[31] **viXra:1805.0408 [pdf]**
*submitted on 2018-05-21 07:56:52*

**Authors:** Zhang Tianshu

**Comments:** 15 Pages.

In this article, the author applies the mathematical induction, classifies positive integers, and passes operations according to the operational rule, to achieve the goal that proves the Collatz conjecture finally.

**Category:** Number Theory

[30] **viXra:1805.0398 [pdf]**
*submitted on 2018-05-21 20:33:48*

**Authors:** Chris Sloane

**Comments:** 20 Pages.

We discovered a way to write the equation x^n+y^n-z^n=0 first studied by Fermat, in powers of 3 other variables defined as; the sum t = x+y-z, the product (xyz) and another term r = x^2+yz-xt-t^2. Once x^n+y^n-z^n is written in powers of t, r and (xyz) we found that 3 cases of a prime factor q of x^2+yz divided t. We realized that from this alternative form of Fermat’s equation if all cases of q divided t that this would lead to a contradiction and solve Fermat’s Last Theorem. Intrigued by this, we then discover that the fourth case, q=3sp+1 also divides t when using a lemma that uniquely defines an aspect of Fermat’s equation resulting in the following theorem:
If x^p +y^p -z^p =0 and suppose x,y,z are pairwise co- prime then any prime factor q of (x^2 +yz) will divide t, where t= x+y-z

**Category:** Number Theory

[29] **viXra:1805.0397 [pdf]**
*submitted on 2018-05-21 22:55:01*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Linear implication, resource interpretation to avoid the frame problem, and the linear transformation property are not tautologous.
In particular, Tony Hoare's 1985 vending machine example as stated below is not tautologous:
"Suppose a candy bar by candy, and a dollar by $1. To state a dollar will buy one candy bar, write the implication $1 ⇒ candy. But in ordinary (classical or intuitionistic) logic, from A and A ⇒ B one can conclude A ∧ B. So, ordinary logic leads us to believe that we can buy the candy bar and keep our dollar!"

**Category:** Number Theory

[28] **viXra:1805.0387 [pdf]**
*submitted on 2018-05-22 08:41:03*

**Authors:** Edgar Valdebenito

**Comments:** 1 Page.

This note presents two integrals involving pi.

**Category:** Number Theory

[27] **viXra:1805.0379 [pdf]**
*submitted on 2018-05-22 15:54:17*

**Authors:** Philip A. Bloom

**Comments:** 3 Pages.

No simple proof of FLT (Fermat's last theorem) has been established for every n > 2. We devise, for positive integral values of n, a detailed algebraic identity, r ^ n + s ^ n = t ^ n, that holds for (r, s, t) such that r, s, t are positive integers - - - which we relate to (x, y, z), such that x, y, z are positive integers, for which x ^ n + y ^ n = z ^ n holds. For integral r ,s ,t ,x ,y ,z we infer that {(r, s, t)} = {(x, y ,z)} by using the unrestricted variable in our identity. For n > 2, we show there exists no (r, s, t) such that r, s, t are integral. Thus, for n > 2, there exists no (x, y, z) such that x, y, z are integral.

**Category:** Number Theory

[26] **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

[25] **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

[24] **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

[23] **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

[22] **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

[21] **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

[20] **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

[19] **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

[18] **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

[17] **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

[16] **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

[15] **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

[14] **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

[13] **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

[12] **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

[11] **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

[10] **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

[9] **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

[8] **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

[7] **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

[6] **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

[5] **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

[4] **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

[3] **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

[2] **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

[1] **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