[21] **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

[20] **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

[19] **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

[18] **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

[17] **viXra:1707.0279 [pdf]**
*replaced on 2018-09-09 04:06:56*

**Authors:** Emmanuil Manousos

**Comments:** 29 Pages.

It holds that every product of natural numbers can also be written as a sum. The inverse does not hold when 1 is excluded from the product. For this reason, the investigation of natural numbers should be done through their sum and not through their product. Such an investigation is presented in the present article. We prove that primes play the same role for odd numbers as the powers of 2 for even numbers, and vice versa. The following theorem is proven: “Every natural number, except for 0 and 1, can be uniquely written as a linear combination of consecutive powers of 2 with the coefficients of the linear combination being -1 or +1.” This theorem reveals a set of symmetries in the internal order of natural numbers which cannot be derived when studying natural numbers on the basis of the product. From such a symmetry a method for identifying large prime numbers is derived. We prove a factorization test for the odd numbers.

**Category:** Number Theory

[16] **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

[15] **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

[14] **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

[13] **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

[12] **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

[11] **viXra:1707.0176 [pdf]**
*replaced on 2017-10-19 18:21:15*

**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

[10] **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

[9] **viXra:1707.0168 [pdf]**
*replaced on 2018-06-12 14:09:45*

**Authors:** Wes Hansen

**Comments:** 105 Pages.

In an earlier paper, “The Q-Naturals: A Recursive Arithmetic Which Extends the ‘Standard’ Model,” we developed a set of non-standard naturals called q-naturals and demonstrated, by construction, the existence of a recursive arithmetical structure which extends the “standard” model. In this paper we extend the q-naturals out to algebraic closure and explore the properties of the various sub-structures along the way. In the process of this development, we realize that the “standard” model of arithmetic and the Q-Natural model are simply the zeroth-order and first-order recursive arithmetics, respectively, in a countable subsumption hierarchy of recursive arithmetics; there exist countably many recursive arithmetical structures.

**Category:** Number Theory

[8] **viXra:1707.0167 [pdf]**
*replaced on 2018-01-02 05:15:39*

**Authors:** Leszek W. Guła

**Comments:** 5 Pages.

The Fermat’s Last Theorem (FLT). The Guła’s Theorem. The Goldbach’s Theorem. The Conclusions. Supplement—two short proofs: of FLT for n=4 and of the Diophantine Inequalities.

**Category:** Number Theory

[7] **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

[6] **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

[5] **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

[4] **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

[3] **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

[2] **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

[1] **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