**Previous months:**

2007 - 0703(3) - 0706(2)

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2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

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2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(18) - 1506(12) - 1507(15) - 1508(14) - 1509(14) - 1510(11) - 1511(9) - 1512(25)

2016 - 1601(14) - 1602(17) - 1603(77) - 1604(54) - 1605(28) - 1606(17) - 1607(19) - 1608(16) - 1609(22) - 1610(22) - 1611(12) - 1612(19)

2017 - 1701(19) - 1702(24) - 1703(25) - 1704(32) - 1705(25) - 1706(25) - 1707(21) - 1708(26) - 1709(17) - 1710(26) - 1711(25) - 1712(34)

2018 - 1801(34) - 1802(22) - 1803(24) - 1804(29) - 1805(26)

Any replacements are listed farther down

[1784] **viXra:1805.0362 [pdf]**
*submitted on 2018-05-19 14:55:32*

**Authors:** Ricardo Gil

**Comments:** 2 Pages. @warlockach

The purpose of this paper is to suggest a process to generate simulations on the UNSW Programmable Quantum Computer.

**Category:** Number Theory

[1783] **viXra:1805.0359 [pdf]**
*submitted on 2018-05-19 16:12:46*

**Authors:** Ricardo Gil

**Comments:** 1 Page. @warlockach

The purpose of this paper is to suggest a Dark Matter Device that can be set off in the Cold Spot to create a new Universe.

**Category:** Number Theory

[1782] **viXra:1805.0325 [pdf]**
*submitted on 2018-05-17 18:25:22*

**Authors:** Wilson Torres Ovejero

**Comments:** 12 Pages.

158 years ago that in the complex analysis a hypothesis was raised, which was used in principle
to demonstrate a theory about prime numbers, but, without any proof; with the passing Over the years, this
hypothesis has become very important, since it has multiple applications to physics, to number theory, statistics,
among others In this article I present a demonstration that I consider is the one that has been dodging all this
time.

**Category:** Number Theory

[1781] **viXra:1805.0296 [pdf]**
*submitted on 2018-05-14 19:44:25*

**Authors:** Zeolla Gabriel Martín

**Comments:** 9 Pages.

The prime numbers greater than 5 have 4 terminations in their unit to infinity (1,3,7,9) and the composite numbers divisible by numbers greater than 3 have 5 terminations in their unit to infinity, these are (1,3,5,7,9). This paper develops an expression to calculate the prime numbers and composite numbers with ending 7.

**Category:** Number Theory

[1780] **viXra:1805.0276 [pdf]**
*submitted on 2018-05-13 10:36:23*

**Authors:** Timothy W. Jones

**Comments:** 2 Pages. It seems likely this angle must have been considered by say Wiles.

A number base uses any whole number greater than one. Scientific notation can be used to express any whole number in any base. As Fermat's Last Theorem concerns whole numbers greater than one to powers of $n$, we can express it using scientific notation.

**Category:** Number Theory

[1779] **viXra:1805.0274 [pdf]**
*submitted on 2018-05-13 11:03:51*

**Authors:** Ricardo Gil

**Comments:** 12 Pages. @Warlockach @Warlockach1

The purpose of this paper is to share hardware and software that can be used in Wall Street in Retrocausal optical computing.

**Category:** Number Theory

[1778] **viXra:1805.0269 [pdf]**
*submitted on 2018-05-13 14:29:57*

**Authors:** Victor Sorokine

**Comments:** 1 Page.

The contradiction:
The Fermat equality does not hold over (k+1)-th digits, where k is the number of zeros at the zero end of the number U=A+B-C=un^k.

**Category:** Number Theory

[1777] **viXra:1805.0268 [pdf]**
*submitted on 2018-05-13 14:31:17*

**Authors:** Victor Sorokine

**Comments:** 1 Page. Russian version

Противоречие: Равенство Ферма не выполняется по (k+1)-м цифрам, где k – число нулей в нулевом окончании числа U=A+B-C=un^k.

**Category:** Number Theory

[1776] **viXra:1805.0259 [pdf]**
*submitted on 2018-05-14 08:47:52*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some definite integrals.

**Category:** Number Theory

[1775] **viXra:1805.0258 [pdf]**
*submitted on 2018-05-14 08:50:54*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some formulas related with the real root of the equation: x^11-x^10-1=0 .

**Category:** Number Theory

[1774] **viXra:1805.0234 [pdf]**
*submitted on 2018-05-11 12:51:36*

**Authors:** Ricardo Gil

**Comments:** 2 Pages. @WARLOCKACH

The purpose of this paper is to suggest the steps for Quantum Programming for the 72 Qubit Bristlecone Quantum Computer.

**Category:** Number Theory

[1773] **viXra:1805.0233 [pdf]**
*submitted on 2018-05-11 13:36:16*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

The purpose of this paper is to show the Topology difference between Einstein and Tesla.

**Category:** Number Theory

[1772] **viXra:1805.0230 [pdf]**
*submitted on 2018-05-11 16:15:28*

**Authors:** David Stacha

**Comments:** 3 Pages.

I will provide the solution of Erdös-Moser equation 1+2^p+3^p+...+(k)^p=(k+1)^p based on the properties of Bernoulli polynomials and prove that there is only one solution satisfying the above-mentioned equation. The Erdös-Moser equation (EM equation), named after Paul Erdös and Leo Moser has been studied by many number theorists through history since combines addition, powers and summation together. The open and very interesting conjecture of Erdös-Moser states that there is no other solution of EM equation than the trivial 1+2=3. Investigation of the properties and identities of the EM equation and ultimately providing the proof of this conjecture is the main purpose of this article.

**Category:** Number Theory

[1771] **viXra:1805.0229 [pdf]**
*submitted on 2018-05-11 16:50:26*

**Authors:** Bertrand Wong

**Comments:** 20 Pages.

This paper explicates the Riemann hypothesis and proves its validity; it explains why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1. Much exact calculations are presented, instead of approximations, for the sake of accuracy or precision, clarity and rigor. (N.B.: New materials have been added to the paper.)

**Category:** Number Theory

[1770] **viXra:1805.0207 [pdf]**
*submitted on 2018-05-10 10:13:33*

**Authors:** Mohamed Ababou

**Comments:** 20 Pages.

The book " Do you know that the digits have an end " is a scientific book, its content is clear from its title. The first thing you will say is " we all know that the digits have an end " but you should read first, my book introduce a bunch of proofs that confirm that the numbers have an end, and the digit is the same thing as the number. The Time in its relation with the numbers is the main idea in my book. This book can change the course of the history of science, it contains the correction for a popular wrong idea that is infinity.
-Mohamed Ababou-

**Category:** Number Theory

[1769] **viXra:1805.0204 [pdf]**
*submitted on 2018-05-10 10:51:51*

**Authors:** Ricardo Gil

**Comments:** 1 Page. @WARLOCKACH

The purpose of this paper is to suggest how matter and antimatter is compactified in 26 dimensions.

**Category:** Number Theory

[1768] **viXra:1805.0187 [pdf]**
*submitted on 2018-05-09 15:00:03*

**Authors:** Stefan Bereza

**Comments:** 7 Pages.

The paper presents an attempt to solve a 300-year-old mathematical problem with minimalistic means of high-school mathematics 1]. As introduction, the Pythagorean equation of right angle triangles a^2 + b^2 = c^2
inscribed in the semicircle is reviewed; then, in an analogue way, the equation a^3+ b^3= c^3
(and then a^n + b^n = c^n) represented by a triangle inscribed in the (vertical) ellipse with its basis c making the minor axis of
the ellipse and the sides of the triangle made by the factors {a,b}. Should the inscribed triangles a^3 + b^3 = c^3(and then a^n + b^n = c^n) represent the integer equations - with {a, b, c, n} positive integers, n > 2 - their sides must
be rational to each other; they must form so called integer triangles. In such triangles, the square of altitude y^2(or the altitude y) must be rational to the sides. An assumption is made that at least one of the inscribed triangles may be
an integral one. A unit is derived from c by dividing it by a natural number m; if the assumption is true, the unit will measure
(= divide) y^2(or y) without leaving an irrational rest behind. The value of y^2(or y) is taken from the equation of the ellipse. Conducted calculations show that y^2(or y) divided by the unit leave always an irrational
rest behind incompatible with c; this proves that y^2(or y) is irrational with the basis c what excludes the existence of the assumed integral triangles and, in consequence, of the discussed integral equations.

**Category:** Number Theory

[1767] **viXra:1805.0185 [pdf]**
*submitted on 2018-05-09 19:09:09*

**Authors:** Gang Li

**Comments:** 13 Pages. Submitted to JNT. This is an improved version of the paper posted at http://vixra.org/abs/1706.0288

We discuss an elementary approach to prove the first case of Fermat's last theorem (FLT). The essence of the proof is to notice that
$a+b+c$ is of order $N^{\alpha}$ if $a^N+b^N+c^N=0$. To prove FLT, we first show that $\alpha$ can not be $2$; we
then show that $\alpha$ can not be $3$, etc. While this is is the standard method of induction, we refer to it here as
the ``infinite ascent'' technique, in contrast to Fermat's original ``infinite descent'' technique. A conjecture, first noted by Ribenboim is used.

**Category:** Number Theory

[1766] **viXra:1805.0184 [pdf]**
*submitted on 2018-05-09 19:15:34*

**Authors:** Gang Li

**Comments:** 5 Pages. to be submitted

We discuss an elementary approach to prove the second case of Fermat’s last theorem (FLT). This is a continuation of an earlier paper that deals with the first case of FLT.

**Category:** Number Theory

[1765] **viXra:1805.0173 [pdf]**
*submitted on 2018-05-08 07:52:55*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some elementary formulas involving pi.

**Category:** Number Theory

[1764] **viXra:1805.0165 [pdf]**
*submitted on 2018-05-08 13:09:17*

**Authors:** Timothy W. Jones

**Comments:** 2 Pages. This is an application of decimal circles developed by the same author to proof the irrationality of zeta values.

Using circles that generate areas of positive integer values, together with the transcendence of pi, we show that x^n + y^n = z^n has no solution in positive integers for n greater than or equal to 3, Fermat's Last Theorem. It fits in a margin.

**Category:** Number Theory

[1763] **viXra:1805.0162 [pdf]**
*submitted on 2018-05-08 16:11:51*

**Authors:** Stephen Marshall

**Comments:** 6 Pages.

Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician. He is remembered today for Goldbach's conjecture. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture: Every even integer which can be written as the sum of two primes (the strong conjecture) He then proposed a second conjecture in the margin of his letter: Every odd integer greater than 7 can be written as the sum of three primes (the weak conjecture). A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers. The “strong” conjecture has been shown to hold up through 4 × 1018, but remains unproven for almost 300 years despite considerable effort by many mathematicians throughout history. In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). In 2013, Harald Helfgott proved Goldbach's weak conjecture. The author would like to give many thanks to Helfgott’s proof of the weak conjecture, because this proof of the strong conjecture is completely dependent on Helfgott’s proof. Without Helfgott’s proof, this elementary proof would not be possible.

**Category:** Number Theory

[1762] **viXra:1805.0152 [pdf]**
*submitted on 2018-05-07 03:22:53*

**Authors:** Preininger Helmut

**Comments:** 24 Pages.

We consider univariate Polynomials, P(s), of the form (a1 * s + b1)*...*(ak * s + bk), where a1,..,ak,b1,..,bk are natural numbers and the variable s is squarefree. We give an algorithm to calculate, for a arbitrary s, the probability that the value of P(s) is squarefree.

**Category:** Number Theory

[1761] **viXra:1805.0076 [pdf]**
*submitted on 2018-05-02 20:27:06*

**Authors:** Zeolla Gabriel Martín

**Comments:** 9 Pages.

The prime numbers greater than 5 have 4 terminations in their unit to infinity (1,3,7,9) and the composite numbers divisible by numbers greater than 3 have 5 terminations in their unit to infinity, these are (1,3,5,7,9). This paper develops an expression to calculate the prime numbers and composite numbers with ending 3.

**Category:** Number Theory

[1760] **viXra:1805.0042 [pdf]**
*submitted on 2018-05-01 13:32:15*

**Authors:** Nazihkhelifa

**Comments:** 2 Pages. Version 1

A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not.
On the next version we will prove the primality tests formula

**Category:** Number Theory

[1759] **viXra:1805.0032 [pdf]**
*submitted on 2018-05-02 07:54:40*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents some integrals involving the Euler-Mascheroni constant: gamma=lim(H(n)-ln(n))=0.577215...

**Category:** Number Theory

[1758] **viXra:1804.0492 [pdf]**
*submitted on 2018-04-30 18:32:49*

**Authors:** Nazihkhelifa

**Comments:** 1 Page. Version 1

Prime Number Formula

**Category:** Number Theory

[1757] **viXra:1804.0474 [pdf]**
*submitted on 2018-04-28 14:33:16*

**Authors:** Elizabeth Gatton-Robey

**Comments:** 2 Pages.

I created an algorithm that guarantees the validity of Goldbach’s Conjecture.
The algorithm eliminates all even integers that are not the sum of an even integer when the number 3 is added to one other prime number.
The visual pattern that emerges from the algorithm maps all prime numbers. This pattern is also applied to individual consecutive primes to eliminate all even numbers that are not the sum of each prime number plus one other prime number.
By layering the pattern to account for all possible “sums of two prime numbers” combinations, it can either be said that “all evens will be eliminated” or “no evens will be eliminated”. What this means is that what happens to one even integer is universal.

**Category:** Number Theory

[1756] **viXra:1804.0470 [pdf]**
*submitted on 2018-04-28 18:44:49*

**Authors:** Colin James III

**Comments:** 1 Page. © 2018 by Colin James III All rights reserved. info@cec-services dot com

The distribution is confirmed as random and refuted as not clumped as in claims by various theoretical methods.

**Category:** Number Theory

[1755] **viXra:1804.0416 [pdf]**
*submitted on 2018-04-27 20:31:50*

**Authors:** Waldemar Puszkarz

**Comments:** 5 Pages. The original pre-Latex version from Feb 11th, 2018.

Computer experiments reveal that primes tend to occur next to squareful numbers more often than next to squarefree numbers compared to what one should expect from a non-biased distribution. The effect is more pronounced for prime pairs than for isolated primes.

**Category:** Number Theory

[1754] **viXra:1804.0409 [pdf]**
*submitted on 2018-04-28 06:38:38*

**Authors:** Andrey B. Skrypnik

**Comments:** 4 Pages.

Now there is a formula for calculating all primes

**Category:** Number Theory

[1753] **viXra:1804.0385 [pdf]**
*submitted on 2018-04-25 21:32:26*

**Authors:** Bing He, Hongcun Zhai

**Comments:** 7 Pages. This is a joint work with Dr. Zhai.

From a very-well-poised _{6}\phi_{5} series formula we deduce a general series expansion formula involving the q-gamma function.
With this formula we can give q-analogues of many Ramanujan-type series.

**Category:** Number Theory

[1752] **viXra:1804.0376 [pdf]**
*submitted on 2018-04-26 06:38:20*

**Authors:** Angel Garcés Doz

**Comments:** 9 Pages.

This modest article shows the connection between the strong Goldbach conjecture and the topological properties of the Klein bottle and the Möbius strip. This connection is established by functions derived from the number of divisors of the two odd integers whose sum is an even number.

**Category:** Number Theory

[1751] **viXra:1804.0366 [pdf]**
*submitted on 2018-04-24 16:44:53*

**Authors:** H.L. Mitchell

**Comments:** 12 Pages.

We introduce a sieve for the number of twin primes less than n by sieving through the set {k ∊ ℤ+ | 6k < n}. We derive formula accordingly using the Euler product and the Brun Sieve.
We then use the Prime Number Theorem and Mertens’ Theorem.
The main results are:
1) A sieve for the twin primes similar to the sieve of Eratosthenes for primes involving only the
values of k, the indices of the multiples of 6, ranging over k = p ,5 ≤ p <√n.It shows the uniform
distribution of the pairs (6k-1,6k+1) that are not twin primes and the decreasing frequency of
multiples of p as p increases.
2) A formula for the approximate number of twin primes less than N in terms of the number of
primes less than n
3) The asymptotic formula for the number of twin primes less than n verifying the Hardy
Littlewood Conjecture.

**Category:** Number Theory

[1750] **viXra:1804.0337 [pdf]**
*submitted on 2018-04-23 23:50:14*

**Authors:** Walter Gress

**Comments:** 12 Pages.

This work expounds upon a theory of peripheral-integers and peripheral-reals, integers and reals that in a modular number line mirror their counterparts. It illustrates the properties of these numbers in hopes to breathe life into research of numbers that go beyond infinity

**Category:** Number Theory

[1749] **viXra:1804.0291 [pdf]**
*submitted on 2018-04-20 19:53:17*

**Authors:** Sergey A. Lazarev

**Comments:** 4 Pages.

Prime number. Its nature, appearance, types, movement, prediction.

**Category:** Number Theory

[1748] **viXra:1804.0289 [pdf]**
*submitted on 2018-04-20 22:09:15*

**Authors:** Walter Gress

**Comments:** 22 Pages.

A Sieve that extracts various properties of numerical sequences, demonstrating patterns in different types of sequences, rational, and irrational numbers.

**Category:** Number Theory

[1747] **viXra:1804.0267 [pdf]**
*submitted on 2018-04-20 03:14:35*

**Authors:** John Atwell Moody

**Comments:** 7 Pages.

Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue -1 at i\infty and one of residue 1 at 1. Let \mu_{pm}:TxH->H be the action of multipying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multipplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha+d\tau)\wedge \mu_-^*)\alpha+d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}. The integral is equal to the stable distance from the origin in a dynamical system where a point is picked up and dropped off with two exponential rates. It spends time orbiting a fixed point or limit cycle before it is dropped off. It is only when c=1/2 that the two rates are equal.

**Category:** Number Theory

[1746] **viXra:1804.0262 [pdf]**
*submitted on 2018-04-20 09:02:41*

**Authors:** Zeolla Gabriel Martín

**Comments:** 9 Pages.

The prime numbers greater than 5 have 4 terminations in their unit to infinity (1,3,7,9) and the composite numbers divisible by numbers greater than 3 have 5 terminations in their unit to infinity, these are (1,3,5,7,9). This paper develops an expression to calculate the prime numbers and composite numbers with ending 1.

**Category:** Number Theory

[1745] **viXra:1804.0259 [pdf]**
*submitted on 2018-04-20 09:59:43*

**Authors:** M. A. Thomas

**Comments:** 3 Pages.

An observation of Diophantine sequences at or near the beginning of Prime sequences

**Category:** Number Theory

[1744] **viXra:1804.0224 [pdf]**
*submitted on 2018-04-16 07:57:36*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas related with Malmsten's integral.

**Category:** Number Theory

[1743] **viXra:1804.0223 [pdf]**
*submitted on 2018-04-16 07:59:39*

**Authors:** Edgar Valdebenito

**Comments:** 6 Pages.

This note presents some definite integrals.

**Category:** Number Theory

[1742] **viXra:1804.0216 [pdf]**
*submitted on 2018-04-16 14:06:25*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The definition of the imaginary number is not tautologous and hence refuted.
The definition as rendered is contingent, the value for falsity.
While the definition can be coerced to be non-contingent, the value for truthity, it is still not tautologous.

**Category:** Number Theory

[1741] **viXra:1804.0192 [pdf]**
*submitted on 2018-04-14 14:31:44*

**Authors:** Mendzina Essomba François

**Comments:** 10 Pages.

I propose in this article the first infinite products of history for inverse sinusoidal functions

**Category:** Number Theory

[1740] **viXra:1804.0183 [pdf]**
*submitted on 2018-04-13 18:31:14*

**Authors:** Zeolla Gabriel Martín

**Comments:** 5 Pages.

This paper develops a modified an old and well-known expression for calculating and obtaining all twin prime numbers greater than three. The conditioning (n) will be the key to make the formula work.

**Category:** Number Theory

[1739] **viXra:1804.0182 [pdf]**
*submitted on 2018-04-13 21:56:20*

**Authors:** Quang Nguyen Van

**Comments:** 4 Pages.

We give some quadratic formulas (including Euler's and Dirichlet's formula) of the equation X^(n-1) ∓ X^(n-2)Y + X^(n-3)Y^(n-2) ∓ … + Y^(n-1) = Z^n(nZ^n) in the cases n = 3, 5 and 7 for finding a solution in integer.

**Category:** Number Theory

[1738] **viXra:1804.0052 [pdf]**
*submitted on 2018-04-03 07:58:59*

**Authors:** Raffaele Cogoni

**Comments:** 20 Pages. Testo di n° 20 pagine in lingua Italiana

Nel presente lavoro viene descritto un algoritmo per determinare la successione dei numeri primi, esso si presenta come una rielaborazione del noto Crivello di Eratostene.

**Category:** Number Theory

[1737] **viXra:1804.0046 [pdf]**
*submitted on 2018-04-03 13:02:17*

**Authors:** Franco Sabino Stoianoff Lindstron

**Comments:** 2 Pages.

The method used in this article is based on analytical geometry, abstract algebra and number theory.

**Category:** Number Theory

[1736] **viXra:1804.0039 [pdf]**
*submitted on 2018-04-02 15:49:23*

**Authors:** Franco Sabino Stoianoff Lindstron

**Comments:** 40 Pages.

The Beal Conjecture has been one of the most interesting problems that existed in number theory since the end of the last century. It was discovered by Andrew Beal during his work on Fermat's Last Theorem. In this paper a detailed review of the conjecture is proposed to end in a possible proof.

**Category:** Number Theory

[1735] **viXra:1804.0038 [pdf]**
*submitted on 2018-04-02 15:51:38*

**Authors:** Philip Aaron Bloom

**Comments:** 2 Pages.

For any given positive integral value of n : We devise an algebraic identity r^n + s^n = t^n holding for positive real r, s, t, to relate to x^n + y^n = z^n holding for positive co-prime x, y, z. The detailed identity has an unrestricted variable, which allows us to directly infer that {r, s, t} = {x, y, z}. We show that for n > 2, there exists no co-prime r, s, t. Consequently, for n > 2, there exists no co-prime nor integral (x, y, z).

**Category:** Number Theory

[1734] **viXra:1804.0037 [pdf]**
*submitted on 2018-04-02 16:53:15*

**Authors:** Zeolla Gabriel Martin

**Comments:** 4 Pages.

This paper shows that the product of the prime numbers adding and subtracting one is always Simple Prime numbers.

**Category:** Number Theory

[1733] **viXra:1804.0036 [pdf]**
*submitted on 2018-04-03 00:26:51*

**Authors:** Radomir Majkic

**Comments:** 10 Pages.

Abstract: The prime numbers set is the three primes addition closed; each prime is the sum of three not necessarily distinct primes. All natural numbers are created on the set of all prime numbers according to the laws of the weak and strong Goldbach's conjectures. Thus all natural numbers are the Goldbach's numbers.

**Category:** Number Theory

[1732] **viXra:1804.0016 [pdf]**
*submitted on 2018-04-02 07:49:27*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some formulas for Catalan's constant.

**Category:** Number Theory

[1731] **viXra:1804.0008 [pdf]**
*submitted on 2018-04-02 12:49:21*

**Authors:** Victor Sorokine

**Comments:** 2 Pages.

Contradiction: Any prime factor r of the number R in the equality A n =A n +B n [...=(A+B)R]
has a single ending 0...01 of infinite length; where r≠n.
All calculations are done with numbers in base n, a prime number greater than 2.

**Category:** Number Theory

[1730] **viXra:1804.0007 [pdf]**
*submitted on 2018-04-02 12:50:23*

**Authors:** Victor Sorokine

**Comments:** 2 Pages. Russian version

Contradiction: Any prime factor r of the number R in the equality A n =A n +B n [...=(A+B)R]
has a single ending 0...01 of infinite length; where r≠n.
All calculations are done with numbers in base n, a prime number greater than 2.
Противоречие: В равенстве A n =A n +B n [...=(A+B)R] любой простой сомножитель r (r≠n,
простое n>2) числа R имеет (в базе n) единичное окончание 0...01 бесконечной длины.
Все вычисления проводятся в системе счисления с простым основанием n>2.

**Category:** Number Theory

[1729] **viXra:1803.0715 [pdf]**
*submitted on 2018-03-30 04:02:45*

**Authors:** Andrey B. Skrypnik

**Comments:** 13 Pages.

Complete solution of Queens Puzzle

**Category:** Number Theory

[1728] **viXra:1803.0689 [pdf]**
*submitted on 2018-03-28 06:44:40*

**Authors:** BERKOUK Mohamed

**Comments:** 27 Pages.

et si nous essayons d'extraire les nombres composés de l'ensemble des entiers naturels , à commencer par trouver la formule qui génère tous les entiers sans les multiple de 2 et 3 ( 1er polynôme ) puis de générer les entiers sans les multiples de 2,3,5 (2eme polynôme ) le but est de trouver l’équation , la formule ou le polynôme qui ne génèrera que les NOMBRES PREMIERS...

**Category:** Number Theory

[1727] **viXra:1803.0668 [pdf]**
*submitted on 2018-03-26 14:56:31*

**Authors:** Haofeng Zhang

**Comments:** 16 Pages.

In this paper the author gives an elementary mathematics method to solve Fermat's
Last Theorem (FLT), in which let this equation become an one unknown number equation, in
order to solve this equation the author invented a method called “Order reducing method for
equations”, where the second order root compares to one order root, and with some necessary
techniques the author successfully proved when x^(n-1)+y^(n-1)- z^(n-1) <= x^(n-2)+y^(n-2)-
z^(n-2) there are no positive solutions for this equation, and also proves with the increasing of x
there are still no positive integer solutions for this equation when x^(n-1)+y^(n-1)- z^(n-1)<=
x^(n-2)+y^(n-2)- z^(n-2) is not satisfied.

**Category:** Number Theory

[1726] **viXra:1803.0654 [pdf]**
*submitted on 2018-03-25 19:15:54*

**Authors:** Zeolla Gabriel Martín

**Comments:** 5 Pages.

This paper develops the formula that calculates the sum of simple composite numbers by golden patterns.

**Category:** Number Theory

[1725] **viXra:1803.0635 [pdf]**
*submitted on 2018-03-23 20:55:27*

**Authors:** Waldemar Puszkarz

**Comments:** 2 Pages.

This note presents some properties of a quadratic polynomial 13n^2 + 53n + 41. One of them is unique, while others are shared with other prime-generating quadratics. The main purpose of this note is to emphasize certain common features of such quadratics that may not have been noted before.

**Category:** Number Theory

[1724] **viXra:1803.0546 [pdf]**
*submitted on 2018-03-23 10:15:03*

**Authors:** Waldemar Puszkarz

**Comments:** 3 Pages.

This note lists all the known prime-generating quadratics with at most two-digit positive coefficients that generate at least 20 primes in a row. The Euler polynomial is the best-known member of this class of six.

**Category:** Number Theory

[1723] **viXra:1803.0493 [pdf]**
*submitted on 2018-03-22 22:28:25*

**Authors:** Elizabeth Gatton-Robey

**Comments:** 22 Pages.

The current mathematical consensus is that Prime numbers, those integers only divisible by one and themselves, follow no standard predictable pattern.
This body of work provides the first formula to predict prime numbers. In doing so, this proves that prime numbers follow a pattern, and proves Goldbach’s Conjecture to be true.
This is done by forming an algorithm that considers all even integers, systematically eliminates some, and the resulting subset of even integers produces all prime numbers once three is subtracted from each.

**Category:** Number Theory

[1722] **viXra:1803.0362 [pdf]**
*submitted on 2018-03-21 07:57:40*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some formulas related with pi.

**Category:** Number Theory

[1721] **viXra:1803.0317 [pdf]**
*submitted on 2018-03-19 20:47:26*

**Authors:** John Atwell Moody

**Comments:** 2 Pages.

Conjecture: d/dc of the magnitude of the integral of e^{(c-1+iw)t} times log(\lambda/q)dt is <0 when c\in (0,1/2) and w>0.

Theorem: The conjecture implies Riemann's hypothesis.

**Category:** Number Theory

[1720] **viXra:1803.0298 [pdf]**
*submitted on 2018-03-20 21:42:49*

**Authors:** Zeolla Gabriel Martin

**Comments:** 3 Pages.

This paper develops the formula that calculates the quantity of simple composite numbers that exist by golden patterns.

**Category:** Number Theory

[1719] **viXra:1803.0289 [pdf]**
*submitted on 2018-03-21 03:18:10*

**Authors:** Bado idriss olivier

**Comments:** 6 Pages.

In this paper we are going to give the proof of Goldbach conjecture by introducing a new lemma which implies Goldbach conjecture .By using Chebotarev-Artin theorem , Mertens formula and Poincare sieve we establish the lemma

**Category:** Number Theory

[1718] **viXra:1803.0265 [pdf]**
*submitted on 2018-03-19 06:45:58*

**Authors:** Yuri Heymann

**Comments:** 20 Pages.

In the present study we use the Dirichlet eta function as an extension of the Riemann zeta function in the strip Re(s) in ]0, 1[. We then determine the domain of admissible complex zeros of the Riemann zeta function in this strip using the symmetries of the function and minimal constraints. We also check for zeros outside this strip. We nd that the admissible domain of complex zeros excluding the trivial zeros is the critical line given by Re(s) = 1/2 as stated in the Riemann hypothesis.

**Category:** Number Theory

[1717] **viXra:1803.0225 [pdf]**
*submitted on 2018-03-15 20:17:04*

**Authors:** Zeolla Gabriel martin

**Comments:** 4 Pages.

This paper develops the formula that calculates the sum of simple prime numbers by golden pattern.

**Category:** Number Theory

[1716] **viXra:1803.0219 [pdf]**
*submitted on 2018-03-16 05:37:57*

**Authors:** Huseyin Ozel

**Comments:** 44 Pages.

The existing definition of imaginary numbers is solely based on the fact that certain mathematical operation, square operation, would not yield certain type of outcome, negative numbers; hence such operational outcome could only be imagined to exist. Although complex numbers actually form the largest set of numbers, it appears that almost no thought has been given until now into the full extent of all possible types of imaginary numbers. A close look into what further non-existing numbers could be imagined help reveal that we could actually expand the set of imaginary numbers, redefine complex numbers, as well as define imaginary and complex mathematical objects other than merely numbers.

**Category:** Number Theory

[1715] **viXra:1803.0192 [pdf]**
*submitted on 2018-03-14 02:45:45*

**Authors:** Andrea Prunotto

**Comments:** 4 Pages.

The equiprobability among two events involving independent extractions of elements from a
finite set is shown to be related to the solutions of Fermat's Diophantine equation.

**Category:** Number Theory

[1714] **viXra:1803.0179 [pdf]**
*submitted on 2018-03-12 18:18:40*

**Authors:** Morgan Osborne

**Comments:** 22 Pages. Keywords: Beal, Diophantine, Continuity (2010 MSC: 11D99, 11D41)

The Beal Conjecture considers positive integers A, B, and C having respective positive integer exponents X, Y, and Z all greater than 2, where bases A, B, and C must have a common prime factor. Taking the general form A^X + B^Y = C^Z, we explore a small opening in the conjecture through reformulation and substitution to create two new variables. One we call 'C dot' representing and replacing C and the other we call 'Z dot' representing and replacing Z. With this, we show that 'C dot' and 'Z dot' are separate continuous functions, with argument (A^X + B^Y), that achieve all positive integers during their continuous non-constant rates of infinite ascent. Possibilities for each base and exponent in the reformulated general equation A^X +B^Y = ('C dot')^('Z dot') are examined using a binary table along with analyzing user input restrictions and 'C dot' values relative to A and B. Lastly, an indirect proof is made, where conclusively we find the continuity theorem to hold over the conjecture.

**Category:** Number Theory

[1713] **viXra:1803.0178 [pdf]**
*submitted on 2018-03-12 18:15:11*

**Authors:** Zeolla Gabriel martin

**Comments:** 3 Pages.

This paper develops the formula that calculates the quantity of simple prime numbers that exist by golden patterns.

**Category:** Number Theory

[1712] **viXra:1803.0150 [pdf]**
*submitted on 2018-03-10 16:37:39*

**Authors:** Pedro Caceres

**Comments:** 21 Pages.

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable z that analytically continues the sum of the Dirichlet series:
ζ(z)=∑_(k=1)^∞ k^(-z)
The Riemann zeta function is a meromorphic function on the whole complex z-plane, which is holomorphic everywhere except for a simple pole at z = 1 with residue 1.
One of the most important advance in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In this paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x), and also provided insights into the roots (zeros) of the zeta function, formulating a conjecture about the location of the zeros of ζ(z) in the critical line Re(z)=1/2.
The Riemann Zeta function is one of the most studied and well known mathematical functions in history. In this paper, we will formulate new propositions to advance in the knowledge of the Riemann Zeta function.
a) A constant C that can be used to express ζ(2n+1)≡a/b*C^(2n+1)
b) An approximation to the values of ζ(s) in R given by ζ(s)=1/(1-π^(-s)-2^(-s))
c) A theorem that states that the infinite sums ∑_(j=1)^∞[ζ(u*k±n)-ζ(v*k±m)] converge
to a value in the interval (-1,1) for all u≥1,v≥1,n,m ∈N such that (u*k±n)>1 and
(v*k±m)>1 for all j∈N
d) A new set of constants CZ_(u,n,v,m)calculated from infinite sums involving ζ(z)
e) A function in C2(x,a,b)= 2*x^(-a)*(∑_(j=1)^(x-1) [j^(-a)*cos(b*(ln(x/j)))]) in R with zeros in(a,b) with a=1/2 and b=Im(z*), with z*=non-trivial zero of ζ(z).
f) A C-transformation that allows for a decomposition of ζ(z) that can be used to study
the Riemann Hypothesis.
g) Linearization of the Harmonic function using Non-Trivial zeros of ζ(z).
h) An expression that links any two Non-Trivial zeros of ζ(z).

**Category:** Number Theory

[1711] **viXra:1803.0121 [pdf]**
*submitted on 2018-03-09 10:43:27*

**Authors:** Zeolla Gabriel martin

**Comments:** 5 Pages.

This paper develops the construction of the Golden Patterns for different prime divisors, the discovery of patterns towards infinity. The discovery of infinite harmony represented in fractal numbers and patterns. The golden pattern works with the simple prime numbers that are known as rough numbers and simple composite number.

**Category:** Number Theory

[1710] **viXra:1803.0110 [pdf]**
*submitted on 2018-03-08 06:44:51*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some trigonometric formulas that involving nested radicals.

**Category:** Number Theory

[1709] **viXra:1803.0108 [pdf]**
*submitted on 2018-03-08 08:14:15*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2017-2018 by Colin James III All rights reserved.

This is the briefest known such proof, and in mathematical logic.

**Category:** Number Theory

[1708] **viXra:1803.0105 [pdf]**
*submitted on 2018-03-07 21:22:05*

**Authors:** Henry Göttler, Chantal Göttler, Heinrich Göttler, Thorsten Göttler, Pei-jung Wu

**Comments:** 7 Pages. Proof of Collatz Conjecture

Over 80 years ago, the German mathematician Lothar Collatz formulated an interesting mathematical problem, which he was afraid to publish, for the simple reason that he could not solve it. Since then the Collatz Conjecture has been around under several names and is still unsolved, keeping people addicted. Several famous mathematicians including Richard Guy stating “Dont try to solve this problem”. Paul Erd¨os even said ”Mathematics is not yet ready for such problems” and Shizuo Kakutani joked that the problem was a Cold War invention of the Russians meant to slow the progress of mathematics in the West. We might have ﬁnally freed people from this addiction.

**Category:** Number Theory

[1707] **viXra:1803.0098 [pdf]**
*submitted on 2018-03-07 09:05:15*

**Authors:** Zeolla Gabriel martin

**Comments:** 6 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-3, the discovery of a pattern to infinity, the demonstration of the inharmonics that are 2,3, and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers. This paper develops a formula to get simple prime number-3 and simple composite number-3
The simple prime numbers-3 is known as the 5-rough numbers.

**Category:** Number Theory

[1706] **viXra:1803.0017 [pdf]**
*submitted on 2018-03-01 10:12:24*

**Authors:** Pablo Hernan Pereyra

**Comments:** 3 Pages.

A discrete condition for twin prime numbers is established by Wilson's theorem. By synchronization is obtained a linear diophantine equation that implies by Bertrand Chebyshev's theorem the existence of infinite twin prime numbers.

**Category:** Number Theory

[1705] **viXra:1802.0433 [pdf]**
*submitted on 2018-02-28 20:58:26*

**Authors:** Clive Jones

**Comments:** 2 Pages.

Featuring the PF5 Function

**Category:** Number Theory

[1704] **viXra:1802.0427 [pdf]**
*submitted on 2018-02-28 07:01:31*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas related with Dottie number.

**Category:** Number Theory

[1703] **viXra:1802.0395 [pdf]**
*submitted on 2018-02-26 20:37:20*

**Authors:** Radomir Majkic

**Comments:** 15 Pages.

Abstract: Goldbach's conjectures are inseparable and both of them stem from an underlying fundamental structure of the natural numbers.Thus,
one of them must consistently imply the other one, and the Goldbach's weak conjecture must imply the Goldbach's strong conjecture. Finally, all natural
numbers are the Goldbach's numbers.

**Category:** Number Theory

[1702] **viXra:1802.0363 [pdf]**
*submitted on 2018-02-26 10:46:52*

**Authors:** Zeolla Gabriel martin

**Comments:** 8 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-13, the discovery of a pattern to infinity, the demonstration of the inharmonics that are 2,3,5,7,11,13 and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers. This paper develops a formula to get simple prime number-13 and simple composite number-13
The simple prime numbers-13 are known as the 17-rough numbers.

**Category:** Number Theory

[1701] **viXra:1802.0353 [pdf]**
*submitted on 2018-02-25 04:25:14*

**Authors:** Dave Ryan T. Cariño

**Comments:** 12 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={2(n+x_p )+3|x_p=x_3+x_5+x_7+x_11+⋯x_p }〗 is true for all integer where for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of adding new expression for .

**Category:** Number Theory

[1700] **viXra:1802.0321 [pdf]**
*submitted on 2018-02-22 07:29:28*

**Authors:** Andrea Prunotto

**Comments:** 2 Pages.

The condition of equiprobability among two events involving independent extractions of elements from a finite set is shown to coincide with Fermat's Diophantine equation. The problem of the division of the stakes, related to the events, is also discussed

**Category:** Number Theory

[1699] **viXra:1802.0309 [pdf]**
*submitted on 2018-02-21 19:03:59*

**Authors:** David Stacha

**Comments:** Pages.

In this article I will provide the solution of Brocard`s problem n!+1=x^2 and I will prove the existence of the finite amount of Brown numbers, where the largest Brown number is (7,71), which represents the equation 7!+1=71^2. Brocard`s problem represents one of the open problem in mathematics from the field of number theory, which has been formulated by Henri Brocard in 1876 and represents the solutions of the following Diophantine equation n!+1=x^2.

**Category:** Number Theory

[1698] **viXra:1802.0303 [pdf]**
*submitted on 2018-02-22 00:31:26*

**Authors:** Pedro Caceres

**Comments:** 56 Pages.

The function x(j,k)=δ+ω(α+βj)^φk in C→C is a generalization of the power function y(α)=α^k in R→R and the exponential function y(k)=α^k in R→R. In this paper we are going to calculate the values of infinite and partial sums and products involving elements of the matrix Xjk=[x(j,k)]∈C
As a result, several new representations will be made for some infinite series, including the Riemann Zeta Function in C.

**Category:** Number Theory

[1697] **viXra:1802.0269 [pdf]**
*submitted on 2018-02-19 17:14:55*

**Authors:** Ayal Sharon

**Comments:** 11 Pages.

The Law of the Excluded Middle holds that either a statement "X" or its opposite "not X" is true. In Boolean algebra form, Y = X XOR (not X). Riemann's analytic continuation of Zeta(s) contradicts the Law of the Excluded Middle, because the Dirichlet series Zeta(s) is proven divergent in the half-plane Re(s)<=1. Further inspection of the derivation of Riemann's analytic continuation of $\zeta(s)$ shows that it is wrongly based on the Cauchy integral theorem, and thus false.

**Category:** Number Theory

[1696] **viXra:1802.0236 [pdf]**
*submitted on 2018-02-18 17:27:55*

**Authors:** Zeolla Gabriel martin

**Comments:** 14 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-11 (1 to 11), the discovery of a pattern to infinity, the demonstration of the Inharmonics that are 2,3,5,7 and 11 and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers.
The simple prime numbers-11 are known as the 13-rough numbers.

**Category:** Number Theory

[1695] **viXra:1802.0213 [pdf]**
*submitted on 2018-02-17 10:38:14*

**Authors:** ANIRILASY Méleste

**Comments:** 2 Pages.

We suggest that there exists, at least, one prime number in four intervals between n² and (n+1)² for any integer n 2 such that :
all intervals are half-open;
the excluded endpoints are multiples of n;
the number of elements in each interval is equal to the least even upper bound for the biggest prime number strictly less than n.
This conjecture is a strong form of Oppermann’s one.

**Category:** Number Theory

[1694] **viXra:1802.0201 [pdf]**
*submitted on 2018-02-15 12:14:36*

**Authors:** Zeolla Gabriel martin

**Comments:** 9 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-5, the discovery of a pattern to infinity, the demonstration of the inharmonics that are 2,3,5 and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers. This paper develops a formula to get simple prime number-5 and simple composite number-5
The simple prime numbers-5 are known as the 7-rough numbers.

**Category:** Number Theory

[1693] **viXra:1802.0198 [pdf]**
*submitted on 2018-02-15 14:55:26*

**Authors:** John Yuk Ching Ting

**Comments:** 67 Pages. This research paper contains rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures.

L-functions form an integral part of the 'L-functions and Modular Forms Database' with far-reaching implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by Bernhard Riemann whereby all nontrivial zeros are [mathematically] conjectured to lie on the critical line of this function. This proposal is equivalently stated in this research paper as all nontrivial zeros are [geometrically] conjectured to exactly match the 'Origin' intercepts of this function. Deeply entrenched in number theory, prime number theorem entails analysis of prime counting function for prime numbers. Solving Riemann hypothesis would enable complete delineation of this important theorem. Involving proposals on the magnitude of prime gaps and their associated sets of prime numbers, Twin prime conjecture deals with prime gap = 2 (representing twin primes) and is thus a subset of Polignac's conjecture which deals with all even number prime gaps = 2, 4, 6,... (representing prime numbers in totality except for the first prime number '2'). Both nontrivial zeros and prime numbers are Incompletely Predictable entities allowing us to employ our novel Virtual Container Research Method to solve the associated hypothesis and conjectures.

**Category:** Number Theory

[1692] **viXra:1802.0176 [pdf]**
*submitted on 2018-02-14 10:10:56*

**Authors:** Philip Gibbs

**Comments:** 11 Pages.

A Diophantine m-tuple is a set of m distinct non-zero integers such that the product of any two elements of the set is one less than a square. The definition can be generalised to any commutative ring. A computational search is undertaken to find Diophantine 5-tuples (quintuples) over the ring of quadratic integers Z[√D] for small positive and negative D. Examples are found for all positive square-free D up to 22, but none are found for the complex rings including the Gaussian integers.

**Category:** Number Theory

[1691] **viXra:1802.0154 [pdf]**
*submitted on 2018-02-13 22:51:29*

**Authors:** Réjean Labrie

**Comments:** 6 Pages.

Let N, n and k be integers larger than 1. Then for all N there exists a minimum threshold k such that for n>=N, if we cut the sequence of consecutive integers from 1 to n*(n+k) into n+k slices of length n, we always find at least a prime number in each slice.
It follows that π(n*(n+k)) > π(n*(n+k-1)) > π(n*(n+k-2)) > π(n*(n+k-3))> ...> π(2n)> π(n) where π(n) is the quantity of prime numbers smaller than or equal to n.

**Category:** Number Theory

[1690] **viXra:1802.0141 [pdf]**
*submitted on 2018-02-12 14:37:55*

**Authors:** Marius Coman

**Comments:** 2 Pages.

The set of Poulet numbers having only odd digits is: 1333333, 1993537, 3911197, 5351537, 5977153, 7759937, 11777599...(22 from the first 7196 Poulet numbers belong to this set). Question: is this sequence infinite? Observations: the numbers n*P + R(P) – n respectively P + n*R(P) - n, where R(P) is the reversal of P and n positive integer, are often primes. Examples: for P = 1333333, the number 1333333 + 3333331 – 1 = 4666663, prime; also 3*1333333 + 3333331 – 3 = 7333327, prime; also 5*1333333 + 3333331 – 5 = 9999991, prime. For the same P, the number 1333333 + 2*3333331 - 2 = 7999993, prime; also the number 1333333 + 4*3333331 - 4 = 14666653, prime.

**Category:** Number Theory

[1689] **viXra:1802.0135 [pdf]**
*submitted on 2018-02-13 02:20:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I noticed that the numbers n*P + R(P) – n respectively P + n*R(P) - n, where P are Poulet numbers having only odd digits, R(P) the reversals of P and n positive integer, are often primes. In this paper I notice that the same is true for primes having only odd digits (see A030096 in OEIS for a list of such primes). Taken thirteen randomly chosen consecutive primes P having nine (odd) digits (from 971111137 to 971111993) I see that for all of them there exist at least a value of n smaller than 15 for which the number n*P + R(P) – n is prime (for 971111591, for instance, there exist four such values of n: 9, 11, 14, 15; for 971111137 three: 2, 4, 7; for 971111551 also three: 1, 2, 6; for 971111959 also three: 1, 9, 10; for 971111993 also three: 5, 6, 14).

**Category:** Number Theory

[1688] **viXra:1802.0134 [pdf]**
*submitted on 2018-02-11 06:47:29*

**Authors:** Ricardo Gil

**Comments:** 1 Page. There are alot of collected papers at Bexar County which I submitted to the Government.

Bexar County Detention Papers
(Topological Number Theory Formula/Equation/Algorithm)
By Ricard.gil@sbcglobal.net
January 9,2017 to Pretrail (Court February 26,2018 CCC4)
The objective of this paper is to show how one can take Gigori Pereleman complex arXiv 39 page paper and make it into a simple topological formula. I am revealing the ide I had in Bexar County Detention on viXra. I want to dedicate the Bexar County Detention papers to my Ex-Wife, Eddie Gil and Ashleigh Gil. (See attached Photo) & I can always be found at 3607 Ticonderoga, San Antonio Texas, where I am a permanent guest/resident.
I. The Topological Formula/Equation/Algorithm
1D=2D=3D/1=1/1D=2D=3D

**Category:** Number Theory

[1687] **viXra:1802.0097 [pdf]**
*submitted on 2018-02-08 06:40:23*

**Authors:** Jesús Álvarez Lobo

**Comments:** 3 Pages. Revista Escolar de la Olimpiada Iberoamericana de Matemática. Volume 34.

In this paper is proved an inequality involving a function of Fibonacci numbers in generic form and its limit at infinity is calculated using the asymptotic relationship given by Barr and Schooling in "The Field" (December 14, 1912).

**Category:** Number Theory

[1686] **viXra:1802.0095 [pdf]**
*submitted on 2018-02-08 06:59:49*

**Authors:** Jesús Álvarez Lobo

**Comments:** 2 Pages. Spanish.

Solution to the problem PMO33.2. Problem of Mathematical Duel 08 (Olomouc, Chorzow, Graz).
Determine all triples (x, y, z) of positive integers verifying the following equation:
3 + x + y + z = xyz

**Category:** Number Theory

[1685] **viXra:1802.0093 [pdf]**
*submitted on 2018-02-08 07:18:35*

**Authors:** Jesús Álvarez Lobo

**Comments:** 1 Page. Revista Escolar de la Olimpiada Iberoamericana de Matemática. Volume 21. Spanish.

Lobo's theorem for heronian triangles:
"Exists at least one heronian triangle such that two sides are consecutive natural numbers and its area is equal to n times the perimeter, for n = 1, 2, 3".
Teorema de Lobo para triángulos heronianos: Existe al menos un triángulo heroniano tal que dos de sus lados son números naturales consecutivos y su área es igual a n veces su perímetro, para n = 1, 2, 3.

**Category:** Number Theory

[1684] **viXra:1802.0039 [pdf]**
*submitted on 2018-02-05 04:50:15*

**Authors:** Andrea Prunotto

**Comments:** 5 Pages.

The condition of equiprobability among two events involving independent extractions of elements from a finite set is shown to be related to the solutions of a Diophantine equation.

**Category:** Number Theory

[1683] **viXra:1801.0416 [pdf]**
*submitted on 2018-01-30 10:29:06*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages. This suggests an entirely different angle on traditional number theory.

A phenomenon is described for analytic number theory. The purpose is to coordinate number theory and to give it a specific goal of modeling the phenomenon.

**Category:** Number Theory

[1682] **viXra:1801.0381 [pdf]**
*submitted on 2018-01-27 16:15:50*

**Authors:** Zeolla Gabriel martin

**Comments:** 5 Pages.

This article develops a formula for calculating the simple prime numbers-7 and the simple composite numbers-7 of the Golden Pattern.

**Category:** Number Theory

[821] **viXra:1805.0230 [pdf]**
*replaced on 2018-05-13 08:59:36*

**Authors:** David Stacha

**Comments:** 4 Pages.

Dear all,
Theorems proved more detailed respectively rigorously.
Thank you
Best regards
David Stacha

**Category:** Number Theory

[820] **viXra:1805.0230 [pdf]**
*replaced on 2018-05-12 08:38:47*

**Authors:** David Stacha

**Comments:** 3 Pages.

I will provide the solution of Erdös-Moser equation 1+2^p+3^p+...+(k)^p=(k+1)^p based on the properties of Bernoulli polynomials and prove that there is only one solution satisfying the above-mentioned equation. The Erdös-Moser equation (EM equation), named after Paul Erdös and Leo Moser has been studied by many number theorists through history since combines addition, powers and summation together. The open and very interesting conjecture of Erdös-Moser states that there is no other solution of EM equation than the trivial 1+2=3. Investigation of the properties and identities of the EM equation and ultimately providing the proof of this conjecture is the main purpose of this article.

**Category:** Number Theory

[819] **viXra:1805.0184 [pdf]**
*replaced on 2018-05-18 02:39:36*

**Authors:** Gang Li

**Comments:** 5 Pages.

We discuss an elementary approach to prove the second case of Fermat's last theorem (FLT). This is a continuation of an earlier paper that deals with the first case of FLT.

**Category:** Number Theory

[818] **viXra:1805.0165 [pdf]**
*replaced on 2018-05-12 07:52:40*

**Authors:** Timothy W. Jones

**Comments:** 2 Pages. A few clarifications.

Using circles that generate areas of positive integer values, together with the transcendence of pi, we show that x^n + y^n = z^n has no solution in positive integers for n greater than or equal to 3, Fermat's Last Theorem. It fits in a margin.

**Category:** Number Theory

[817] **viXra:1805.0162 [pdf]**
*replaced on 2018-05-09 11:30:03*

**Authors:** Stephen Marshall

**Comments:** 6 Pages.

Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician. He is remembered today for Goldbach's conjecture. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture: Every even integer which can be written as the sum of two primes (the strong conjecture) He then proposed a second conjecture in the margin of his letter: Every odd integer greater than 7 can be written as the sum of three primes (the weak conjecture). A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers. The “strong” conjecture has been shown to hold up through 4 × 1018, but remains unproven for almost 300 years despite considerable effort by many mathematicians throughout history. In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). In 2013, Harald Helfgott proved Goldbach's weak conjecture. The author would like to give many thanks to Helfgott’s proof of the weak conjecture, because this proof of the strong conjecture is completely dependent on Helfgott’s proof. Without Helfgott’s proof, this elementary proof would not be possible.

**Category:** Number Theory

[816] **viXra:1804.0416 [pdf]**
*replaced on 2018-05-15 17:05:10*

**Authors:** Waldemar Puszkarz

**Comments:** 6 Pages. New section added, abstract slightly changed.

Computer experiments reveal that twin primes tend to center on squareful multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of squareful multiples of 6 to the number of squarefree multiples of 6 equal $\pi^2/3-1$, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427 (for the first $10^{10}$ primes), a relative difference of ca $6.0\%$ measured against the expected value. A deviation from the expected value of this ratio, ca $1.9\%$, exists also for isolated primes. These numbers show that primes are drawn excessively to squareful numbers, a phenomenon most likely previously unknown.

**Category:** Number Theory

[815] **viXra:1804.0416 [pdf]**
*replaced on 2018-05-05 18:14:28*

**Authors:** Waldemar Puszkarz

**Comments:** 6 Pages. New abstract, conclusion extended, small cosmetic changes.

Computer experiments reveal that twin primes tend to center on squareful multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of squareful multiples of 6 to the number of squarefree multiples of 6 equal pi^2/3-1, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427 (for the first 10^10 primes), meaning that on average for every 1000 twin primes centered on squarefree multiples of 6, there are ca 137 twins that favor squareful multiples over squarefree multiples, a bias of ca 6.0%. The same kind of bias, though a bit weaker, ca 1.9%, exists for isolated primes.

**Category:** Number Theory

[814] **viXra:1804.0416 [pdf]**
*replaced on 2018-05-03 22:03:38*

**Authors:** Waldemar Puszkarz

**Comments:** 5 Pages. Conclusion extended plus some largely cosmetic modifications.

Computer experiments reveal that primes tend to occur next to squareful numbers more often than next to squarefree numbers compared to what one should expect from a non-biased distribution. The effect is more pronounced for prime pairs than for isolated primes.

**Category:** Number Theory

[813] **viXra:1804.0416 [pdf]**
*replaced on 2018-05-01 15:11:21*

**Authors:** Waldemar Puszkarz

**Comments:** 5 Pages. Latex version, slightly modified compared to the original.

Computer experiments reveal that primes tend to occur next to squareful numbers more often than next to squarefree numbers compared to what one should expect from a non-biased distribution. The effect is more pronounced for prime pairs than for isolated primes.

**Category:** Number Theory

[812] **viXra:1804.0267 [pdf]**
*replaced on 2018-05-05 14:16:48*

**Authors:** John Atwell Moody

**Comments:** 14 Pages.

Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue 1 at i\infty and one of residue -1 at 1. The ratio [\alpha: i\pi dtau] tends to 1 at the upper limit of [0,i\infty). Let \mu_{pm}:TxH->H be the action of multiplying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multiplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha-\pi d\tau)\wedge \mu_-^*)\alpha-i\pi d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}.

The rate of change of the magnitude is given by an integral involving a unitary character. Conjecturally the rate seminegative on the region 0

The form descends to the real projective line, it is locally meromorphic there with one pole and integrates to \pi e^{i\pi ({3\over 2}s + 1}. The value \zeta(s)=0 if and only if the integral along the arc from 0 to \infty not passing 1 is zero. This implies the arc passing through 1 equals a residue. We begin to relate the equality with the condition Re(s)=1/2.

**Category:** Number Theory

[811] **viXra:1804.0267 [pdf]**
*replaced on 2018-04-29 20:28:18*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue 1 at i\infty and one of residue -1 at 1. The ratio [\alpha: i\pi dtau] tends to 1 at the upper limit of [0,i\infty). Let \mu_{pm}:TxH->H be the action of multiplying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multiplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha-\pi d\tau)\wedge \mu_-^*)\alpha-i\pi d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}.

The rate of change of the magnitude is given by an integral involving a unitary character. Conjecturally the rate seminegative on the region 0

The form descends to the real projective line, it is locally meromorphic there with one pole and integrates to \pi e^{i\pi ({3\over 2}s + 1}. The value \zeta(s)=0 if and only if the integral along the arc from 0 to \ifnty is zero. The real part Re(s) is equal to 1/2 if and only if the real part of the square of the integral over the remaining part (the arc passing through 1) is zero.

**Category:** Number Theory

[810] **viXra:1804.0267 [pdf]**
*replaced on 2018-04-24 16:07:40*

**Authors:** John Atwell Moody

**Comments:** 6 Pages.

Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue 1 at i\infty and one of residue -1 at 1. The ratio [\alpha: i\pi dtau] tends to 1 at the upper limit of [0,i\infty). Let \mu_{pm}:TxH->H be the action of multiplying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multiplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha-\pi d\tau)\wedge \mu_-^*)\alpha-i\pi d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}.

The rate of change of the magnitude is given by an integral involving a unitary character. Conjecturally the rate seminegative on the region 0**Category:** Number Theory

[809] **viXra:1804.0267 [pdf]**
*replaced on 2018-04-23 14:20:11*

**Authors:** John Atwell Moody

**Comments:** 5 Pages.

Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue -1 at i\infty and one of residue 1 at 1. Let \mu_{pm}:TxH->H be the action of multipying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multipplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha-d\tau)\wedge \mu_-^*)\alpha-d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}.

The integral is equal to the stable distance from the origin in a dynamical system where a point is picked up and dropped off with two exponential rates which match if and only if Re(s)=1/2.

For 0**Category:** Number Theory

[808] **viXra:1804.0267 [pdf]**
*replaced on 2018-04-21 05:08:23*

**Authors:** John Atwell Moody

**Comments:** 7 Pages.

Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue -1 at i\infty and one of residue 1 at 1. Let \mu_{pm}:TxH->H be the action of multipying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multipplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha+d\tau)\wedge \mu_-^*)\alpha+d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}. The integral is equal to the stable distance from the origin in a dynamical system where a point is picked up and dropped off with two exponential rates. It spends time orbiting a fixed point or limit cycle before it is dropped off. It is only when c=1/2 that the two rates are equal.

**Category:** Number Theory

[807] **viXra:1804.0267 [pdf]**
*replaced on 2018-04-20 14:12:33*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue -1 at i\infty and one of residue 1 at 1. Let \mu_{pm}:TxH->H be the action of multipying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multipplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha+d\tau)\wedge \mu_-^*)\alpha+d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}. The integral is equal to the stable distance from the origin in a dynamical system where a point is picked up and dropped off with two exponential rates. It spends time orbiting a fixed point or limit cycle before it is dropped off. It is only when c=1/2 that the two rates are equal.

**Category:** Number Theory

[806] **viXra:1804.0038 [pdf]**
*replaced on 2018-05-16 17:09:45*

**Authors:** Philip Aaron Bloom

**Comments:** 3 Pages.

No simple proof of FLT has been established for every n > 2. We devise, for positive integral values of n, an elaborate algebraic identity, r^n + s^n = t^n, that holds for positive integral (r, s, t), a triple that we relate to positive integral (x, y, z) for which x^n + y^n = z^n holds. We infer that integral (r, s, t) equals integral (x, y, z) by using the unrestricted variable in our identity. For n > 2, we demonstrate that there exists no integral (r, s, t). Hence, for n > 2, there exists no integral (x, y, z).

**Category:** Number Theory

[805] **viXra:1804.0038 [pdf]**
*replaced on 2018-05-09 20:15:39*

**Authors:** Philip Aaron Bloom

**Comments:** 3 Pages.

No simple proof of FLT has been established for every n > 2. We devise, for positive integral values of n, an elaborate algebraic identity, r^n + s^n = t^n, that holds for positive integral (r, s, t), a triple that we relate to positive integral (x, y, z) for which x^n + y^n = z^n holds. We infer that integral (r, s, t) equals integral (x, y, z) by using the unrestricted variable in our identity. For n > 2, we demonstrate that there exists no integral (r, s, t). Hence, for n > 2, there exists no integral (x, y, z).

**Category:** Number Theory

[804] **viXra:1804.0038 [pdf]**
*replaced on 2018-04-27 19:59:38*

**Authors:** Philip Aaron Bloom

**Comments:** 3 Pages.

No simple proof of FLT has been established for every n > 2. We devise, for positive integral values of n, an elaborate algebraic identity, r^n + s^n = t^n, that holds for positive integral (r, s, t), a triple that we relate to positive integral (x, y, z) for which x^n + y^n = z^n holds. We infer that integral (r, s, t) equals integral (x, y, z) by using the unrestricted variable in our identity. For n > 2, we demonstrate that there exists no integral (r, s, t). Hence, for n > 2, there exists no integral (x, y, z).

**Category:** Number Theory

[803] **viXra:1804.0038 [pdf]**
*replaced on 2018-04-22 23:37:17*

**Authors:** Philip Aaron Bloom

**Comments:** 3 Pages.

No simple proof of FLT has been established for every n >2 . To prove FLT we devise, for positive integral n, a detailed algebraic identity, r^n + s^n = t^n, that holds for positive real (r, s, t), which we can relate to x^n + y^n = z^n holding for positive integral (x, y, z). We show for n > 2 that there exists no positive integral (r, s, t). We infer that integral (r, s, t) equals integral (x, y, z) by using our identity's unrestricted variable. So, for n > 2, there exists no integral (x, y, z).

**Category:** Number Theory

[802] **viXra:1804.0038 [pdf]**
*replaced on 2018-04-11 21:14:55*

**Authors:** Philip Aaron Bloom

**Comments:** 3 Pages.

No simple proof of FLT has been established for every n >2 . To prove FLT we devise, for positive integral n, a detailed algebraic identity, r^n + s^n = t^n, that holds for positive real (r, s, t), which we can relate to x^n + y^n = z^n holding for positive integral (x, y, z). We show for n > 2 that there exists no positive integral (r, s, t). We infer that integral (r, s, t) equals integral (x, y, z) by using our identity's unrestricted variable. So, for n > 2, there exists no integral (x, y, z).

**Category:** Number Theory

[801] **viXra:1804.0038 [pdf]**
*replaced on 2018-04-10 16:26:14*

**Authors:** Philip Aaron Bloom

**Comments:** 3 Pages.

No simple proof of FLT has been established for every n >2 . To prove FLT we devise, for positive integral n, a detailed algebraic identity, r^n + s^n = t^n, that holds for positive real (r, s, t), which we can relate to x^n + y^n = z^n holding for positive integral (x, y, z). We show for n > 2 that there exists no positive integral (r, s, t). We infer that integral (r, s, t) equals integral (x, y, z) by using our identity's unrestricted variable. So, for n > 2, there exists no integral (x, y, z).

**Category:** Number Theory

[800] **viXra:1804.0038 [pdf]**
*replaced on 2018-04-08 17:42:58*

**Authors:** Philip Aaron Bloom

**Comments:** 2 Pages.

**Category:** Number Theory

[799] **viXra:1804.0038 [pdf]**
*replaced on 2018-04-06 23:31:09*

**Authors:** Philip Aaron Bloom

**Comments:** 2 Pages.

For positive integral values of n : We design an algebraic identity r^n + s^n = t^n holding for positive real values r, s, t to relate to x^n + y^n = z^n holding for positive co-prime values x, y, z. We show for n > 2 that there exists no values of (r, s, t) in the co-prime subset of real numbers. We can infer that such {r, s, t} equals such {x, y, z} since our identity has an unrestricted variable. Hence, for n > 2, there exists no co-prime nor integral values of (x, y, z).

**Category:** Number Theory

[798] **viXra:1804.0038 [pdf]**
*replaced on 2018-04-05 22:40:40*

**Authors:** Philip Aaron Bloom

**Comments:** 2 Pages.

For positive integral values of n : We design an algebraic identity r^n + s^n = t^n holding for positive real values r, s, t to relate to x^n + y^n = z^n holding for positive co-prime values x, y, z. We show for n > 2 that there exists no values of (r, s, t) in the co-prime subset of real numbers. We can infer that such {r, s, t} equals such {x, y, z} since our identity has an unrestricted variable. Hence, for n > 2, there exists no co-prime nor integral values of (x, y, z).

**Category:** Number Theory

[797] **viXra:1804.0036 [pdf]**
*replaced on 2018-04-12 23:10:20*

**Authors:** Radomir Majkic

**Comments:** 10 Pages. Apart from small technical and obvious language corrections, there is no difference between this and previously submitted version of the Goldbach’s Conjectures paper.

Abstract: The prime numbers set is the three primes addition closed; each prime is the sum of three not necessarily distinct primes. All natural numbers are created on the set of all prime numbers according to the laws of the weak and strong Goldbach's conjectures. Thus all natural numbers are the Goldbach's numbers.

**Category:** Number Theory

[796] **viXra:1803.0668 [pdf]**
*replaced on 2018-04-18 10:58:01*

**Authors:** Haofeng Zhang

**Comments:** 19 Pages.

In this paper the author gives an elementary mathematics method to solve Fermat's
Last Theorem (FLT), in which let this equation become an one unknown number equation, in order to solve this equation the author invented a method called “Order reducing method for equations”, where the second order root compares to one order root, and with some necessary techniques the author successfully proved when x^(n-1)+y^(n-1)- z^(n-1) <= x^(n-2)+y^(n-2)-z^(n-2) there are no positive solutions for this equation, and also proves with the increasing of x there are still no positive integer solutions for this equation when x^(n-1)+y^(n-1)- z^(n-1)<=x^(n-2)+y^(n-2)- z^(n-2) is not satisfied.

**Category:** Number Theory

[795] **viXra:1803.0668 [pdf]**
*replaced on 2018-03-29 06:10:41*

**Authors:** Haofeng Zhang

**Comments:** 18 Pages.

In this paper the author gives an elementary mathematics method to solve Fermat's
Last Theorem (FLT), in which let this equation become an one unknown number equation, in
order to solve this equation the author invented a method called “Order reducing method for
equations”, where the second order root compares to one order root, and with some necessary
techniques the author successfully proved when x^(n-1)+y^(n-1)- z^(n-1) <= x^(n-2)+y^(n-2)-
z^(n-2) there are no positive solutions for this equation, and also proves with the increasing of x there are still no positive integer solutions for this equation when x^(n-1)+y^(n-1)- z^(n-1)<=
x^(n-2)+y^(n-2)- z^(n-2) is not satisfied.

**Category:** Number Theory

[794] **viXra:1803.0317 [pdf]**
*replaced on 2018-04-09 06:07:22*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

Let p(c,r,v)=e^{(c-1)(r+2v)} log({{\lambda(r+v)}\over{q(r+v)}}) log({{\lambda(v)}\over{q(v)}}),
f(c,r)=\int_{-\infty}^\infty p(c,r,v)+p(c,-r,v) dv. Let c be a real number such that 0

f(c,r)<0 and {{\partial}\over{\partial r}}f(c,r)>0 for all $r\ge 0$
while {{\partial}\over{\partial c}}f(c,r)<0 and {{\partial^2}\over {\partial c \partial r}}f(c,r)>0 for all r>0.

Then \zeta(c+i\omega) \ne 0 for all \omega.

**Category:** Number Theory

[793] **viXra:1803.0317 [pdf]**
*replaced on 2018-04-08 04:55:12*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

Let p(c,r,v)=e^{(c-1)(r+2v)} log({{\lambda(r+v)}\over{q(r+v)}}) log({{\lambda(v)}\over{q(v)}}),
f(c,r)=\int_{-\infty}^\infty p(c,r,v)+p(c,-r,v) dv. Let c be a real number such that 0

f(c,r)<0 and {{\partial}\over{\partial r}}f(c,r)>0 for all $r\ge 0$
while {{\partial}\over{\partial c}}f(c,r)<0 and {{\partial^2}\over {\partial c \partial r}}f(c,r)>0 for all r>0.

Then \zeta(c+i\omega) \ne 0 for all \omega.

**Category:** Number Theory

[792] **viXra:1803.0317 [pdf]**
*replaced on 2018-04-06 12:23:48*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

Let p(c,r,v)=e^{(c-1)(r+2v)} log({{\lambda(r+v)}\over{q(r+v)}}) log({{\lambda(v)}\over{q(v)}}),
f(c,r)=\int_{-\infty}^\infty p(c,r,v)+p(c,-r,v) dv. Let c be a real number such that 0

f(c,r)<0 and {{\partial}\over{\partial r}}f(c,r)>0 for all $r\ge 0$
while {{\partial}\over{\partial c}}f(c,r)<0 and {{\partial^2}\over {\partial c \partial r}}f(c,r)>0 for all r>0.

Then \zeta(c+i\omega) \ne 0 for all \omega.

**Category:** Number Theory

[791] **viXra:1803.0317 [pdf]**
*replaced on 2018-03-27 20:41:31*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

Let p(c,r,v)=e^{(c-1)(r+2v)} log({{\lambda(r+v)}\over{q(r+v)}}) log({{\lambda(v)}\over{q(v)}}),

f(c,r)=\int_{-\infty}^\infty p(c,r,v)+p(c,-r,v) dv.

Let c be a real number such that 0<c<1/2. Suppose that for all $r\ge 0$

f(c,r)<0, {{\partial}\over{\partial r}}f(c,r)>0, {{\partial}\over{\partial c}}f(c,r)<0, {{\partial^2}\over {\partial c \partial r}}f(c,r)>0.

Then \zeta(c+i\omega) \ne 0 for all \omega.

**Category:** Number Theory

[790] **viXra:1803.0265 [pdf]**
*replaced on 2018-05-03 05:16:57*

**Authors:** Yuri Heymann

**Comments:** 23 Pages.

In the present study we used the Dirichlet eta function as an extension of the Riemann zeta function in the strip Re(s) in ]0, 1[. We then determined the domain of admissible complex zeros of the Riemann zeta function in this strip using minimal constraints and the symmetries of the function. We also checked for zeros outside this strip. We found that the admissible domain of complex zeros excluding the trivial zeros is the critical line given by Re(s) = 1/2 as stated in the Riemann hypothesis.

**Category:** Number Theory

[789] **viXra:1803.0178 [pdf]**
*replaced on 2018-03-16 15:50:40*

**Authors:** Zeolla Gabriel Martin

**Comments:** 3 Pages.

This paper develops the formula that calculates the quantity of simple prime numbers that exist by golden patterns.

**Category:** Number Theory

[788] **viXra:1803.0017 [pdf]**
*replaced on 2018-04-03 12:13:21*

**Authors:** Pablo Hernan Pereyra

**Comments:** 4 Pages.

A discrete condition for twin prime numbers is established by Wilson's theorem. By synchronization is obtained a linear diophantine equation that implies by Bertrand Chebyshev's theorem the existence of infinite twin prime numbers.

**Category:** Number Theory

[787] **viXra:1803.0017 [pdf]**
*replaced on 2018-03-21 15:48:41*

**Authors:** Pablo Hernan Pereyra

**Comments:** 4 Pages.

A discrete condition for twin prime numbers is established by Wilson's theorem. By synchronization is obtained a linear diophantine equation that implies by Bertrand Chebyshev's theorem the existence of infinite twin prime numbers.

**Category:** Number Theory

[786] **viXra:1803.0017 [pdf]**
*replaced on 2018-03-09 13:03:48*

**Authors:** Pablo Hernan Pereyra

**Comments:** 4 Pages.

**Category:** Number Theory

[785] **viXra:1803.0017 [pdf]**
*replaced on 2018-03-07 11:30:17*

**Authors:** Pablo Hernan Pereyra

**Comments:** 4 Pages.

**Category:** Number Theory

[784] **viXra:1803.0017 [pdf]**
*replaced on 2018-03-06 14:56:08*

**Authors:** Pablo Hernan Pereyra

**Comments:** 3 Pages.

**Category:** Number Theory

[783] **viXra:1802.0353 [pdf]**
*replaced on 2018-03-08 04:44:34*

**Authors:** Dave Ryan T. Cariño

**Comments:** 17 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={3+2(n+x_p )├|x_p=x_3+x_5+x_7+x_11+⋯x_p ┤}〗 is true for all integer where n>0 for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of expanding term x_p.

**Category:** Number Theory

[782] **viXra:1802.0353 [pdf]**
*replaced on 2018-02-28 01:16:15*

**Authors:** Dave Ryan T. Cariño

**Comments:** 17 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={3+2(n+x_p )├|x_p=x_3+x_5+x_7+x_11+⋯x_p ┤}〗 is true for all integer where n>0 for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of expanding term x_p.

**Category:** Number Theory

[781] **viXra:1802.0353 [pdf]**
*replaced on 2018-02-27 05:19:51*

**Authors:** Dave Ryan T. Cariño

**Comments:** 16 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={3+2(n+x_p )├|x_p=x_3+x_5+x_7+x_11+⋯x_p ┤}〗 is true for all integer where n>0 for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of expanding term x_p.

**Category:** Number Theory

[780] **viXra:1802.0353 [pdf]**
*replaced on 2018-02-27 00:43:09*

**Authors:** Dave Ryan T. Cariño

**Comments:** 16 Pages.

**Category:** Number Theory

[779] **viXra:1802.0353 [pdf]**
*replaced on 2018-02-25 22:33:51*

**Authors:** Dave Ryan T. Cariño

**Comments:** 12 Pages.

Function and method for solving the distribution of prime numbers accurately using the combination of step functions, polynomial functions, inverse functions and continuous functions. Equation 〖lim┬(n→∞) p(n)〗〖={3+2(n+x_p )├|x_p=x_3+x_5+x_7+x_11+⋯x_p ┤}〗 is true for all integer where n>0 for the distribution and generation of exact values of prime numbers without exception. This formula is efficient by means of modern supercomputers for the task of adding new expression for x_p.

**Category:** Number Theory

[778] **viXra:1802.0309 [pdf]**
*replaced on 2018-02-22 14:33:51*

**Authors:** David Stacha

**Comments:** 9 Pages.

In this article I will provide the solution of Brocard`s problem n!+1=x^2 and I will prove the existence of the finite amount of Brown numbers, where the largest Brown number is (7,71), which represents the equation 7!+1=71^2. Brocard`s problem represents one of the open problem in mathematics from the field of number theory, which has been formulated by Henri Brocard in 1876 and represents the solutions of the following Diophantine equation n!+1=x^2.

**Category:** Number Theory

[777] **viXra:1802.0198 [pdf]**
*replaced on 2018-02-18 05:08:06*

**Authors:** John Yuk Ching Ting

**Comments:** 65 Pages. Targeting the General Public - Rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

L-functions form an integral part of the 'L-functions and Modular Forms Database' with far-reaching implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by Bernhard Riemann whereby all nontrivial zeros are [mathematically] conjectured to lie on the critical line of this function. This proposal is equivalently stated in this research paper as all nontrivial zeros are [geometrically] conjectured to exactly match the 'Origin' intercepts of this function. Deeply entrenched in number theory, prime number theorem entails analysis of prime counting function for prime numbers. Solving Riemann hypothesis would enable complete delineation of this important theorem. Involving proposals on the magnitude of prime gaps and their associated sets of prime numbers, Twin prime conjecture deals with prime gap = 2 (representing twin primes) and is thus a subset of Polignac's conjecture which deals with all even number prime gaps = 2, 4, 6,... (representing prime numbers in totality except for the first prime number '2'). Both nontrivial zeros and prime numbers are Incompletely Predictable entities allowing us to employ our novel Virtual Container Research Method to solve the associated hypothesis and conjectures.

**Category:** Number Theory

[776] **viXra:1802.0198 [pdf]**
*replaced on 2018-02-17 12:33:19*

**Authors:** John Yuk Ching Ting

**Comments:** 65 Pages. Targeting the General Public - Rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

L-functions form an integral part of the 'L-functions and Modular Forms Database' with far-reaching implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by Bernhard Riemann whereby all nontrivial zeros are [mathematically] conjectured to lie on the critical line of this function. This proposal is equivalently stated in this research paper as all nontrivial zeros are [geometrically] conjectured to exactly match the 'Origin' intercepts of this function. Deeply entrenched in number theory, prime number theorem entails analysis of prime counting function for prime numbers. Solving Riemann hypothesis would enable complete delineation of this important theorem. Involving proposals on the magnitude of prime gaps and their associated sets of prime numbers, Twin prime conjecture deals with prime gap = 2 (representing twin primes) and is thus a subset of Polignac's conjecture which deals with all even number prime gaps = 2, 4, 6,... (representing prime numbers in totality except for the first prime number '2'). Both nontrivial zeros and prime numbers are Incompletely Predictable entities allowing us to employ our novel Virtual Container Research Method to solve the associated hypothesis and conjectures.

**Category:** Number Theory

[775] **viXra:1802.0198 [pdf]**
*replaced on 2018-02-16 19:15:04*

**Authors:** John Yuk Ching Ting

**Comments:** 66 Pages. This research paper contains rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

**Category:** Number Theory

[774] **viXra:1802.0154 [pdf]**
*replaced on 2018-02-14 09:08:19*

**Authors:** Réjean Labrie

**Comments:** 6 Pages.

Abstract: Let N, n and k be integers larger than 1. Then for all N there exists a minimum threshold k such that for n>=N, if we cut the sequence of consecutive integers from 1 to n*(n+k) into n+k slices of length n, we always find at least a prime number in each slice.
It follows that π(n*(n+k)) > π(n*(n+k-1)) > π(n*(n+k-2)) > π(n*(n+k-3))> ...> π(2n)> π(n) where π(n) is the quantity of prime numbers smaller than or equal to n.

**Category:** Number Theory