[26] **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

[25] **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

[24] **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

[23] **viXra:1708.0380 [pdf]**
*replaced on 2017-12-04 09:06:19*

**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

[22] **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

[21] **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

[20] **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

[19] **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

[18] **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

[17] **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

[16] **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

[15] **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

[14] **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

[13] **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

[12] **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

[11] **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

[10] **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

[9] **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

[8] **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

[7] **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

[6] **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

[5] **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

[4] **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

[3] **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

[2] **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

[1] **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