**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(1)

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)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(2) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(7) - 1203(6) - 1204(6) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(14) - 1211(10) - 1212(4)

2013 - 1301(5) - 1302(9) - 1303(16) - 1304(15) - 1305(12) - 1306(12) - 1307(25) - 1308(11) - 1309(8) - 1310(13) - 1311(15) - 1312(21)

2014 - 1401(20) - 1402(10) - 1403(26) - 1404(10) - 1405(17) - 1406(20) - 1407(33) - 1408(51) - 1409(47) - 1410(16) - 1411(16) - 1412(18)

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(17) - 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(23) - 1804(29) - 1805(33) - 1806(17) - 1807(11)

Any replacements are listed farther down

[1815] **viXra:1807.0283 [pdf]**
*submitted on 2018-07-15 09:39:24*

**Authors:** Andrey B. Skrypnik

**Comments:** 2 Pages.

This is the third result of applying Formula of Disjoint Sets of Odd Numbers

**Category:** Number Theory

[1814] **viXra:1807.0266 [pdf]**
*submitted on 2018-07-14 05:44:43*

**Authors:** Victor Sorokine

**Comments:** 2 Pages.

The study of digits of numbers related to the Fermat's equation shows Fermat's equality is impossible, for n>2.

**Category:** Number Theory

[1813] **viXra:1807.0265 [pdf]**
*submitted on 2018-07-14 05:47:47*

**Authors:** Victor Sorokine

**Comments:** 1 Page. Russian version

Изучение цифр чисел, зависящих от уравнения Теоремы Ферма, показывает, что уравнение невозможно для n>2.
The study of digits of numbers related to the Fermat's equation shows Fermat's equality is impossible, for n>2.

**Category:** Number Theory

[1812] **viXra:1807.0256 [pdf]**
*submitted on 2018-07-13 06:32:07*

**Authors:** Andrey B. Skrypnik

**Comments:** 2 Pages.

This is the second result of applying Formula of Disjoint Sets of Odd Numbers

**Category:** Number Theory

[1811] **viXra:1807.0182 [pdf]**
*submitted on 2018-07-10 06:58:50*

**Authors:** Andrey B. Skrypnik

**Comments:** 4 Pages.

This is the first result of applying Formula of Disjoint Sets of Odd Numbers

**Category:** Number Theory

[1810] **viXra:1807.0116 [pdf]**
*submitted on 2018-07-04 08:46:12*

**Authors:** Andrey B. Skrypnik

**Comments:** 4 Pages.

Destroyed another fortress of unproven tasks

**Category:** Number Theory

[1809] **viXra:1807.0100 [pdf]**
*submitted on 2018-07-05 03:34:36*

**Authors:** Robert Spoljaric

**Comments:** 1 Page.

Expressing the even perfect numbers as the sum of powers of 2

**Category:** Number Theory

[1808] **viXra:1807.0099 [pdf]**
*submitted on 2018-07-05 04:35:17*

**Authors:** Mihir Kumar Jha

**Comments:** 2 Pages.

The motive of this paper is to put forward a new approach to find the value of infinite sum series, given by S = (1+2+3+4+5+6-----) and to show that, the series converges at the value equal to zero.

**Category:** Number Theory

[1807] **viXra:1807.0055 [pdf]**
*submitted on 2018-07-02 12:18:14*

**Authors:** Zeolla Gabriel Martín

**Comments:** 8 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers (1 to 9), this paper is the continuation of the Golden Pattern. In this summary I will show that the inverted sum ordered by columns maintains amazing equivalences and proportions

**Category:** Number Theory

[1806] **viXra:1807.0053 [pdf]**
*submitted on 2018-07-02 16:18:53*

**Authors:** Alfredo Olmos hHernández

**Comments:** 3 Pages.

In this article, we proceed to study and apply the properties of the Gamma function to obtain a formula that allows us to calculate the sum of the first n factorial numbers.

**Category:** Number Theory

[1805] **viXra:1807.0048 [pdf]**
*submitted on 2018-07-03 02:58:32*

**Authors:** Simon Plouffe

**Comments:** 48 Pages.

These are scans of my pages of the inverter in 1988, they were done on a Mac 512 with diskettes and paper (about 1 foot thick). The main catalog was on Hypercard. The format at the time was 41 digits in scientific notation and some acronyms for constants
ND = golden ratio
sqr = sqrt
EG = exp(gamma)
s = sin
c = cos

**Category:** Number Theory

[1804] **viXra:1806.0427 [pdf]**
*submitted on 2018-06-29 05:37:45*

**Authors:** Pedro Hugo García Peláez

**Comments:** 3 Pages.

The sum of the numbers interspersed between two Fibonacci numbers has a curious property. The Fibonacci numbers act as border points of subsets of numbers that have a curious property.

**Category:** Number Theory

[1803] **viXra:1806.0423 [pdf]**
*submitted on 2018-06-27 07:50:08*

**Authors:** Victor Sorokine

**Comments:** 1 Page.

The number D=A^n+B^n-C^n < 0.

**Category:** Number Theory

[1802] **viXra:1806.0422 [pdf]**
*submitted on 2018-06-27 07:51:11*

**Authors:** Victor Sorokine

**Comments:** 1 Page. Russian version

Число D=A^n+B^n-C^n < 0.

**Category:** Number Theory

[1801] **viXra:1806.0420 [pdf]**
*submitted on 2018-06-27 08:03:57*

**Authors:** Andrey B. Skrypnik

**Comments:** 5 Pages.

Here is the only possible Proof of the Last Theorem of the Fermat in the requirements of the Fermat of 1637. - The theorem is proved universally for all numbers. - The theorem is proved on the apparatus of Diofont arithmetic. - The proof takes no more than two notebook pages of handwritten text. - The proof is clear to the pupil of the school. - The real meaning of Fermat's words about the margins of the book page is revealed. The secret of the Last Theorem of Fermat is discovered!

**Category:** Number Theory

[1800] **viXra:1806.0375 [pdf]**
*submitted on 2018-06-26 05:08:21*

**Authors:** Andrey B. Skrypnik

**Comments:** 5 Pages.

Here is the only possible Proof of the Last Theorem of the Fermat in the requirements of the Fermat of 1637.
- The theorem is proved universally for all numbers.
- The theorem is proved on the apparatus of Diofont arithmetic.
- The proof takes no more than two notebook pages of handwritten text.
- The proof is clear to the pupil of the school.
- The real meaning of Fermat's words about the margins of the book page is revealed.
The secret of the Last Theorem of Fermat is discovered!

**Category:** Number Theory

[1799] **viXra:1806.0353 [pdf]**
*submitted on 2018-06-25 02:58:27*

**Authors:** Andrei Lucian Dragoi

**Comments:** 11 Pages.

This paper presents a new conjecture on the divisor summatory function (also in relation with prime numbers), offering a much higher prediction accuracy than Dirichlet's divisor problem approach. Keywords: conjecture; divisor function; divisor summatory function; prime numbers; Dirichlet's divisor problem

**Category:** Number Theory

[1798] **viXra:1806.0330 [pdf]**
*submitted on 2018-06-22 11:08:32*

**Authors:** Andrey B. Skrypnik

**Comments:** 13 Pages.

Complete solution of Queens Puzzle

**Category:** Number Theory

[1797] **viXra:1806.0272 [pdf]**
*submitted on 2018-06-15 11:22:27*

**Authors:** Andrey B. Skrypnik

**Comments:** 4 Pages.

Now there is a formula for calculating all primes

**Category:** Number Theory

[1796] **viXra:1806.0219 [pdf]**
*submitted on 2018-06-19 08:57:13*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents two integrals.

**Category:** Number Theory

[1795] **viXra:1806.0175 [pdf]**
*submitted on 2018-06-12 09:19:38*

**Authors:** Yuki Yoshino

**Comments:** 13 Pages.

The number 0 has no distinction between positive and negative as -0 =+0, it is a number with special properties.
In this paper, we define a new concept of numbers that seems to be special, like 0, it's name is Ami.And I propose new axioms of real numbers extended by adding Ami to Hilbert 's real axiom.

**Category:** Number Theory

[1794] **viXra:1806.0095 [pdf]**
*submitted on 2018-06-07 07:25:11*

**Authors:** Vladimir Ushakov

**Comments:** 1 Page.

The key problem of MIT (matrix individualism theory) is to find a way to fill a square matrix of size N by numbers 1 to N in such a way that no row or column or diagonal contains two equal numbers, diagonal here is any line in matrix with +-45% slope. This definition of diagonal (+-45%) refers only to 1-st order individualism, later I will give a clear definition of higher order matrix individualism as well

**Category:** Number Theory

[1793] **viXra:1806.0062 [pdf]**
*submitted on 2018-06-05 08:43:10*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents a nontrivial identity that involve the number pi: pi=3.1415926535...

**Category:** Number Theory

[1792] **viXra:1806.0061 [pdf]**
*submitted on 2018-06-05 08:46:07*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some definite integrals.

**Category:** Number Theory

[1791] **viXra:1806.0052 [pdf]**
*submitted on 2018-06-06 02:29:12*

**Authors:** Kunle Adegoke

**Comments:** 18 Pages.

We derive several identities for arbitrary homogeneous second order recurrence sequences with constant coefficients. The results are then applied to present a harmonized study of six well known integer sequences, namely the Fibonacci sequence, the sequence of Lucas numbers, the Jacobsthal sequence, the Jacobsthal-Lucas sequence, the Pell sequence and the Pell-Lucas sequence.

**Category:** Number Theory

[1790] **viXra:1806.0051 [pdf]**
*submitted on 2018-06-06 02:32:45*

**Authors:** Kunle Adegoke

**Comments:** 9 Pages.

We derive various weighted summation identities, including binomial and double binomial identities, for Tribonacci numbers. Our results contain some previously known results as special cases.

**Category:** Number Theory

[1789] **viXra:1806.0046 [pdf]**
*submitted on 2018-06-06 05:11:33*

**Authors:** S Fushida-Hardy

**Comments:** 3 Pages.

We construct an isomorphism between the category of Ababou Constants and the category of affine bundles. We explore some special cases, namely the image of the integers equipped with the distinguished Ababou constant under the isomorphism, and prove that the distinguished Ababou constant is composite.

**Category:** Number Theory

[1788] **viXra:1806.0022 [pdf]**
*submitted on 2018-06-02 06:26:51*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

In this paper we define a novel kind of prime “p” with "m+n" digits whose first “m” digits represent a prime and the next “n” digits also represent a prime in just one possible way. These primes which we call as precious primes relate three different primes and therefore products of precious primes may allow representation of complex structures such as graphs.

**Category:** Number Theory

[1787] **viXra:1805.0544 [pdf]**
*submitted on 2018-05-31 13:45:07*

**Authors:** Zeolla Gabriel Martín

**Comments:** 7 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers-17, the discovery of a pattern to infinity, the demonstration of the inharmonics that are 2,3,5,7,11,13,17 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-17 and simple composite number-17
The simple prime numbers-17 are known as the 19-rough numbers.

**Category:** Number Theory

[1786] **viXra:1805.0443 [pdf]**
*submitted on 2018-05-25 04:29:03*

**Authors:** jean pierre MORVAN

**Comments:** 4 Pages.

Why the guess of COLLATZ is the true

**Category:** Number Theory

[1785] **viXra:1805.0431 [pdf]**
*submitted on 2018-05-23 17:07:06*

**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 9.

**Category:** Number Theory

[1784] **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

[1783] **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

[1782] **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

[1781] **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

[1780] **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

[1779] **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

[1778] **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

[1777] **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

[1776] **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

[1775] **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

[1774] **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

[1773] **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

[1772] **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

[1771] **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

[1770] **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

[1769] **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

[1768] **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

[1767] **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

[1766] **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

[1765] **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

[1764] **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

[1763] **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

[1762] **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

[1761] **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

[1760] **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

[1759] **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

[1758] **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

[1757] **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

[1756] **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

[1755] **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

[1754] **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

[1753] **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

[1752] **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

[1751] **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

[1750] **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

[1749] **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

[1748] **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

[1747] **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

[1746] **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

[1745] **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

[1744] **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

[1743] **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

[1742] **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

[1741] **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

[1740] **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

[1739] **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

[1738] **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

[1737] **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

[1736] **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

[1735] **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

[1734] **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

[1733] **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

[1732] **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

[1731] **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

[1730] **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

[1729] **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

[1728] **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

[1727] **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

[1726] **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

[1725] **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

[1724] **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

[1723] **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

[1722] **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

[1721] **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

[1720] **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

[1719] **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

[873] **viXra:1807.0116 [pdf]**
*replaced on 2018-07-08 02:27:39*

**Authors:** Andrey B. Skrypnik

**Comments:** 4 Pages.

Destroyed another fortress of unproven tasks

**Category:** Number Theory

[872] **viXra:1807.0100 [pdf]**
*replaced on 2018-07-14 00:12:41*

**Authors:** Robert Spoljaric

**Comments:** 2 Pages.

In this note we show which rows in Pascal’s Triangle sum to Perfect Numbers. We end this note with an algorithm allowing us to trivially calculate all the proper divisors for any Perfect Number greater than 6.

**Category:** Number Theory

[871] **viXra:1807.0100 [pdf]**
*replaced on 2018-07-13 02:53:24*

**Authors:** Robert Spoljaric

**Comments:** 2 Pages.

In this note we show which rows in Pascal’s Triangle sum to Perfect Numbers. We end this note with a conjectured algorithm allowing us to calculate all the proper divisors for any Perfect Number greater than 6.

**Category:** Number Theory

[870] **viXra:1807.0100 [pdf]**
*replaced on 2018-07-09 19:57:01*

**Authors:** Robert Spoljaric

**Comments:** 2 Pages.

In this note we show which rows in Pascal’s Triangle sum to Perfect Numbers. We end this note with a conjectured algorithm allowing us to calculate all the proper divisors for any Perfect Number greater than 6.

**Category:** Number Theory

[869] **viXra:1807.0100 [pdf]**
*replaced on 2018-07-08 18:53:02*

**Authors:** Robert Spoljaric

**Comments:** 2 Pages.

In this note we show which rows in Pascal’s Triangle sum to Perfect Numbers.

**Category:** Number Theory

[868] **viXra:1807.0100 [pdf]**
*replaced on 2018-07-08 02:34:08*

**Authors:** Robert Spoljaric

**Comments:** 2 Pages.

In this note we show which rows in Pascal’s Triangle sum to Perfect Numbers.

**Category:** Number Theory

[867] **viXra:1807.0100 [pdf]**
*replaced on 2018-07-07 05:15:24*

**Authors:** Robert Spoljaric

**Comments:** 2 Pages.

In this note we show that the even Perfect Numbers can be found in Pascal’s Triangle by expressing the even Perfect Numbers as sums of powers of 2

**Category:** Number Theory

[866] **viXra:1806.0420 [pdf]**
*replaced on 2018-07-03 11:51:45*

**Authors:** Andrey B. Skrypnik

**Comments:** 5 Pages.

Here is the only possible Proof of the Last Theorem of the Fermat in the requirements of the Fermat of 1637. - The theorem is proved universally for all numbers. - The theorem is proved on the apparatus of Diofont arithmetic. - The proof takes no more than two notebook pages of handwritten text. - The proof is clear to the pupil of the school. - The real meaning of Fermat's words about the margins of the book page is revealed. The secret of the Last Theorem of Fermat is discovered!

**Category:** Number Theory

[865] **viXra:1806.0046 [pdf]**
*replaced on 2018-06-10 07:22:13*

**Authors:** Yellocord soc.

**Comments:** 3 Pages.

We construct an isomorphism between the category of Ababou constants and the category of affine bundles. We explore a special case, namely the image of the integers equipped with the distinguished Ababou constant under the isomorphism. Using our new machinery we prove that the distinguished Ababou constant is composite.

**Category:** Number Theory

[864] **viXra:1806.0022 [pdf]**
*replaced on 2018-06-04 20:47:27*

**Authors:** Prashanth R. Rao, Tirumal Rao

**Comments:** 2 Pages.

In this paper we define a novel kind of prime “p” with (m+n) digits whose first “m” digits represent a prime and the next “n” digits also represent a prime in just one possible way. These primes which we call as precious primes relate three different primes and therefore products of precious primes may allow representation of complex structures such as graphs.

**Category:** Number Theory

[863] **viXra:1805.0544 [pdf]**
*replaced on 2018-06-06 19:02:43*

**Authors:** Zeolla Gabriel Martin

**Comments:** 10 Pages. The previous file was damaged

This paper develops the divisibility of the so-called Simple Primes numbers-17, the discovery of a pattern to infinity, the demonstration of the inharmonics that are 2,3,5,7,11,13,17 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-17 and simple composite number-17
The simple prime numbers-17 are known as the 19-rough numbers.

**Category:** Number Theory

[862] **viXra:1805.0443 [pdf]**
*replaced on 2018-05-25 07:15:29*

**Authors:** Jean Pierre Morvan

**Comments:** 4 Pages.

Pourquoi la conjecture de COLLATZ est vraie.

**Category:** Number Theory

[861] **viXra:1805.0398 [pdf]**
*replaced on 2018-05-27 19:21:30*

**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

[860] **viXra:1805.0379 [pdf]**
*replaced on 2018-07-09 21:33:43*

**Authors:** Philip A. Bloom

**Comments:** 3 Pages.

There is no confirmed, simple proof of Fermat's last theorem (FLT) for each integral n > 2. Our proposed, simple proof of FLT is based on our algebraic identity, a function of two variables, denoted for convenience as r ^ n + s ^ n = t ^ n. For positive integral values of n, we relate ( r, s ,t ) for which r ^ n + s ^ n = t ^ n holds, with ( x, y, z ) for which x ^ n + y ^ n = z ^ n holds. From these true equations we infer by direct argument ( not by way of contradiction ), that { ( r, s, t ) | r, s, t in Z, r ^ n + s ^ n = t ^ n } = { ( x, y, z )| r, s, t in Z, x ^ n + y ^ n = z ^ n } for any given n for which these sets are nonempty. Also, we show, for n > 2, that {( r, s, t ) | r, s, t in Z } is null. Hence, for n > 2, set { ( x, y, z ) | x, y, z in Z } is null

**Category:** Number Theory

[859] **viXra:1805.0379 [pdf]**
*replaced on 2018-06-29 23:35:11*

**Authors:** Philip A. Bloom

**Comments:** 3 Pages.

There is no confirmed, simple proof of Fermat's last theorem (FLT) for each integral n > 2. Our proposed, simple proof of FLT is based on our algebraic identity, a function of two variables, denoted for convenience as r ^ n + s ^ n = t ^ n. For positive integral values of n, we relate ( r, s ,t ) for which r ^ n + s ^ n = t ^ n holds, with ( x, y, z ) for which x ^ n + y ^ n = z ^ n holds. From these true equations we infer by direct argument ( not by way of contradiction ), that { ( r, s, t ) | r, s, t in Z, r ^ n + s ^ n = t ^ n } = { ( x, y, z )| r, s, t in Z, x ^ n + y ^ n = z ^ n } for any given n for which these sets are nonempty. Also, we show, for n > 2, that {( r, s, t ) | r, s, t in Z } is null. Hence, for n > 2, set { ( x, y, z ) | x, y, z in Z } is null

**Category:** Number Theory

[858] **viXra:1805.0379 [pdf]**
*replaced on 2018-06-05 13:36:54*

**Authors:** Philip A. Bloom

**Comments:** Pages.

There is no confirmed, simple proof of Fermat's last theorem (FLT) for each integral n > 2. Our proposed, simple proof of FLT is based on our algebraic identity, a function of two variables, denoted for convenience as r ^ n + s ^ n = t ^ n. For positive integral values of n, we relate ( r, s ,t ) for which r ^ n + s ^ n = t ^ n holds, with ( x, y, z ) for which x ^ n + y ^ n = z ^ n holds. From these true equations we infer by direct argument ( not by way of contradiction ), that { ( r, s, t ) | r, s, t in Z, r ^ n + s ^ n = t ^ n } = { ( x, y, z )| r, s, t in Z, x ^ n + y ^ n = z ^ n } for any given n for which these sets are nonempty. Also, we show, for n > 2, that {( r, s, t ) | r, s, t in Z } is null. Hence, for n > 2, set { ( x, y, z ) | x, y, z in Z } is null

**Category:** Number Theory

[857] **viXra:1805.0379 [pdf]**
*replaced on 2018-05-30 11:22:41*

**Authors:** Philip A. Bloom

**Comments:** Pages.

There is no confirmed simple proof of FLT for each integral n > 2. Our proposed, simple proof of FLT is based on r ^ n + s ^ n = t ^ n, our algebraic identity that is, for integral n > 1 , a function of two variables. This statement is true for (r, s ,t) with integral r ,s, t > 1 , which we relate to (x, y, z) for which x, y, z > 1 are integers such that x ^ n + y ^ n = z ^ n holds. From these two true equations we infer by direct argument (not by way of contradiction), for any given value of n, the equality {(r, s, t)} = {(x, y, z)}. In addition, we demonstrate, for {n > 2}, that (r , s, t) with integral r, s, t is null. So, for {n > 2}, set {(x, y, z)} with integral x, y, z is null.

**Category:** Number Theory

[856] **viXra:1805.0379 [pdf]**
*replaced on 2018-05-24 23:48:32*

**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

[855] **viXra:1805.0362 [pdf]**
*replaced on 2018-05-23 10:37:31*

**Authors:** Ricardo Gil

**Comments:** 1 Page. @warlockach

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

**Category:** Number Theory

[854] **viXra:1805.0269 [pdf]**
*replaced on 2018-05-25 01:31:15*

**Authors:** Victor Sorokine

**Comments:** 2 Pages.

All calculations are done with numbers in base n, a prime number greater than 2.
If in the Fermat equality A^n+B^n-C^n=0, the number U=A+B-C=un^k ends by k zeroes and
the factor u is not ending by digit 1, after discarding k-digit endings in the numbers A, B, C,
the Fermat equality turns into an inequality, which is NOT transformed back into an equality
after the restoration of k-digit endings.

**Category:** Number Theory

[853] **viXra:1805.0268 [pdf]**
*replaced on 2018-05-25 01:32:16*

**Authors:** Victor Sorokine

**Comments:** 2 Pages. Russian version

Доказательство проводится в системе счисления с простым основанием n>2.
Если в равенстве Ферма A^n+B^n-C^n=0 число U=A+B-C=un^k оканчивается на k
нулей и его сомножитель u не оканчивается на цифру 1, то после отбрасывания k-
значных окончаний в числах A, B, C равенство Ферма превращается в неравенство,
которое после восстановления k-значных окончаний в равенство уже НЕ превращается.

**Category:** Number Theory

[852] **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

[851] **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

[850] **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

[849] **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

[848] **viXra:1804.0416 [pdf]**
*replaced on 2018-07-02 20:34:06*

**Authors:** Waldemar Puszkarz

**Comments:** 7 Pages. Last version submitted to viXra. First version submitted to arXiv.

Computer experiments reveal that twin primes tend to center on nonsquarefree multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of nonsquarefree 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, 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. This shows that the distribution of primes is biased towards nonsquarefree numbers, a phenomenon most likely previously unknown. For twins, this leads to nonsquarefree numbers gaining an excess of $1.2\%$ of the total number of twins. In the case of isolated primes, this excess for nonsquarefree numbers amounts to $0.4\%$ of the total number of such primes. The above numbers are for the first $10^{10}$ primes, with the bias showing a tendency to grow, at least for isolated primes.

**Category:** Number Theory

[847] **viXra:1804.0416 [pdf]**
*replaced on 2018-06-27 19:48:55*

**Authors:** Waldemar Puszkarz

**Comments:** 7 Pages. New references added.

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, 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. This shows that the distribution of primes is biased towards squareful numbers, a phenomenon most likely previously unknown. For twins, this leads to squareful numbers gaining an excess of 1.2% of the total number of twins. In the case of isolated primes, this excess for squareful numbers amounts to 0.4% of the total number of such primes. The above numbers are for the first 10^10 primes, with the bias showing a tendency to grow, at least for isolated primes.

**Category:** Number Theory

[846] **viXra:1804.0416 [pdf]**
*replaced on 2018-05-26 16:55:29*

**Authors:** Waldemar Puszkarz

**Comments:** 7 Pages. Slightly modified and extended.

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, 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. This shows that the distribution of primes is biased towards squareful numbers, a phenomenon most likely previously unknown. For twins, this leads to squareful numbers gaining an excess of 1.2% of the total number of twins. In the case of isolated primes, this excess for squareful numbers amounts to 0.4% of the total number of such primes. The above numbers are for the first 10^10 primes, with the bias showing a tendency to grow, at least for isolated primes.

**Category:** Number Theory

[845] **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

[844] **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

[843] **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

[842] **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

[841] **viXra:1804.0409 [pdf]**
*replaced on 2018-06-27 04:06:28*

**Authors:** Andrey B. Skrypnik

**Comments:** 4 Pages.

Now there is a formula for calculating all primes

**Category:** Number Theory

[840] **viXra:1804.0267 [pdf]**
*replaced on 2018-06-06 04:30:05*

**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

[839] **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

[838] **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

[837] **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

[836] **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

[835] **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

[834] **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

[833] **viXra:1804.0259 [pdf]**
*replaced on 2018-05-21 14:45:07*

**Authors:** M. A. Thomas

**Comments:** 3 Pages.

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

**Category:** Number Theory

[832] **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

[831] **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

[830] **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

[829] **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

[828] **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

[827] **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

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

**Authors:** Philip Aaron Bloom

**Comments:** 2 Pages.

**Category:** Number Theory

[825] **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

[824] **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

[823] **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

[822] **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

[821] **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