments:** 1 Page.

Conway’s problem is correct.

**Category:** Number Theory

[1256] **viXra:1606.0184 [pdf]**
*submitted on 2016-06-17 22:38:57*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 18 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Relative Metric That Generates A Field Super-Set To The Fields Generated By
Various Distinct Relative Metrics’.

**Category:** Number Theory

[1255] **viXra:1606.0173 [pdf]**
*submitted on 2016-06-17 08:50:55*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 16 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Holistic Beauty Primality Tree Of Any Set, Universal Growth Of Any Given Set’.

**Category:** Number Theory

[1254] **viXra:1606.0154 [pdf]**
*submitted on 2016-06-15 07:31:44*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 14 Pages.

In this research investigation, the author has presented a theory of ‘The Universal
Irreducible Any Field Generating Metric’.

**Category:** Number Theory

[1253] **viXra:1606.0144 [pdf]**
*submitted on 2016-06-15 00:52:10*

**Authors:** Anthony J. Browne

**Comments:** 5 Pages.

A form of the exponential Mangoldt function is derived using indicator functions. The function's relationship to other important number theoretic functions are derived and discussed.

**Category:** Number Theory

[1252] **viXra:1606.0136 [pdf]**
*submitted on 2016-06-14 00:55:44*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 13 Pages.

In this research investigation, the author has presented a theory of ‘The Universal Irreducible Any Field Generating Metric’.

**Category:** Number Theory

[1251] **viXra:1606.0135 [pdf]**
*submitted on 2016-06-14 00:58:08*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 13 Pages.

In this research investigation, the author has presented a theory of ‘The Universal Irreducible Any Field Generating Metric’.

**Category:** Number Theory

[1250] **viXra:1606.0118 [pdf]**
*submitted on 2016-06-13 03:17:38*

**Authors:** Reuven Tint

**Comments:** 6 Pages. written Russian

Let us prove that invariant identity is used for the proof of the FLT and Beal conjecture.

**Category:** Number Theory

[1249] **viXra:1606.0109 [pdf]**
*submitted on 2016-06-11 15:07:52*

**Authors:** William R. Blickos

**Comments:** 12 Pages.

This paper provides a method using periodic functions to check for primality, count factors, list factors,
calculate the exact prime distribution, and determine the Nth prime. It describes this method in a straightforward
manner, from one equation to the next, using graphs between each key step to help quickly visualize the
reasoning and result of each maneuver. It concludes with brief afterthoughts and ongoing questions about the
technique.

**Category:** Number Theory

[1248] **viXra:1606.0108 [pdf]**
*submitted on 2016-06-11 15:09:58*

**Authors:** William R. Blickos

**Comments:** 4 Pages.

A Trigonometric Series is crafted that gives the number of factors of x for all x. It is then manipulated to list all those factors. It is then used to give the exact distribution of the primes. It is then changed into a recursive sequence that generates the nth prime. Finally, a Product Polynomial Series is suggested as an improved alternative to the original. There is a series of questions in the second section regarding the functions.

**Category:** Number Theory

[1247] **viXra:1606.0088 [pdf]**
*submitted on 2016-06-09 13:43:39*

**Authors:** Hitesh Jain

**Comments:** 4 Pages.

To find out necessary conditions for any number to be expressed as difference of 4th powers of 2 natural numbers and maximum number of ways.

**Category:** Number Theory

[1246] **viXra:1606.0072 [pdf]**
*submitted on 2016-06-07 11:02:32*

**Authors:** William R. Blickos

**Comments:** 7 Pages.

Surfaces representing the primes and composites, the upper twin prime, the difference between squares, and the relations among them, are used to select input for a quadratic in such a way as to always generate a Twin Prime. Because each input generates a unique Twin, and because the input set is infinite, there are infinitely many twin primes.

**Category:** Number Theory

[1245] **viXra:1606.0065 [pdf]**
*submitted on 2016-06-06 21:27:48*

**Authors:** Igor Turkanov

**Comments:** 29 Pages.

This theorem is based on the study of holomorphy functions and on the fact that near the singularity point of the imaginary part of some rational function can accept an arbitrary preassigned value.

**Category:** Number Theory

[1244] **viXra:1605.0311 [pdf]**
*submitted on 2016-05-31 10:35:22*

**Authors:** José de Jesús Camacho Medina

**Comments:** 6 Pages.

This article disseminates a series of formulas that generate primes numbers as a product of the investigations of the author since the year 2011. This document describes general patterns on the entities primales and conceived an order in its distribution.

**Category:** Number Theory

[1243] **viXra:1605.0310 [pdf]**
*submitted on 2016-05-31 12:07:57*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 47 Pages. In French. Paper submitted to the Journal Algebra and Number Theory.

The paper gives a complete proof of the conjecture of BEAL followed by some numerical examples.

**Category:** Number Theory

[1242] **viXra:1605.0247 [pdf]**
*submitted on 2016-05-24 00:51:23*

**Authors:** Hajime Mashima

**Comments:** 4 Pages.

This is the expectation that "two or more of the natural number 4/n will be represented by the sum of the three unit fractions".

**Category:** Number Theory

[1241] **viXra:1605.0245 [pdf]**
*submitted on 2016-05-23 13:23:29*

**Authors:** Terubumi Honjou

**Comments:** 4 Pages.

The new equation by the elementary particle pulsation hypothesis.
Arrow of time turns. In response to it, mass m changes with +1, 0, -1.
Arrow of time turns.
It is the same as the arrow of time of the Schrodinger equation.
Arrow of time turns.
It is the same as a material wave of the elementary particle pulsation hypothesis.
Current physics is always +mc²>0. Elementary particle mass m does not change.
A pulsation principle suggests a super-high-speed change of elementary particle mass m.

**Category:** Number Theory

[1240] **viXra:1605.0229 [pdf]**
*submitted on 2016-05-22 07:48:37*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

As non-trivial Riemann zero's grow larger on the 1/2 critical line so does the distance grow proportionally between 0 and 1. Therefore, a non-trivial Riemann zero will not be outside the critical strip or off of the critical line.

**Category:** Number Theory

[1239] **viXra:1605.0221 [pdf]**
*submitted on 2016-05-21 11:18:19*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

The purpose of this paper is to share a simple way to solve a Sodoku in three steps. (P = NP ).

**Category:** Number Theory

[1238] **viXra:1605.0201 [pdf]**
*submitted on 2016-05-19 07:29:23*

**Authors:** Terubumi Honjou

**Comments:** 6 Pages.

Proof (article) for the Lehman expectation by Mr.Dobranju became the topic.
I think. A prime number and an illustration of commentary of the fusion of the physics make up for prism interpretation. (Mr.Dobranju)
Various wavelengths of the material wave appear as light (rainbow) of each color in the horizon (three-dimensional space).
The summary of the article is called 0 points of prism interpretation.
Light is branched into the light of various colors by a prism and is projected on a straight line.
The figure of prime number, physics fusion by the elementary particle pulsation principle is similar to 0 points of prism interpretation.
A material wave and the point of intersection with the horizon (straight line of Lehman) are zero points of the Lehman expectation.
A prime number and elementary particle pulsation principle hypothesis.(2).
(Fusion of a prime number and the physics. )

**Category:** Number Theory

[1237] **viXra:1605.0195 [pdf]**
*submitted on 2016-05-18 18:45:13*

**Authors:** A. A. Frempong

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

Fermat's last theorem has been proved on a single page. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, .the general equation, c^n = a^n + b^n holds, followed by showing that if
n > 2, the general equation, c^n = a^n + b^n does not hold. Let a,, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also let the acute angle between the hypotenuse and the horizontal be denoted by theta. The proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin space for it in his paper.

**Category:** Number Theory

[1236] **viXra:1605.0164 [pdf]**
*submitted on 2016-05-14 20:42:59*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

I introduce a congruence that restates the characterization of primes that have 2 as a quadratic residue, non-residue.

**Category:** Number Theory

[1235] **viXra:1605.0162 [pdf]**
*submitted on 2016-05-15 02:34:43*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

This short paper presents a new form of the Liouville function. To the author's best knowledge, this relationship is previously unknown.

**Category:** Number Theory

[1234] **viXra:1605.0160 [pdf]**
*submitted on 2016-05-14 14:20:12*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

In this short paper I present a closed form formula for the right half of Pascal's triangle.

**Category:** Number Theory

[1233] **viXra:1605.0140 [pdf]**
*submitted on 2016-05-13 11:27:31*

**Authors:** José de Jesús Camacho Medina

**Comments:** 2 Pages.

This article disseminates a new mathematical concept designed by the author in the year 2015 to which entitled: "Magic Dart", this mathematical entity has a property exquisite not before discovered by the mathematical community, which will be of great pleasure for those who are interested in the educational and recreational mathematics.
One of the activities that are carried out by the author in his daily work is the teaching practice and it is there where attempts to combine the objectives entrusted to him in the delivery of the classes with the implementation of the Recreational Mathematics, always seeking to expand and disclose in an entertaining way the mathematical knowledge that are manifest in a classroom.

**Category:** Number Theory

[1232] **viXra:1605.0134 [pdf]**
*submitted on 2016-05-12 19:39:00*

**Authors:** Terubumi Honjou

**Comments:** 6 Pages.

Online posting to YOUTUBE.
Einstein continued looking for four-dimensional space.
I discovered 4-dimensional space.
I upload it in 2010/09/02.
Contributor excellent Hikari368.
It is 215533 access (as of 2016.5.12).
As the key which realized a general theory, Einstein of later years continued looking for four-dimensional space, but was not able to carry it out. An elementary particle pulsation principle discovered the four-dimensional space. The elementary particle pulsates in the four-dimensional space.

**Category:** Number Theory

[1231] **viXra:1605.0131 [pdf]**
*submitted on 2016-05-13 01:10:28*

**Authors:** Terubumi Honjou

**Comments:** 11 Pages.

Online posting to YOUTUBE.
When the mystery of the prime number is removed, the elementary particle pulsation principle will be recognized as the theory of all things.
I assume real number axis, cross axle an imaginary number axis with a vertical axis and insert a figure of elementary particle pulse motion principle energy wave pattern in the complex number coordinate which developed the non-self-evident zero point of a prime number and the zeta function.
I assumed the straight line that 0 points formed a line the horizon which showed the vacuum space of the figure of elementary particle pulsation principle energy wave pattern . I turn a prime number and a figure of that Mr. Sugimoto made which showed 0 points 90 degrees. Non-self-evident straight line and gap 1/2 with the imaginary number axis where 0 points form a line of the zeta function show energy density (zero point energy) of the vacuum space.
I set the top (a mountain and valley) of the wave pattern of the elementary particle pulsation with the prime number on the real number axis. Furthermore, I set it as if a wave pattern passes on a non-self-evident zero point.
I assumed a prime number an orbital pole in a period and assumed a point zero eigenvalue.
The wave (peculiar empty ¬ interval) of the material wave pulsating every 0 points accompanies it. This wave pattern shows elementary particle pulsation, and a minus number particle, a wave pattern do a horizontal state with a wave in the grain ¬ child whom the energy of the place centered on in the heap of wave patterns, the valley of the wave pattern and repeat particle (+mc²), wave (0)¬, conversion (pulsation) of minus number particle (-mc²) super in (Planck time) for a short time.

**Category:** Number Theory

[1230] **viXra:1605.0123 [pdf]**
*submitted on 2016-05-11 23:17:44*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We have spotted an error of Euler's proof, so that the used infinite descent is impossible in his proof

**Category:** Number Theory

[1229] **viXra:1605.0120 [pdf]**
*submitted on 2016-05-11 13:47:07*

**Authors:** Terubumi Honjou

**Comments:** 20 Pages.

The author announced the elementary particle pulsation hypothesis in the Physical Society of Japan of 1980.
To date, I study an elementary particle pulsation hypothesis.
The author contributed the article that let a prime number and physics fuse. youtube (December, 2012).
This page is the gravity in the elementary particle pulsation hypothesis and an article about the electromagnetic force.

**Category:** Number Theory

[1228] **viXra:1605.0113 [pdf]**
*submitted on 2016-05-11 04:59:52*

**Authors:** Anthony J. Browne

**Comments:** 5 Pages.

An equivalent form of the Goldbach Conjecture is stated using manipulation of characteristic equations and simple logical arguments that lead to an equation which restates the conjecture. A new form of the number of unordered partitions of an even number into two primes is presented.

**Category:** Number Theory

[1227] **viXra:1605.0105 [pdf]**
*submitted on 2016-05-10 13:36:32*

**Authors:** Terubumi Honjo

**Comments:** 20 Pages.

The author announced the elementary particle pulsation hypothesis in the Physical Society of Japan of 1980.
To date, I study an elementary particle pulsation hypothesis.
The author contributed the article that let a prime number and physics fuse. youtube (December, 2012).
This page is the gravity in the elementary particle pulsation hypothesis and an article about the electromagnetic force.

**Category:** Number Theory

[1226] **viXra:1605.0104 [pdf]**
*submitted on 2016-05-10 13:58:07*

**Authors:** José de Jesús Camacho Medina

**Comments:** 4 Pages.

This article educational and recreational mathematics, broadcasts a new category of numbers designed by the author in the year 2014 to which I have called "Numbers Fresnillenses" which permeates this title to a sense of belonging because I have is a native from the city of Fresnillo Zacatecas Mexico. This new classification of numbers you have an exquisite property not previously discovered by the mathematical community; it will be a great pleasure for fans of the numbers.

**Category:** Number Theory

[1225] **viXra:1605.0103 [pdf]**
*submitted on 2016-05-10 14:09:27*

**Authors:** José de Jesús Camacho Medina

**Comments:** 11 Pages.

This article offers a solution in relation to the distribution of the primes, in this research we provide simple formulas and unpublished with a new approach that allow us to assimilate and conclude that the entities primales are sorted as regular as possible. We provide a new vision for addressing what that since ancient times has been a real challenge for the minds linked to the world mathematician, we deliver the reader a key to unravel the structure and behavior of the primes without open the door to the complexity.

**Category:** Number Theory

[1224] **viXra:1605.0095 [pdf]**
*submitted on 2016-05-10 06:59:15*

**Authors:** Terubumi Honjou

**Comments:** 6 Pages.

1) A hypothesis of a prime number and the physics fusion.
The prime number is energy, mcc and a concentric circle. The center is 1/2.
2). The prime number is related deeply with Planck's constant, (h).
3). Natural number is 1* integer. The energy is h X integer.
4). I do it with integer, 1 = 1h.
In m, a radius is a circle of nh. The prime number can express a radius with the circle of the prime number.
5). The equation of the prime number. I accord with energy nh and the prime number of the prime number stairs n turn. (expectation).
6). The prime number distance of the prime number stairs n turn accords with distance of the nuclear resonance energy nh neighborhood. (expectation).

**Category:** Number Theory

[1223] **viXra:1605.0082 [pdf]**
*submitted on 2016-05-09 02:55:23*

**Authors:** Jaykov Foukzon

**Comments:** 23 Pages.

In this paper possible completion of the Robinson non-Archimedean field *R constructed by Dedekind sections. As interesting example I show how, a
few simple ideas from non-archimedean analysis on the pseudo-ring *R_d gives a short clear nonstandard reconstruction for the Euler’s original proof of the Goldbach-Euler theorem.

**Category:** Number Theory

[1222] **viXra:1605.0066 [pdf]**
*submitted on 2016-05-06 02:22:44*

**Authors:** Terubumi Honjou

**Comments:** 7 Pages.

1)Equations of Euler product multiplied by PI, the equation for the area of a circle.
2) All these equations become concentric.
3) Is the Riemann Zeta function can be represented as a sum of an infinite number of simple sin, cos waves by Fourier transform.
4) And sine curve and cosine curve in circular motion, all in concentric circles that share the same center line.
5) Non-trivial Riemann zero was zero and area of a circle of sin and cosine curve that is equivalent to the Riemann Zeta-function
To be equivalent.
6) And the area of a circle of sine curve and cosine curve to zero all become concentric axis of rotation (1/2 straight).
7) This is Riemann's "line other than zero point does not exist" would prove that. And Honjo expected.

**Category:** Number Theory

[1221] **viXra:1605.0061 [pdf]**
*submitted on 2016-05-05 06:55:33*

**Authors:** T.Nakashima

**Comments:** 5 Pages.

This paper is all cycle case resolve.But sequence goes to infinity case I can not prove not possibillity.

**Category:** Number Theory

[1220] **viXra:1605.0056 [pdf]**
*submitted on 2016-05-04 11:34:05*

**Authors:** Terubumi Honjou

**Comments:** 3 Pages.

A fusion of prime number and quantum physics.(1)
This figure shows a prime number of the Lehman expectation and the relations with 0 points.
The zeta function is expressed as the sum of simple sin function and cosine function.
The complicated period function is expressed for the sum of sin function and cosine function that are the mathematical expression of a simple wave.

**Category:** Number Theory

[601] **viXra:1609.0360 [pdf]**
*replaced on 2016-09-25 14:06:25*

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

The odd Goldbach conjecture states that every odd integer greater than seven may be expressed as the sum of three odd primes whereas the even Goldbach conjecture states that every even integer greater than four may be expressed as the sum of two odd primes. Harald Helfgott has provided a convincing proof of the Odd Goldbach conjecture. The even Goldbach conjecture, however remains unproven. We start with the assumption that first counterexamples of both the even and odd Goldbach conjectures exist. We identify a relationship between the first counterexample of the odd conjecture with the counterexample of the even. Invoking Helfgott’s proof of the Odd Goldbach conjecture, we study the value of the first possible even counter-example and it must be infinitely large so that the first odd counter-example may also be infinitely large thereby non-existent.

**Category:** Number Theory

[600] **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

[599] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-18 23:32:38*

**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

[598] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-16 01:43:10*

**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 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 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

[597] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-13 13:35:35*

**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 where a, b, and c are relatively prime positive integers. 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

[596] **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

[595] **viXra:1609.0080 [pdf]**
*replaced on 2016-09-13 21:17:38*

**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, 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

[594] **viXra:1609.0030 [pdf]**
*replaced on 2016-09-12 11:35:51*

**Authors:** Alexander K

**Comments:** 1 Page.

Beal conjecture, fermat last theorem

**Category:** Number Theory

[593] **viXra:1609.0030 [pdf]**
*replaced on 2016-09-11 16:47:25*

**Authors:** Alexander K

**Comments:** 2 Pages.

Beal conjecture, fermat last theorem

**Category:** Number Theory

[592] **viXra:1608.0429 [pdf]**
*replaced on 2016-09-09 09:27:31*

**Authors:** Gyeongmin Yang

**Comments:** 5 Pages.

This article is based on how to look for a closed-form expression related to the odd zeta function values and explained what meaning of the expansion of the Euler zigzag numbers is.

**Category:** Number Theory

[591] **viXra:1608.0375 [pdf]**
*replaced on 2016-08-30 23:38:11*

**Authors:** Nathan Sponder

**Comments:** 10 Pages.

We discuss the asymptotics of the sum $\sum_{k=1}^{m} e^{ \frac{{\ln(k)}^n}{k} }-1$ for $n \geq 0 $. Our main interest is to show the asymptotics of this sum and show expressions for the constants tied to the asymptotics of the sum as well as in particular show the properties of the constants associated with the sum.

**Category:** Number Theory

[590] **viXra:1607.0359 [pdf]**
*replaced on 2016-07-20 07:41:06*

**Authors:** Matilda Walter

**Comments:** 2 Pages.

We present a simple sieve algorithm for finding all existing solutions to the binary Goldbach
problem for a given even number 2N > 4.

**Category:** Number Theory

[589] **viXra:1607.0087 [pdf]**
*replaced on 2016-07-25 10:05:25*

**Authors:** Ryan Zielinski

**Comments:** 84 Pages. This work is licensed under the CC BY 4.0, a Creative Commons Attribution License.

What is a general expression for the sum of the first n integers, each raised to the mth power, where m is a positive integer? Answering this question will be the aim of the paper....We will take the unorthodox approach of presenting the material from the point of view of someone who is trying to solve the problem himself. Keywords: analogy, Johann Faulhaber, finite sums, heuristics, inductive reasoning, number theory, George Polya, problem solving, teaching of mathematics.

**Category:** Number Theory

[588] **viXra:1607.0072 [pdf]**
*replaced on 2016-07-15 07:54:41*

**Authors:** Korn Rakpradit

**Comments:** 47 Pages.

The opinions of this work are revising, stalking and proving in details the derivation of Riemann Zeta Function and Riemann Hypothesis, which Riemann did roughly for more than 150 years ago without proof, and correcting all mistakes about the boundaries of the integrals that was found and those undefined (and/or multiplied by zero) functional equations which caused very big problems to this Riemann Hypothesis.
Proof or disproof of Riemann Hypothesis’s derivation will be very useful for many mathematicians and physicists nowadays because the Hypothesis is widely used in many subjects and works, unaware of risks, thought it is not officially proved right or wrong.

**Category:** Number Theory

[587] **viXra:1607.0072 [pdf]**
*replaced on 2016-07-08 11:10:23*

**Authors:** Korn Rakpradit

**Comments:** 48 Pages.

The opinions of this work are stalking and proving in details the derivation of Riemann Zeta Function and Riemann Hypothesis, which Riemann did roughly for more than 150 years ago without proof, and correcting all mistakes about the boundaries of the integrals and the undefined or divergent functional equations which caused very big problems to this Riemann Hypothesis.
Proof or disproof of Riemann Hypothesis’s derivation will be very useful for mathematicians and physicists because it is widespread in use though it is unproved officially for a long time.

**Category:** Number Theory

[586] **viXra:1606.0302 [pdf]**
*replaced on 2016-07-21 20:07:29*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

reration of recurring decimal and primitive root

**Category:** Number Theory

[585] **viXra:1606.0144 [pdf]**
*replaced on 2016-06-24 21:17:25*

**Authors:** Anthony J. Browne

**Comments:** 5 Pages.

A form of the exponential Mangoldt function is derived using indicator functions. The function's relationship to other important number theoretic functions are derived and discussed.

**Category:** Number Theory

[584] **viXra:1606.0118 [pdf]**
*replaced on 2016-06-13 05:09:24*

**Authors:** Reuven Tint

**Comments:** 12 Pages. Original written Russian

Let us prove that invariant identity is used for the proof of the FLT and Beal conjecture.

**Category:** Number Theory

[583] **viXra:1605.0229 [pdf]**
*replaced on 2016-06-07 15:21:17*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

As non-trivial Riemann zero's grow larger on the 1/2 critical line so does the distance grow proportionally between 0 and 1. Therefore, a non-trivial Riemann zero will not be outside the critical strip or off of the critical line.My Visual Proof by Automorphism asserts Symmetry. If 1/2 10^13 is inside (0,1) then 1/2 10^100 is inside (0,1). In other words if there is Symmetry about the real axis and Symmetry about the critical line then there is Symmetry between 1/2 10^13 is inside (0,1) and 1/2 10^100 which is inside (0,1).

**Category:** Number Theory

[582] **viXra:1605.0195 [pdf]**
*replaced on 2016-09-01 00:46:59*

**Authors:** A. A. Frempong

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

Honorable Pierre de Fermat was truthful. He 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

[581] **viXra:1605.0195 [pdf]**
*replaced on 2016-08-07 00:34:31*

**Authors:** A. A. Frempong

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

Honorable Pierre de Fermat was truthful. He could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Two similar versions of the proof are presented, using a single page for each version. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. It is shown by contradiction that the uniqueness of this 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, the general equation, c^n = a^n + b^n holds, followed by showing that if n > 2 (n an integer), the general equation, c^n = a^n + b^n does not hold. For the first version, let a, b and c be three relatively prime positive integers which are the lengths of the sides of the right triangle ABC, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle at vertex A be denoted by theta. For the second version of the proof, ratio terms were used to begin the construction of the proof, without reference to a triangle. The second version confirmed the proof in the first version. It is also exemplified that if some of the lengths are not positive integers but positive radicals, the derived necessary condition for c^n = a^n + b^n to hold is applicable. Each proof is very simple, and even high school students can learn it. 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

[580] **viXra:1605.0195 [pdf]**
*replaced on 2016-07-24 01:38:27*

**Authors:** A. A. Frempong

**Comments:** 9 Pages. Copyright © by A. A. Frempong

Honorable Pierre de Fermat was truthful. He could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Five similar versions of the proof are presented, using a single page for each version. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. Using a common sense approach, one will first show that if
n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if n > 2 (n an integer), the general equation, c^n = a^n + b^n does not hold. For the first three versions, one applies a polar coordinate system as follows. Let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. For the fourth and fifth versions of the proof, ratio terms were used to begin the construction of the proof. The fourth and fifth versions confirmed the proofs in the first three versions. It is also exemplified that if some of the lengths are not positive integers but positive radicals, the derived necessary condition for c^n = a^n + b^n to hold is applicable. Each proof is very simple, and even high school students can learn it. 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 is $715,000, then the prize for a single page proof (considering the advantages) using inverse proportion, is $107,250,000.

**Category:** Number Theory

[579] **viXra:1605.0195 [pdf]**
*replaced on 2016-06-03 03:24:22*

**Authors:** A. A. Frempong

**Comments:** 9 Pages. Copyright © by A. A. Frempong

Honorable Pierre de Fermat was truthful. He could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Five similar versions of the proof are presented, using a single page for each version. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if n > 2 (n an integer), the general equation, c^n = a^n + b^n does not hold. For the first three versions, one applies a polar coordinate system as follows. Let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. For the fourth and fifth versions of the proof, ratio terms were used to begin the construction of the proof. It is also exemplified that if some of the lengths are not positive integers but positive radicals, the derived necessary condition for c^n = a^n + b^n to hold is applicable. Each proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper.

**Category:** Number Theory

[578] **viXra:1605.0195 [pdf]**
*replaced on 2016-06-01 00:31:28*

**Authors:** A. A. Frempong

**Comments:** 9 Pages. Copyright © A. A. Frempong

Honorable Pierre de Fermat was truthful. He could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Five similar versions of the proof are presented, using a single page for each version. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if n > 2 (n an integer), the general equation, c^n = a^n + b^n does not hold. For the first three versions, one applies a polar coordinate system as follows. Let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. For the fourth and fifth versions of the proof, ratio terms were used to begin the construction of the proof. Each proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper.

**Category:** Number Theory

[577] **viXra:1605.0195 [pdf]**
*replaced on 2016-05-29 00:16:35*

**Authors:** A. A. Frempong

**Comments:** 8 Pages. Copyright © by A. A. Frempong

Honorable Pierre de Fermat was truthful. He could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Four similar versions of the proof are presented, using a single page for each version. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if n > 2 (n an integer), the general equation, c^n = a^n + b^n does not hold. For the first three versions, one applies a polar coordinate system as follows. Let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. For the fourth version of the proof, ratio terms were used to begin the construction of the proof. Each proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper.

**Category:** Number Theory

[576] **viXra:1605.0195 [pdf]**
*replaced on 2016-05-25 03:37:01*

**Authors:** A. A. Frempong

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

Honorable Pierre de Fermat was truthful. He could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if n > 2 (n an integer), the general equation, c^n = a^n + b^n does not hold. Applying a polar coordinate system, let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. Three similar versions of the proof are presented. The proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper.

**Category:** Number Theory

[575] **viXra:1605.0195 [pdf]**
*replaced on 2016-05-23 01:58:04*

**Authors:** A. A. Frempong

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

Honorable Pierre de Fermat was truthful. He could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if n > 2, the general equation, c^n = a^n + b^n does not hold. Applying a polar coordinate system, let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. Three similar versions of the proof are presented. The proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper.

**Category:** Number Theory

[574] **viXra:1605.0195 [pdf]**
*replaced on 2016-05-19 16:43:49*

**Authors:** A. A. Frempong

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

Fermat's last theorem has been proved on a single page. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if
n > 2, the general equation, c^n = a^n + b^n does not hold. Applying a polar coordinate system, let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. The proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin space for it in his paper.

**Category:** Number Theory

[573] **viXra:1605.0123 [pdf]**
*replaced on 2016-05-15 23:40:15*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We have spotted an error of Euler's proof, so that the used infinite descent is impossible in his proof

**Category:** Number Theory

[572] **viXra:1605.0113 [pdf]**
*replaced on 2016-05-21 11:04:35*

**Authors:** Anthony J. Browne

**Comments:** 6 Pages.

An equivalent form of the Goldbach Conjecture is stated using manipulation of characteristic equations and simple logical arguments that lead to an equation which restates the conjecture. A new form of the number of unordered partitions of an even number into two primes is presented.

**Category:** Number Theory

[571] **viXra:1605.0113 [pdf]**
*replaced on 2016-05-12 00:27:47*

**Authors:** Anthony J. Browne

**Comments:** 6 Pages.

An equivalent form of the Goldbach Conjecture is stated using manipulation of characteristic equations and simple logical arguments that lead to an equation which restates the conjecture. A new form of the number of unordered partitions of an even number into two primes is presented.

**Category:** Number Theory

[570] **viXra:1605.0061 [pdf]**
*replaced on 2016-05-12 06:55:09*

**Authors:** T.Nakashima

**Comments:** 5 Pages.

This paper is all cycle case resolve.But sequence goes to infinity case I can not prove not possibillity.

**Category:** Number Theory

[569] **viXra:1605.0061 [pdf]**
*replaced on 2016-05-10 22:16:21*

**Authors:** T.Nakashima

**Comments:** 5 Pages.

This paper is all cycle case resolve.But sequence goes to infinity case I can not prove not possibillity.

**Category:** Number Theory

[568] **viXra:1605.0061 [pdf]**
*replaced on 2016-05-07 04:18:28*

**Authors:** T.Nakashima

**Comments:** 6 Pages.

**Category:** Number Theory