**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(3) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(10) - 1203(6) - 1204(8) - 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(34) - 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(12) - 1511(9) - 1512(25)

2016 - 1601(14) - 1602(17) - 1603(77) - 1604(54) - 1605(28) - 1606(17) - 1607(19) - 1608(16) - 1609(22) - 1610(23) - 1611(12) - 1612(20)

2017 - 1701(19) - 1702(24) - 1703(29) - 1704(32) - 1705(25) - 1706(26) - 1707(21) - 1708(26) - 1709(12)

Any replacements are listed further down

[1573] **viXra:1709.0375 [pdf]**
*submitted on 2017-09-24 18:23:32*

**Authors:** Matanari Shimoinuda

**Comments:** 28 Pages.

This article is the summary of the spectral interpretation of critical zeroes of an L-function by Alain Connes. I try to examine the subject from the view of the representation theory and add some comments.

**Category:** Number Theory

[1572] **viXra:1709.0312 [pdf]**
*submitted on 2017-09-22 01:36:01*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we analyze the behavior of prime numbers.

**Category:** Number Theory

[1571] **viXra:1709.0295 [pdf]**
*submitted on 2017-09-20 06:46:03*

**Authors:** Faisal Amin Yassein Abdelmohssin

**Comments:** 4 Pages.

Fermat’s zero theorem is stated as follows: It is impossible to separate a square of a difference of two natural numbers into two squares of differences, or a cube power of a difference into two cube powers of differences, or a fourth power of a difference into two fourth powers, or in general, any power higher than the first, into two like powers of differences.

**Category:** Number Theory

[1570] **viXra:1709.0288 [pdf]**
*submitted on 2017-09-19 04:55:48*

**Authors:** Ranganath G. Kulkarni

**Comments:** 2 Pages.

A quadratic equation for prime numbers is assumed to be true that satisfy the following four rules. Some prime numbers violate these rules. Whereas some non prime numbers satisfy the four rules. They are not prime, therefore to make them violate the fourth rule we need to study how to choose the value of m and n so as make the quadratic equation as the primes generating formula.

**Category:** Number Theory

[1569] **viXra:1709.0258 [pdf]**
*submitted on 2017-09-17 13:33:28*

**Authors:** Victor Sorokine

**Comments:** 2 Pages.

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between
the second digits of the factors of the number А.

**Category:** Number Theory

[1568] **viXra:1709.0257 [pdf]**
*submitted on 2017-09-17 13:35:16*

**Authors:** Victor Sorokine

**Comments:** 2 Pages. French version

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between
the second digits of the factors of the number А.

L'égalité de Fermat est contradictoire entre les deuxièmes chiffres des facteurs du nombre A.

**Category:** Number Theory

[1567] **viXra:1709.0256 [pdf]**
*submitted on 2017-09-17 13:36:32*

**Authors:** Victor Sorokine

**Comments:** 2 Pages. Russian version

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between
the second digits of the factors of the number А.

Суть противоречия. Равенство Ферма противоречиво по вторым цифрам сомножителей числа А.

**Category:** Number Theory

[1566] **viXra:1709.0227 [pdf]**
*submitted on 2017-09-15 05:23:56*

**Authors:** José Francisco García Juliá

**Comments:** 2 Pages.

It is obtained a minor theorem related with the Fermat conjecture.

**Category:** Number Theory

[1565] **viXra:1709.0128 [pdf]**
*submitted on 2017-09-11 05:10:30*

**Authors:** Faisal Amin Yassein Abdelmohssin

**Comments:** 3 Pages.

Relationships among natural numbers constituting a Pythagorean triple (PT) and between these natural numbers constituting the Pythagorean triples (PTs) and Prime Numbers (PNs) have been found. These relationships are formulated as theorems; first theorem is that the natural numbers constituting a Pythagorean triple (PT) satisfy a certain equation related to sum of their differences; second theorem is that differences of sum of the natural numbers constituting a Pythagorean triple (PT) are prime numbers.

**Category:** Number Theory

[1564] **viXra:1709.0092 [pdf]**
*submitted on 2017-09-08 12:19:26*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas for pi.

**Category:** Number Theory

[1563] **viXra:1709.0039 [pdf]**
*submitted on 2017-09-04 11:17:48*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

Here is defined, following the RSA cryptosystem, a method of cryptography for polynomials over finite rings.

**Category:** Number Theory

[1562] **viXra:1709.0013 [pdf]**
*submitted on 2017-09-02 03:44:43*

**Authors:** Zhang Tianshu

**Comments:** 25 Pages.

Let us regard positive integers which have a common prime factor as a kind, then the positive half line of the number axis consists of infinite many recurring line segments which have same permutations of c kinds of integers’ points, where c≥1. In this article we shall prove Grimm’s conjecture by the method which changes stepwise symbols of each kind of composite numbers’ points at the original number axis, so as to form consecutive composite numbers’ points inside the limited field of proven Legendre- Zhang conjecture as the true.

**Category:** Number Theory

[1561] **viXra:1709.0003 [pdf]**
*submitted on 2017-09-01 07:17:32*

**Authors:** T.Nakashima

**Comments:** 1 Page.

In this paper, we prove Conway's problem.

**Category:** Number Theory

[1560] **viXra:1708.0421 [pdf]**
*submitted on 2017-08-28 08:10:40*

**Authors:** Edgar Valdebenito

**Comments:** 21 Pages.

This note presents a collection of double integrals for some classical constants.

**Category:** Number Theory

[1559] **viXra:1708.0411 [pdf]**
*submitted on 2017-08-28 10:09:24*

**Authors:** Preininger Helmut

**Comments:** 9 Pages.

In this paper we give an implementation of a Core(c) Number Sieve (for a given c=1,2,3,.. we sift out numbers that have in there factorization a prime with a power >= c). For c=2 we have a squarefree number sieve. (Note, that, for c=1, our implementation compute the usual prime number sieve.) Our goal is to use only one codebase and avoid extra algorithms for every c.
We use some well known algorithms and adopt it for our purpose.

**Category:** Number Theory

[1558] **viXra:1708.0400 [pdf]**
*submitted on 2017-08-28 00:19:33*

**Authors:** Lulu Karami

**Comments:** 48 Pages.

This submission demonstrates how to use the analytic class number formula to express certain quotients of Dedekind's Eta function as a unit raised to the power of a quoteint of class numbers, for particular number fields. It includes a loose derivation for some special cases of reciprocity laws and the Fourier series of particular Eisenstein series.

**Category:** Number Theory

[1557] **viXra:1708.0380 [pdf]**
*submitted on 2017-08-27 09:31:10*

**Authors:** I Gede Putra Yasa ''Gus Satya''

**Comments:** 24 Pages.

Division by 0 is not defined in mathematics. Mathematics suggests solutions by work around methods. However they give only approximate, not the actual or exact, results. Through this paper we propose methods to solve those problems. One characteristic of our solution methods is that they produce actual or exact results. They are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.

**Category:** Number Theory

[1556] **viXra:1708.0255 [pdf]**
*submitted on 2017-08-21 21:17:24*

**Authors:** Jun Chen

**Comments:** 5 Pages.

A new idea of the Goldbach conjecture has been studied, it is that the even number is more bigger, the average form of the sum of two primes are more larger too. And then, we prove that every sufficiently large even number is the sum of two primes.

**Category:** Number Theory

[1555] **viXra:1708.0234 [pdf]**
*submitted on 2017-08-19 11:06:29*

**Authors:** Ramaswamy Krishnan

**Comments:** 3 Pages.

This proof is based on an assumption that value of an infinite series cannot be obtained from a finite number of terms of the series. For all possible factors of (x + y -z) which are not factors of x or y or z, 3 infinite series can be developed, 2 convergent and 1 divergent. In all the 3 cases, the value of the infinite series can be obtained by considering only a finite number of terms. This gives the value for (x + y -z) = p to to the power of alpha * (p1) (p2) (p3). Thus proving Fermat's last theorem.

**Category:** Number Theory

[1554] **viXra:1708.0231 [pdf]**
*submitted on 2017-08-19 11:51:51*

**Authors:** Edigles Guedes, Cícera Guedes

**Comments:** 11 Pages.

In this paper, we construct a relation involving q-Pochhammer symbol, q-bracket, q-factorial and q-binomial coefficient among other things.

**Category:** Number Theory

[1553] **viXra:1708.0230 [pdf]**
*submitted on 2017-08-19 11:56:45*

**Authors:** Edigles Guedes, Cícera Guedes

**Comments:** 4 Pages.

In this paper, we construct a identity for a q-hypergeometric series.

**Category:** Number Theory

[1552] **viXra:1708.0221 [pdf]**
*submitted on 2017-08-19 03:46:20*

**Authors:** Mendzina Essomba Francois

**Comments:** 6 Pages.

I found a formula due to Ramanjan, I have given a generalization in this article

**Category:** Number Theory

[1551] **viXra:1708.0220 [pdf]**
*submitted on 2017-08-19 03:47:39*

**Authors:** Ranganath G. Kulkarni

**Comments:** 1 Page.

An equation for distribution of prime numbers is found that agree well with actual values of prime numbers in the range x. We find that Riemann's formula is approximate one. We need to study the variation of prime numbers with given number x and new variable r.

**Category:** Number Theory

[1550] **viXra:1708.0206 [pdf]**
*submitted on 2017-08-17 05:19:11*

**Authors:** Ramaswamy Krishnan

**Comments:** 3 Pages. Title has been changed a little bit. Instead of mod p, it should be mod p cubed

If\quad { 2 }^{ p-1 }\quad \equiv \quad 1\quad mod\quad ({ p }^{ 3 })\quad then\quad 2,-1,{ 2 }^{ p-2 }\quad are\quad solutions\quad to\quad the\quad equation\\ f(a)\quad =\quad 1\quad -\quad { a }^{ p }\quad -\quad { (1-a) }^{ p\quad \quad }\equiv \quad o\quad mod({ p }^{ 3 }).\quad Using\quad this\quad fact\quad and\quad an\quad expression\quad for\\ { (x+y) }^{ n }\quad \quad in\quad terms\quad of\quad xy\quad ,\quad (x+y)\quad ,\quad ({ x }^{ 2 }+xy+{ y }^{ 2 })\quad it\quad is\quad prooved\quad that\\ { 2 }^{ p-1 }\quad \ncong \quad 1\quad mod({ p }^{ 3 })\quad for\quad any\quad prime\quad 'p'.

**Category:** Number Theory

[1549] **viXra:1708.0204 [pdf]**
*submitted on 2017-08-17 05:25:20*

**Authors:** Ramaswamy Krishnan

**Comments:** 2 Pages.

If\quad f(a)\quad =\quad 1\quad -\quad { a }^{ p }\quad -\quad { (1-a) }^{ p }\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad and\quad even\quad if\quad { a }^{ 2 }-a+1\quad \ncong \quad 0\quad mod(p)\\ it\quad is\quad prooved\quad that\quad f({ a }_{ r })\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad .Then\quad using\quad the\quad fact\quad that\quad if\\ { 3 }^{ p-1 }\quad \equiv \quad 1\quad mod({ p }^{ 3 })\quad ,\quad { a }^{ 2 }+a+1\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad is\quad also\quad a\quad solution\quad to\quad \\ f(a)\quad \equiv \quad 0\quad mod({ p }^{ 3 }).

**Category:** Number Theory

[1548] **viXra:1708.0187 [pdf]**
*submitted on 2017-08-16 12:56:13*

**Authors:** Edgar Valdebenito

**Comments:** 6 Pages.

This note presents some formulas for pi.

**Category:** Number Theory

[1547] **viXra:1708.0181 [pdf]**
*submitted on 2017-08-16 06:53:04*

**Authors:** Kurmet Sultan

**Comments:** 29 pages, Written in Russian

The article presents the proof of the lonely runner conjecture.

**Category:** Number Theory

[1546] **viXra:1708.0177 [pdf]**
*submitted on 2017-08-16 00:48:13*

**Authors:** Kurmet Sultan

**Comments:** 49 Pages.

The article provides with the evidence of the Collatz conjecture.

**Category:** Number Theory

[1545] **viXra:1708.0158 [pdf]**
*submitted on 2017-08-14 08:00:35*

**Authors:** Faisal Amin Yassein Abdelmohssin

**Comments:** 3 Pages.

I found a pattern in the Pythagorean triples formed of the natural number12;{(12,5,13), (12,9,15), (12,16, 20), (12,35,37)}. The pattern is the decreasing value of the difference z - y for the triples such that the differences form a sequence of
the even numbers {8,6,4,2} in that order. The existence of such sequence for other natural numbers transforms the Pythagorean equation into a linear
equation in y .

**Category:** Number Theory

[1544] **viXra:1708.0141 [pdf]**
*submitted on 2017-08-13 04:27:47*

**Authors:** Hervé G.

**Comments:** 4 Pages.

Yet another proof that zeta(2)=Pi^2/6

**Category:** Number Theory

[1543] **viXra:1708.0114 [pdf]**
*submitted on 2017-08-10 21:43:45*

**Authors:** Joseph Dise

**Comments:** 3 Pages.

A minimum number of paired composite sums is shown for all 2N. By logical extension, it proves the existence of paired prime sums for all 2N.

**Category:** Number Theory

[1542] **viXra:1708.0108 [pdf]**
*submitted on 2017-08-10 09:44:07*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

The P-alic numbers are defined and we show a formula giving the decomposition of a polynomial according the valuations.

**Category:** Number Theory

[1541] **viXra:1708.0103 [pdf]**
*submitted on 2017-08-09 13:24:14*

**Authors:** Joseph Dise

**Comments:** 1 Page.

6x±1 are twin primes where x=6nm±(n±m) has no solution for positive integers x, n, and m. This paper follows that definition to its conclusion.

**Category:** Number Theory

[1540] **viXra:1708.0082 [pdf]**
*submitted on 2017-08-08 05:09:33*

**Authors:** François Mendzina Essomba

**Comments:** 12 Pages.

I propose new formulas for the transcendent functions that I discovered.

**Category:** Number Theory

[1539] **viXra:1708.0076 [pdf]**
*submitted on 2017-08-08 07:03:16*

**Authors:** Zhang Tianshu

**Comments:** 22 Pages.

Positive integers which are able to be operated to 1 by set operational rule of the Collatz conjecture and positive integers got via operations by the operational rule versus the set operational rule are one-to-one the same, thus we refer to converse operational routes, apply the mathematical induction, next classify positive integers to prove the Collatz conjecture by substeps according to beforehand prepared two theorems concerned.

**Category:** Number Theory

[1538] **viXra:1708.0063 [pdf]**
*submitted on 2017-08-06 18:22:13*

**Authors:** Mendzina Essomba Francois

**Comments:** 1 Page.

A strange formula for the calculation of the natural logarithm of any real number which I guessed almost effortlessly.
This formula has the advantage of being very efficient for the calculation of the logarithm of large numbers

**Category:** Number Theory

[1537] **viXra:1708.0048 [pdf]**
*submitted on 2017-08-04 15:49:56*

**Authors:** Mendzina Essomba Francois

**Comments:** 3 Pages.

I present some formulas found for the calculation of phy. Some of these formulas accelerate the calculation of decimals and others are written as a function of e and pi by continuous fractions

**Category:** Number Theory

[1536] **viXra:1708.0046 [pdf]**
*submitted on 2017-08-05 03:08:52*

**Authors:** Mendzina Essomba Francois

**Comments:** 2 Pages.

some of my many pi formulae

**Category:** Number Theory

[1535] **viXra:1708.0044 [pdf]**
*submitted on 2017-08-04 11:37:00*

**Authors:** Antoine Balan

**Comments:** 6 pages, written in french

Foundations of numbers theory are reviewed and some basic definitions are studied.

**Category:** Number Theory

[1534] **viXra:1707.0410 [pdf]**
*submitted on 2017-07-31 10:45:34*

**Authors:** Victor Sorokine

**Comments:** 6 Pages. English version

The essence of the proof:
From the known properties of the Fermat’s equality An+Bn=Cn follows:
If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero, then the new reduced numbers A°, B°, C° become /stay/ infinitely large.

**Category:** Number Theory

[1533] **viXra:1707.0395 [pdf]**
*submitted on 2017-07-29 16:17:02*

**Authors:** Ramón Ruiz

**Comments:** 36 Pages. This document is written in Spanish

Goldbach's Conjecture: “Every even integer greater than 2 can be expressed as the sum of two primes”.
In this document I used the prime numbers theorem enunciated by Carl Friedrich Gauss and the prime numbers theorem in arithmetic progressions. These two theorems applied to a combination of two arithmetic progressions of module 30 and that contain prime numbers, allows us to develop a nonprobability general formula to calculate, approximately, the number of prime pairs that adding up an even number x.
This research is based on a approach designed solely to demonstrate the Binary Goldbach Conjecture and the Twin Prime Conjecture.

**Category:** Number Theory

[1532] **viXra:1707.0392 [pdf]**
*submitted on 2017-07-30 03:32:36*

**Authors:** L. Castillo

**Comments:** 9 Pages.

I explore a method to characterize all the real numbers a,b such that all of $a - b, a^2 - b^2,...,a^n - b^n$ are integers for a given n and paying particular attention to the special case when neither of a and b are integers themselves.

**Category:** Number Theory

[1531] **viXra:1707.0335 [pdf]**
*submitted on 2017-07-26 03:43:24*

**Authors:** Idriss Olivier Bado

**Comments:** In 4 pages i give the proof to that conjecture

In this paper we give the proof of even gap conjecture whose can be expressed by it exists infinitely prime p such that p+n is prime for an even integer n and we deduce Polignac conjecture

**Category:** Number Theory

[1530] **viXra:1707.0279 [pdf]**
*submitted on 2017-07-20 13:09:45*

**Authors:** Emmanuil Manousos

**Comments:** 29 Pages.

Natural numbers have a strictly defined internal structure that is being revealed in the present article. This structure is inherent of the natural numbers and is not derived through the introduction of any axioms for the set of natural numbers. In the present article, we prove the fundamental theorems that determine this structure. As a consequence of this structure, a mathematical expression for the set of odd numbers that are not primes is derived. Given the set of odd numbers, we can identify the set of prime numbers. Additionally, a new method for expressing odd composite numbers as the product of powers of prime numbers is derived.

**Category:** Number Theory

[1529] **viXra:1707.0258 [pdf]**
*submitted on 2017-07-18 15:08:59*

**Authors:** Rédoane Daoudi

**Comments:** 5 Pages.

In this short paper we propose a new result about prime numbers: lim n→+∞ n/(p(n) − n(ln n + ln ln n − 1)) = +∞ .

**Category:** Number Theory

[1528] **viXra:1707.0241 [pdf]**
*submitted on 2017-07-17 13:26:16*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

This note presents formulas and fractals related with Ramanujan's trigonometric formula.

**Category:** Number Theory

[1527] **viXra:1707.0240 [pdf]**
*submitted on 2017-07-17 14:34:07*

**Authors:** François Mendzina Essomba

**Comments:** 5 Pages.

I present in this article some of my many formulas discovered on pi

**Category:** Number Theory

[1526] **viXra:1707.0237 [pdf]**
*submitted on 2017-07-17 22:43:49*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We have found the possible max- difference between two successive prime numbers, and by them, Lengendre's conjecture is verified.

**Category:** Number Theory

[1525] **viXra:1707.0217 [pdf]**
*submitted on 2017-07-15 15:40:09*

**Authors:** Mendzina Essomba Francois

**Comments:** 1 Page.

How to prove that an integer number is prime with the factoriels.
We give in this article which is not complete a property of the facoral which allows in an interval of given length to verify if the number is prime

**Category:** Number Theory

[1524] **viXra:1707.0176 [pdf]**
*submitted on 2017-07-13 03:54:39*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.

**Category:** Number Theory

[1523] **viXra:1707.0174 [pdf]**
*submitted on 2017-07-12 07:32:30*

**Authors:** Victor Sorokine

**Comments:** 4 Pages.

The proof of Fermat's last theorem for the base case /
Доказательство ВТФ для базового случая
ABSTRACT
The essence of the proof:
From the known properties of the Fermat’s equality An+Bn=Cn follows:
If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero, then the new reduced numbers A°, B°, C° become /remain/ infinitely large.
Суть доказательства:
Из базового равенства Ферма An+Bn=Cn следует:
Если вторые цифры всех простых сомножителей чисел А, В, С УМЕНЬШИТЬ до нуля, то новые уменьшенные числа А°, В°, С° становятся /остаются/ бесконечно большими.
(See also http://vixra.org/abs/1707.0092)

**Category:** Number Theory

[1522] **viXra:1707.0168 [pdf]**
*submitted on 2017-07-11 17:00:37*

**Authors:** Wes Hansen

**Comments:** 100 Pages.

In an earlier paper, “Q-Naturals: A Counter-Example to Tennenbaum’s Theorem,” we developed a set of non-standard naturals called q-naturals and demonstrated a counter-example to Tennenbaum’s Theorem. In this paper we extend the q-naturals to the Q-Universe and explore the properties of the various subsets along the way. In the process of this development, we realize that the standard Universe and the Q-Universe are simply the zeroth-order and first-order Universes, respectively, in a countable subsumption hierarchy of recursive Universes; there exist countably many counter-examples to Tennenbaum’s Theorem.

**Category:** Number Theory

[1521] **viXra:1707.0167 [pdf]**
*submitted on 2017-07-11 18:24:29*

**Authors:** Leszek Włodzimierz Guła

**Comments:** 7 Pages. In this work we have a new deductions.

The proof of the Fermat’s Last Theorem. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2. The proof of the Goldbach’s Conjecture.

**Category:** Number Theory

[1520] **viXra:1707.0152 [pdf]**
*submitted on 2017-07-10 13:09:21*

**Authors:** Rédoane Daoudi

**Comments:** 5 Pages.

In this paper we propose a conjecture about prime numbers. Based on the result of Pierre Dusart stating that the n th prime number is smaller than n(ln n + ln ln n − 0.9484) for n ≥ 39017 we propose that the n th prime number is smaller than n(ln n + ln ln n − 1+) when n → +∞.

**Category:** Number Theory

[1519] **viXra:1707.0092 [pdf]**
*submitted on 2017-07-06 04:02:34*

**Authors:** Victor Sorokine

**Comments:** 6 Pages. The text is in French

English. The essence of the proof
From the known properties of the Fermat’s equality A n +B n =C n follows:
If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero,
then the new reduced numbers A°, B°, C° become infinitely large.
From which follows the truth of the FLT?
Français. L'essence de la preuve :
A partir des propriétés connues de l'égalité de Fermat A n +B n =C n il suit:
Si les deuxièmes chiffres de tous les facteurs premiers des nombres A, B, C, réduit à zéro,
alors les nombres de nouveaux A°, B°, C°, devenir infiniment grand.
Ce qui implique la vérité du DTF ?
Русский. Суть доказательства:
Из базового равенства Ферма A n +B n =C n следует:
Если вторые цифры всех простых сомножителей чисел А, В, С УМЕНЬШИТЬ до нуля,
то новые уменьшенные числа А°, В°, С° становятся бесконечно большими.
Из чего следует истинность ВТФ?

**Category:** Number Theory

[1518] **viXra:1707.0086 [pdf]**
*submitted on 2017-07-05 14:31:20*

**Authors:** François Mendzina Essomba

**Comments:** 5 Pages.

I have come to the conclusion, after finishing a first reection on infinite sums, that all the functions which are written in the form of an infinite sum are written according to the famous Zeta function, this statement is explicitly presented in this article.

**Category:** Number Theory

[1517] **viXra:1707.0048 [pdf]**
*submitted on 2017-07-05 03:01:29*

**Authors:** Muneer Jebreel Karama

**Comments:** 2 Pages.

A positive integer is called Fixed Happy Cube Numbers (FHCN) in case, if you are cubing its digits and adding them together one time you got the same number. For example the number 153 is happy cube because;
153= 1^3+5^3+3^3, in fact this paper will address new propriety of this extraordinary happy cube number .

**Category:** Number Theory

[1516] **viXra:1707.0020 [pdf]**
*submitted on 2017-07-02 05:41:19*

**Authors:** I Gede Putra Yasa ``Gus Satya''

**Comments:** 20 Pages.

Division by 0 is not defined in mathematics.
Mathematics suggests solutions by work around methods. However those solution methods give only approximate, not the actual or exact, results.
Through this paper we propose methods to solve those problems.
One characteristic of our solution methods is that the methods produce actual or exact results. The solution methods are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.

**Category:** Number Theory

[1515] **viXra:1706.0543 [pdf]**
*submitted on 2017-06-28 23:55:17*

**Authors:** Liu Ran

**Comments:** 6 Pages.

Thank ancient philosopher Zeno, who brought such an interesting and meaningful paradox. It imply that the limit is reachable. Then we can deduct the infinity is about 618724203×10^26,

**Category:** Number Theory

[1514] **viXra:1706.0531 [pdf]**
*submitted on 2017-06-29 05:39:25*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 7 Pages.

In this research investigation, the author has presented a Recursive Past Equation and a Recursive Future Equation based on the Ananda-Damayanthi Normalized Similarity Measure considered to Exhaustion [1] (please see the addendum of [1] as well).

**Category:** Number Theory

[1513] **viXra:1706.0507 [pdf]**
*submitted on 2017-06-26 13:13:35*

**Authors:** Thomas Pierre Nicolas Jean Brouard

**Comments:** 3 Pages.

Using the η function, we show that the real part of the non-trivial zeros of the Riemann zeta fuction is 1 . Then, we calculate two big primes using the Riemann hypothesis as true. These two big primes have respectively more than one hundred millions digits and more than one billion digits.

**Category:** Number Theory

[1512] **viXra:1706.0479 [pdf]**
*submitted on 2017-06-25 18:25:43*

**Authors:** Kunle Adegoke

**Comments:** 11 Pages.

We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of a certain form.

**Category:** Number Theory

[1511] **viXra:1706.0457 [pdf]**
*submitted on 2017-06-23 13:24:49*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

In this note we present an integral for the constant pi:pi=3.1415926535...

**Category:** Number Theory

[1510] **viXra:1706.0414 [pdf]**
*submitted on 2017-06-21 05:00:47*

**Authors:** Marius Coman

**Comments:** 20 Pages.

A selection of forty sequences regarding primes and Fermat pseudoprimes from my yet unpublished papers, presented in "OEIS style", with definition of the terms of a sequence, examples, few first terms, notes and conjectures.

**Category:** Number Theory

[1509] **viXra:1706.0410 [pdf]**
*submitted on 2017-06-21 00:28:19*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: There exist an infinity of primes p having the property that concatenating s(p) – d(1) with s(p) – d(2) and repeatedly up to s(p) – d(k), where s(p) is the sum of digits of p and d(1),...,d(k) are the digits of p, is obtained a prime q. Example: such prime p is 127 because concatenating 9 (= 10 – 1) with 8 (= 10 – 2) and with 3 (= 10 – 7) is obtained a prime q = 983.

**Category:** Number Theory

[1508] **viXra:1706.0408 [pdf]**
*submitted on 2017-06-21 02:28:55*

**Authors:** Choe Ryong Gil

**Comments:** 12 pages, 2 tables

The aim of this paper is to show a new sufficient condition (NSC) by the Euler function for the Riemann hypothesis and its possibility. We build the NSC for any natural numbers ≥ 2 from well-known Robin theorem, and prove that the NSC holds for all odd and some even numbers while, the NSC holds for any even numbers under a certain condition, which would be called the condition (d).

**Category:** Number Theory

[1507] **viXra:1706.0407 [pdf]**
*submitted on 2017-06-21 02:30:07*

**Authors:** Choe Ryong Gil

**Comments:** 27 pages, 6 tables

In this paper, it is obtained a new estimate for the error term E(t) of the Mertens' formula sum_{p≤t}{p^{-1}}=loglogt+b+E(t), where t>1 is a real number, p is the prime number and b is the well-known Mertens' constant. We , first, provide an upper bound, not a lower bound, of E(p) for any prime number p≥3 and, next, give one in the form as E(t)<logt/√t for any real number t≥3. This is an essential improvement of already known results. Such estimate is very effective in the study of the distribution of the prime numbers.

**Category:** Number Theory

[1506] **viXra:1706.0381 [pdf]**
*submitted on 2017-06-18 22:43:41*

**Authors:** Lahcen Aghray

**Comments:** 5 Pages.

We obtain a parameterization of a Diophantine equation of degree 4

**Category:** Number Theory

[1505] **viXra:1706.0380 [pdf]**
*submitted on 2017-06-18 23:13:05*

**Authors:** Lahcen Aghray

**Comments:** 2 Pages.

The resolution of a Diophantine equation by calculating the intersection of a curve of degree 3 with a line

**Category:** Number Theory

[1504] **viXra:1706.0288 [pdf]**
*submitted on 2017-06-15 07:46:58*

**Authors:** Gang Li

**Comments:** 14 Pages.

An attempt of using elementary approach to prove Fermat's last theorem (FLT)
is given. For infinitely many prime numbers, Case I of the FLT can be proved
using this approach. Furthermore, if a conjecture proposed in this paper is
true (k-3 conjecture), then case I of the FLT is proved for all prime numbers.
For case II of the FLT, a constraint for possible solutions is obtained.

**Category:** Number Theory

[1503] **viXra:1706.0206 [pdf]**
*submitted on 2017-06-13 13:41:27*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

In this note we recall some formulas related with continued fractions , numbers , sequences and the constant pi.

**Category:** Number Theory

[1502] **viXra:1706.0205 [pdf]**
*submitted on 2017-06-13 13:45:55*

**Authors:** Edgar Valdebenito

**Comments:** 10 Pages.

In this note we briefly explore the equation: z^5+z^4-1=0

**Category:** Number Theory

[1501] **viXra:1706.0197 [pdf]**
*submitted on 2017-06-14 09:53:22*

**Authors:** Ryan Zielinski

**Comments:** 1 Page. This work is licensed under the CC BY 4.0, a Creative Commons Attribution License.

In this note we will use Faulhaber's Formula to explain why the odd Bernoulli numbers are equal to zero.

**Category:** Number Theory

[1500] **viXra:1706.0196 [pdf]**
*submitted on 2017-06-14 15:11:32*

**Authors:** Mendzina Essomba Francois

**Comments:** 2 Pages.

J present
two algorithms for calculating the natural logarithm of any real number. The first is an algorithm obtained by the
method of Archimedes for the calculation of pi and the second the product of a succession of rad
icals.

**Category:** Number Theory

[1499] **viXra:1706.0192 [pdf]**
*submitted on 2017-06-15 02:06:54*

**Authors:** Leszek W. Guła

**Comments:** 5 Pages. Certainly no scientist was working under such conditions. Nobody will ever announce to the world my creative proposals.

1. The truly marvellous proof of The Fermat's Last Theorem (FLT).
2. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2.

**Category:** Number Theory

[1498] **viXra:1706.0134 [pdf]**
*submitted on 2017-06-09 07:24:09*

**Authors:** Kolosov Petro

**Comments:** 12 pages, arXiv:1603.02468, MSC 2010: 40C15, 32A05

This paper describes a method of natural exponented power's $y=x^n, \ \forall(x,n) \in {\mathbb{N}}$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, based on induction, except above described theorems.
Keywords: power, power function, monomial, polynomial, power series, third power, series, finite difference, divided difference, high order finite difference, derivative, binomial coefficient, binomial theorem, Newton's binomial theorem, binomial expansion, n-th difference of n-th power, number theory, cubic number, cube, Euler number, exponential function, Pascal triangle, Pascal’s triangle, mathematics, math, maths, algebra, science, arxiv, preprint, series representation, series expansion, open scicence, calculus

**Category:** Number Theory

[1497] **viXra:1706.0112 [pdf]**
*submitted on 2017-06-07 14:51:48*

**Authors:** Kolosov Petro

**Comments:** 12 pages, 6 figures, arXiv:1705.02516

Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown.
Keywords: derivative, differential calculus, differentiation, Taylor's theorem, Taylor's formula, Taylor's series, Taylor's polynomial, power function, Binomial theorem, smooth function, real calculus, Newton's interpolation formula, finite difference, q-derivative, Jackson derivative, q-calculus, quantum calculus, (p,q)-derivative, (p,q)-Taylor formula, mathematics, math, maths, science, arxiv, preprint

**Category:** Number Theory

[1496] **viXra:1706.0111 [pdf]**
*submitted on 2017-06-07 19:48:40*

**Authors:** Kolosov Petro

**Comments:** 12 pages, 1 figure, arXiv:1608.00801

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials.
Keywords: finite difference, divided difference, high order finite difference, derivative, ode, pde, partial derivative, partial difference, power, power function, polynomial, monomial, power series, high order derivative, mathematics, differential calculus, math, maths, science, arxiv, preprint, algebra, calculus, open science, differential equations

**Category:** Number Theory

[1495] **viXra:1706.0102 [pdf]**
*submitted on 2017-06-06 11:10:53*

**Authors:** Marius Coman

**Comments:** 2 Pages.

This paper is inspired by one of my previous papers, namely “Large primes obtained concatenating the numbers P - d(k) where d(k) are the prime factors of the Poulet number P”, where I conjectured that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Because some of these Poulet numbers are 3-Poulet numbers of the form (6k + 1)*(6h + 1)*(6j + 1) I extend in this paper that idea conjecturing that for any prime p of the form 6k + 1 there exist an infinity of pairs of primes [q, r], of the form 6h + 1 and 6j + 1, such that the number obtained concatenating p*q*r – p with p*q*r – q with p*q*r – r then with p*q*r is prime.

**Category:** Number Theory

[1494] **viXra:1706.0097 [pdf]**
*submitted on 2017-06-06 04:10:22*

**Authors:** Marius Coman

**Comments:** 2 Pages.

This paper is inspired by one of my previous papers, namely “Large primes obtained concatenating the numbers P - d(k) where d(k) are the prime factors of the Poulet number P”, where I conjectured that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Because some of these Poulet numbers are 2-Poulet numbers of the form (6k + 1)*(6h + 1) I extend in this paper that idea conjecturing that for any prime p of the form 6k + 1 there exist an infinity of primes q of the form 6h + 1 such that the number obtained concatenating p*q – p with p*q – q then with p*q is prime.

**Category:** Number Theory

[1493] **viXra:1706.0037 [pdf]**
*submitted on 2017-06-05 05:55:08*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Example: using the sign “//” with the meaning “concatenated to”, for the Poulet number 129921 (= 3*11*31*127), the number (129921 – 3)//(129921 – 11)//(129921 – 31)//(129921 – 127)//129921 = 129918129910129890129794129921 is prime. Note that such primes are obtained for 10 from the first 90 Poulet numbers!

**Category:** Number Theory

[1492] **viXra:1706.0033 [pdf]**
*submitted on 2017-06-04 11:53:02*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: for many squares of primes (I conjecture that for an infinity of them) the numbers obtained concatenating 30 – d(1), 30 – d(2),..., 30 – d(k), where d(1),..., d(k) are the digits of a square of a prime, are primes. Example: for 1369 (= 37^2) the number obtained concatenating 29 = 30 – 1 with 27 = 30 – 3 with 24 = 30 – 6 with 21 = 30 – 9, i.e. the number 29272421, is prime. Note that for 35 from the first 200 squares of primes the numbers obtained this way are primes!

**Category:** Number Theory

[1491] **viXra:1706.0032 [pdf]**
*submitted on 2017-06-04 12:51:12*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following observation: for many Poulet numbers (I conjecture that for an infinity of them) the numbers obtained concatenating 30 – d(1), 30 – d(2),..., 30 – d(n), where d(1),..., d(n) are the digits of a n-digits Poulet number, are primes. Example: for 8911 the number obtained concatenating 22 = 30 – 8 with 21 = 30 – 9 with 29 = 30 – 1 with 29 = 30 – 1, i.e. the number 22212929, is prime.

**Category:** Number Theory

[1490] **viXra:1706.0029 [pdf]**
*submitted on 2017-06-04 02:09:22*

**Authors:** Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

**Comments:** 24 Pages.

Introduction of new trigonometric functions and mathematical constants.
The same mathematical equation connects the circle to the square, the sphere to the cube, the hyper-sphere to the hyper-cube, another also connects the ellipse to the rectangle, the ellipsoid to a rectangular parallelepiped, the hyper-ellipsoid To the rectangular hyper-parallelepiped.

**Category:** Number Theory

[1489] **viXra:1705.0461 [pdf]**
*submitted on 2017-05-29 12:33:16*

**Authors:** Edgar Valdebenito

**Comments:** 12 Pages.

This note presents some formulas and fractals related with the equation : x^3+x^2+1=0.

**Category:** Number Theory

[1488] **viXra:1705.0460 [pdf]**
*submitted on 2017-05-29 07:06:59*

**Authors:** John Yuk Ching Ting

**Comments:** 66 Pages. Rigorous proofs for Riemann hypothesis, Polignac's conjecture and Twin prime conjecture

L-functions form an integral part of the 'L-functions and Modular Forms Database' which is associated with far-reaching applications and implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by German mathematician Bernhard Riemann whereby all nontrivial zeros of Riemann zeta function are conjectured to be located on the critical line. This proposal is equivalently stated in this research paper as all nontrivial zeros are conjectured to exactly match the 'Origin' intercepts of Riemann zeta function. Deeply entrenched in number theory, prime number theorem involves analysis of the prime counting function for prime numbers. Solving Riemann hypothesis would result in a crucial primary by-product whereby absolute and full delineation of this important prime number theorem will occur. Involving the study of prime numbers [which are Incompletely Predictable entities], Twin prime conjecture involves the analysis of prime gap = 2 [representing all twin primes] and is thus a subset of Polignac's conjecture which involves the analysis of all even number prime gaps = 2, 4, 6,... [representing prime numbers in totality except for the first prime number '2']. Nontrivial zeros of Riemann zeta function are also Incompletely Predictable entities. With the common presence of Incompletely Predictable entities and with this helpful presence considered a major asset; the task to solve the above mentioned intractable open problems of Riemann hypothesis, Polignac's and Twin prime conjectures is conveniently amalgamated together in this paper. We employ our novel Virtual Container Research Method which acts essentially as foundation for the mathematical framework enabling successful completion of this monumental task.

**Category:** Number Theory

[1487] **viXra:1705.0395 [pdf]**
*submitted on 2017-05-28 03:10:36*

**Authors:** Oleg Cherepanov

**Comments:** 6 Pages. http://www.trinitas.ru/rus/doc/0016/001d/2254-chr.pdf

The discovered algorithm for extracting prime numbers from the natural series is alternative to both the Eratosthenes lattice and Sundaram and Atkin's sentences. The distribution of prime numbers does not have a formula, but if the number is one less than the prime number is an exponent of the integers, then there are no two scalar scalars whose sum is equal to the third integer in the same degree. This is the sound of P. Fermat's Great Theorem, the proof of which he could begin by using the Minor theorem known to him. The first part of the proof is here restored. But how did P. Fermat finish it?

**Category:** Number Theory

[1486] **viXra:1705.0393 [pdf]**
*submitted on 2017-05-27 18:13:53*

**Authors:** Caitherine Gormaund

**Comments:** 2 Pages.

Herein we introduce the subject of the Gormaund numbers, and prove a fundamental property thereof.

**Category:** Number Theory

[1485] **viXra:1705.0390 [pdf]**
*submitted on 2017-05-27 07:41:40*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present a method to obtain from a given prime p1 larger primes, namely inserting before of a digit of p1 a power of 3, and, once a prime p2 is obtained, repeating the operation on p2 and so on. By this method I obtained from a prime with 9 digits a prime with 36 digits (the steps are showed in this paper) using just the numbers 3, 9(3^2), 27(3^3) and 243(3^5).

**Category:** Number Theory

[1484] **viXra:1705.0379 [pdf]**
*submitted on 2017-05-26 06:15:45*

**Authors:** Ricardo.gil

**Comments:** 1 Page. Email solutions or suggestions to Ricardo.gil@sbcglobal.net

In math the the 7 Clay Math unsolved problems? Another problem is the question if there is a God(s)? In my paper the purpose is to explain that in the end we all meet our maker and that man does not have the power to cheat death. Like the Riemann Zeta function that remains unsolved and when solved will give insight to distribution of the Primes, giving or solving this open-end problem will help me solve a problem. This is the only problem I have not been able to solve and I am open sourcing it.

**Category:** Number Theory

[1483] **viXra:1705.0360 [pdf]**
*submitted on 2017-05-25 05:19:12*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page. Identity Proof FLT

This Identity proofs direct Fermats Last Theorem

**Category:** Number Theory

[1482] **viXra:1705.0343 [pdf]**
*submitted on 2017-05-22 13:32:15*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas for pi constant

**Category:** Number Theory

[1481] **viXra:1705.0342 [pdf]**
*submitted on 2017-05-22 13:36:50*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

This note presents some formulas related with the number z=LambertW(i),where LambertW(x) is the Lambert function.

**Category:** Number Theory

[1480] **viXra:1705.0293 [pdf]**
*submitted on 2017-05-19 11:18:33*

**Authors:** Prashanth Rao

**Comments:** 1 Page.

If p is any odd prime number and c is any odd number less than p, then there must exist a positive number c’ less than p, such that cc’= -2modp

**Category:** Number Theory

[1479] **viXra:1705.0289 [pdf]**
*submitted on 2017-05-19 07:56:37*

**Authors:** Helmut Preininger

**Comments:** 14 Pages.

In this paper we take a closer look to the distribution of the residues of squarefree natural numbers and explain an algorithm to compute those distributions.
We also give some conjectures about the minimal number of cycles in the squarefree arithmetic progression and explain an algorithm to compute this minimal numbers.

**Category:** Number Theory

[1478] **viXra:1705.0277 [pdf]**
*submitted on 2017-05-19 01:01:55*

**Authors:** Shaban A. Omondi Aura

**Comments:** 30 Pages. Preferably for journals, academies and conferences

This paper is concerned with formulation and demonstration of new versions of equations that can help us resolve problems concerning maximal gaps between consecutive prime numbers, the number of prime numbers at a given magnitude and the location of nth prime number. There is also a mathematical argument on why prime numbers as elementary identities on their own respect behave the way they do. Given that the equations have already been formulated, there are worked out examples on numbers that represent different cohorts. This paper has therefore attempted to formulate an equation that approximates the number of prime numbers at a given magnitude, from N=3 to N=〖10〗^25. Concerning the location of an nth prime number, the paper has devised a method that can help us locate a given prime number within specified bounds. Nonetheless, the paper has formulated an equation that can help us determine extremely bounded gaps. Lastly, using trans-algebraic number theory method, the paper has shown that unpredictable behaviors of prime numbers are due to their identity nature.

**Category:** Number Theory

[1477] **viXra:1705.0242 [pdf]**
*submitted on 2017-05-16 03:47:59*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any pair of twin primes [p, q], p ≥ 11, there exist a number n having the sum of its digits equal to 12 such that inserting n after the first digit of p respectively q are obtained two primes (almost always twins, as in the case [1481, 1483] where n = 48 is inserted in [11, 13], beside the case that the first digit of twins is different, as in the case [5669, 6661] where n = 66 is inserted in [59, 61]).

**Category:** Number Theory

[1476] **viXra:1705.0224 [pdf]**
*submitted on 2017-05-15 02:29:14*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p, p ≥ 7, there exist a prime q obtained inserting a number n with the sum of digits equal to 12 before the last digit of p.

**Category:** Number Theory

[1475] **viXra:1705.0221 [pdf]**
*submitted on 2017-05-15 03:41:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p, p ≥ 5, there exist a prime q obtained inserting a number n with the sum of digits equal to 12 after the first digit of p.

**Category:** Number Theory

[692] **viXra:1709.0312 [pdf]**
*replaced on 2017-09-24 03:38:04*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we find the axiomatic pattern of prime numbers.

**Category:** Number Theory

[691] **viXra:1708.0220 [pdf]**
*replaced on 2017-08-23 05:44:34*

**Authors:** Ranganath G. Kulkarni

**Comments:** 2 Pages.

An equation for distribution of prime numbers is found that agree well with actual values of prime numbers in the range x. We find that Riemann hypothesis may be wrong. We need to study the variation of new variable r with the given number x.

**Category:** Number Theory

[690] **viXra:1708.0108 [pdf]**
*replaced on 2017-08-12 18:30:40*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

We define here a p-adic like field, called the P-alic field. We replace in fact the ring Z by Q[X] and we show a formula of decomposition of a polynomial.

**Category:** Number Theory

[689] **viXra:1708.0108 [pdf]**
*replaced on 2017-08-11 13:40:29*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

Following the definition of p-adic numbers, we apply the definitions to the ring Z[X] instead of Z, we show a formula for a polynomial.

**Category:** Number Theory

[688] **viXra:1708.0108 [pdf]**
*replaced on 2017-08-10 18:38:28*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

The P-alic numbers are defined following the definition of p-adic numbers. We show a formula for a polynomial.

**Category:** Number Theory

[687] **viXra:1708.0103 [pdf]**
*replaced on 2017-09-24 12:48:05*

**Authors:** Joseph Dise

**Comments:** 1 Page.

6x±1 are twin primes where x=6nm±(n±m) has no solution for positive integers x, n, and m. This paper follows that definition to its conclusion.

**Category:** Number Theory

[686] **viXra:1707.0335 [pdf]**
*replaced on 2017-07-26 07:25:36*

**Authors:** Idriss Olivier Bado

**Comments:** In 4 pages i give the proof

In this paper i give the proof of Polignac conjecture and even gap cobjecture by using Chebotarev Artin theorem

**Category:** Number Theory

[685] **viXra:1707.0176 [pdf]**
*replaced on 2017-09-08 04:10:25*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.

**Category:** Number Theory

[684] **viXra:1707.0176 [pdf]**
*replaced on 2017-07-21 06:58:15*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.

**Category:** Number Theory

[683] **viXra:1707.0176 [pdf]**
*replaced on 2017-07-15 02:12:22*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

**Category:** Number Theory

[682] **viXra:1707.0176 [pdf]**
*replaced on 2017-07-14 06:25:46*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

**Category:** Number Theory

[681] **viXra:1707.0176 [pdf]**
*replaced on 2017-07-13 07:41:09*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

**Category:** Number Theory

[680] **viXra:1707.0168 [pdf]**
*replaced on 2017-08-05 15:40:50*

**Authors:** Wes Hansen

**Comments:** 100 Pages.

In an earlier paper, “Q-Naturals: A Counter-Example to Tennenbaum’s Theorem,” we developed a set of non-standard naturals called q-naturals and demonstrated a counter-example to Tennenbaum’s Theorem. In this paper we extend the q-naturals to the Q-Universe and explore the properties of the various subsets along the way. In the process of this development, we realize that the standard Universe and the Q-Universe are simply the zeroth-order and first-order Universes, respectively, in a countable subsumption hierarchy of recursive Universes; there exist countably many counter-examples to Tennenbaum’s Theorem.

**Category:** Number Theory

[679] **viXra:1707.0167 [pdf]**
*replaced on 2017-08-03 18:41:24*

**Authors:** Leszek Włodzimierz Guła

**Comments:** 10 Pages. This is the new variant of my work.

The proof of the Fermat’s Last Theorem. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2. The proof of the Goldbach’s Conjecture. The proof of the Beal’s Conjecture.

**Category:** Number Theory

[678] **viXra:1707.0023 [pdf]**
*replaced on 2017-07-18 03:50:04*

**Authors:** Juan Moreno Borrallo

**Comments:** 13 Pages.

In this paper it is proved the existence of a prime number in the interval between the square of any natural number greater than one, and the number resulting from adding or subtracting this natural number to its square (Oppermann’s Conjecture). As corollaries of this proof, they are proved three classical prime number’s conjectures: Legendre’s, Brocard’s, and Andrica’s. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated as corollary the existence of infinite prime numbers equal to the square of a natural number, plus a natural number inferior to that natural number, and minus a natural number inferior to that natural number.

**Category:** Number Theory

[677] **viXra:1707.0023 [pdf]**
*replaced on 2017-07-11 07:05:28*

**Authors:** Juan Moreno Borrallo

**Comments:** 12 Pages.

In this paper it is proved the existence of a prime number in the interval between the square of any natural number greater than one, and the number resulting from adding or subtracting this natural number to its square (Oppermann’s Conjecture). As corollaries of this proof, they are proved three classical prime number’s conjectures: Legendre’s, Brocard’s, and Andrica’s. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated as corollary the existence of infinite prime numbers equal to the square of a natural number, plus a natural number inferior to that natural number, and minus a natural number inferior to that natural number.

**Category:** Number Theory

[676] **viXra:1707.0023 [pdf]**
*replaced on 2017-07-05 03:11:26*

**Authors:** Juan Moreno Borrallo

**Comments:** 12 Pages.

In this paper it is proved the existence of a prime number in the interval between the square of any natural number greater than one, and the number resulting from adding or subtracting this natural number to its square (Oppermann’s Conjecture). As corollaries of this proof, they are proved three classical prime number’s conjectures: Legendre’s, Brocard’s, and Andrica’s. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated as corollary the existence of infinite prime numbers equal to the square of a natural number, plus a natural number inferior to that natural number, and minus a natural number inferior to that natural number.

**Category:** Number Theory

[675] **viXra:1707.0020 [pdf]**
*replaced on 2017-08-04 01:55:55*

**Authors:** I Gede Putra Yasa ``Gus Satya''

**Comments:** 23 Pages.

Division by 0 is not defined in mathematics. Mathematics suggests solutions by work around methods. However they give only approximate, not the actual or exact, results. Through this paper we propose methods to solve those problems. One characteristic of our solution methods is that they produce actual or exact results. They are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.

**Category:** Number Theory

[674] **viXra:1707.0010 [pdf]**
*replaced on 2017-07-30 11:06:38*

**Authors:** John Atwell Moody

**Comments:** 115 Pages.

Contents:

Analytic primality testing 1

Lefshetz numbers of modular curves 23

Grothendieck sections and rational points of modular curves 29

Rational points of modular curves 33

Conclusion about modular forms 41

Outline geometric proof of Mordell’s conjecture 48

Example:the Fermat curves
63

The residue calculation 69

The meaning of positive and negative 81

**Category:** Number Theory

[673] **viXra:1706.0507 [pdf]**
*replaced on 2017-06-27 12:34:17*

**Authors:** Thomas Pierre Nicolas Jean Brouard

**Comments:** 3 Pages.

Using the η function, we show that the real part of the non-trivial zeros of the Riemann zeta function is 1/2. Then, we calculate two big primes using the Riemann hypothesis as true. These two big primes have respectively more than one hundred millions digits and more than one billion digits.

**Category:** Number Theory

[672] **viXra:1706.0507 [pdf]**
*replaced on 2017-06-27 08:43:31*

**Authors:** Thomas Pierre Nicolas Jean Brouard

**Comments:** 3 Pages.

Using the η function, we show that the real part of the non-trivial zeros of the Riemann zeta fuction is 1/2. Then, we calculate two big primes using the Riemann hypothesis as true. These two big primes have respectively more than one hundred millions digits and more than one billion digits.

**Category:** Number Theory

[671] **viXra:1706.0197 [pdf]**
*replaced on 2017-06-28 09:57:27*

**Authors:** Ryan Zielinski

**Comments:** 4 Pages. Version 2 is an extended version of the original paper. Both works are licensed under the CC BY 4.0, a Creative Commons Attribution License.

In this note we will use Faulhaber's Formula to explain why the odd Bernoulli numbers are equal to zero.

**Category:** Number Theory

[670] **viXra:1706.0192 [pdf]**
*replaced on 2017-07-04 17:51:05*

**Authors:** Leszek W. Guła

**Comments:** 6 Pages.

1. The proper proof of The Fermat's Last Theorem (FLT).
2. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2.

**Category:** Number Theory

[669] **viXra:1706.0134 [pdf]**
*replaced on 2017-09-07 17:27:09*

**Authors:** Kolosov Petro

**Comments:** 13 pages, 5 figures, arXiv:1603.02468, Keywords: Power function, Monomial, Polynomial, Power series, Finite difference, Derivative, Differential calculus, Differentiation, Binomial coefficient, Newton's Binomial Theorem, Exponential function

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Binomial sum. Additionally, in section 4 exponential function’s $e^x$ representation is shown. In Application 3 generalized calculus of finite differences, based on expression (1.9) is shown.

**Category:** Number Theory

[668] **viXra:1706.0134 [pdf]**
*replaced on 2017-08-01 15:52:40*

**Authors:** Petro Kolosov

**Comments:** 11 pages, 5 figures, arXiv:1603.02468, MSC 2010: 40C15, 32A05.

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Binomial sum. Additionally, in section 4 exponential function’s $e^x$ representation is shown.

**Category:** Number Theory

[667] **viXra:1705.0460 [pdf]**
*replaced on 2017-08-09 04:23:37*

**Authors:** John Yuk Ching Ting

**Comments:** 69 Pages. Rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

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

**Category:** Number Theory

[666] **viXra:1705.0460 [pdf]**
*replaced on 2017-07-23 07:26:29*

**Authors:** John Yuk Ching Ting

**Comments:** 68 Pages. Rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

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

**Category:** Number Theory

[665] **viXra:1705.0360 [pdf]**
*replaced on 2017-05-25 07:46:44*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

This Identity proofs direct Fermats Last Theorem

**Category:** Number Theory