[20] **viXra:1810.0497 [pdf]**
*replaced on 2018-11-20 01:44:54*

**Authors:** M. Sghiar

**Comments:** Copyright © iosrjournals.org . Published In IOSR-JM Journal. 5 pages.

If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 110 pages by A. Wiles [4], the puspose of this article is to give a simple demonstration and deduce a proof of the Beal conjecture.

**Category:** Number Theory

[19] **viXra:1810.0487 [pdf]**
*submitted on 2018-10-29 13:14:19*

**Authors:** Ankur Shukla, Anurag Singh

**Comments:** 20 Pages.

This article provide unique methods of creating perfect magic squares of order 4 and using certain conditions the magic square can be made more precise and varied. The article contains rules which connects all the perfect squares of order 4 and bring them under one roof. The rules contained in this article deals with various methods of arranging a square maintaining the perfectness and bringing new appearance every time using symmetry. The main objective is to create a method for generation of perfect formula of magic square.

**Category:** Number Theory

[18] **viXra:1810.0479 [pdf]**
*submitted on 2018-10-28 10:11:24*

**Authors:** Victor Sorokine

**Comments:** 1 Page.

The number D=[(A+B)^n-(C-B)^n-(C-A)^n+(A^n+B^n-C^n)] ends with k+2 zeroes, where k is the number of zeroes at the end of the number A (and in the number A+B-C). However, after the opening of the binomials in the number D its (k+2)-th digit is not equal to zero.

**Category:** Number Theory

[17] **viXra:1810.0478 [pdf]**
*submitted on 2018-10-28 10:12:26*

**Authors:** Victor Sorokine

**Comments:** 1 Page. Russian version

Число D=[(A+B)^n-(C-B)^n-(C-A)^n+(A^n+B^n-C^n)] оканчивается на k+2 нулей, где k – число нулей на конце числа А (и в числе A+B-C). Однако после раскрытия биномов в числе D его (k+2)-я цифра нулю не равна.

**Category:** Number Theory

[16] **viXra:1810.0459 [pdf]**
*submitted on 2018-10-27 12:00:17*

**Authors:** Di Pietro Gabriele

**Comments:** 9 Pages.

This paper gives us an application of Eratosthenes sieve to distribution mean
distance between primes using first and upper orders of Gauss integral log-
arithm Li(x).We define function Υ in section 5. Sections 1 − 4 give us an
introduction to the terminology and a clarification on Υ terms. Section 6
reassumes foregoing explanations and gives us two theorems using first and
upper integral logarithm orders.

**Category:** Number Theory

[15] **viXra:1810.0442 [pdf]**
*submitted on 2018-10-26 15:59:26*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

The conjectured proof of the Riemann hypothesis using the excluded middle is refuted by the Meth8/VŁ4 modal logic model checker.

**Category:** Number Theory

[14] **viXra:1810.0423 [pdf]**
*replaced on 2018-11-21 23:25:13*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

I calculated ζ (3),ζ(5). ζ (7),ζ(9)……… ζ (23).
And the formula indicated.
For example, in ζ (3)
For example, in ζ (5)
And ultimately the following formula is required
n and m are positive integer.

**Category:** Number Theory

[13] **viXra:1810.0418 [pdf]**
*submitted on 2018-10-24 06:14:11*

**Authors:** Matan Cohen

**Comments:** 9 Pages.

In this paper, I will describe and demonstrate a new method to prove Goldbach’s Conjecture. The idea behind my method is to organize all natural numbers in a binary tree, and to find the connections between the even numbers and the prime numbers by using the characteristics of the tree structure.

**Category:** Number Theory

[12] **viXra:1810.0390 [pdf]**
*submitted on 2018-10-23 07:31:38*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents a definite integral for pi.

**Category:** Number Theory

[11] **viXra:1810.0368 [pdf]**
*submitted on 2018-10-22 14:22:06*

**Authors:** A.I.Somsikov

**Comments:** 2 Pages. -

Examples of the seeming contradictions of mathematical transformations

**Category:** Number Theory

[10] **viXra:1810.0335 [pdf]**
*replaced on 2018-11-29 10:43:21*

**Authors:** Timothy W. Jones

**Comments:** 10 Pages. Some reviewers pointed out an error. It's fixed. Clarifications.

We look at some of the details of Cantor's Diagonal Method and argue that the swap function given does not have to exclude 9 and 0, base 10. We also puzzle out why the convergence of the constructed number, its value, is of no concern. We next review general properties of decimals and prove the existence of an irrational number with a modified version of Cantor's diagonal method. Finally, we show, with yet another modification of the method, that Zeta(2) is irrational.

**Category:** Number Theory

[9] **viXra:1810.0288 [pdf]**
*submitted on 2018-10-19 02:09:09*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

If prime, we found that the remainder is a prime number including 1 when divided by 30, 60 and 90.
This is called toshichan-man's small theorem.
It seems to be useful for prime number determination (especially huge prime number determination).

**Category:** Number Theory

[8] **viXra:1810.0282 [pdf]**
*replaced on 2018-11-06 12:17:49*

**Authors:** Vyacheslav Telnin

**Comments:** 122 Pages. Format A5. Content is at the end of the paper.

In this paper, we analyze the construction of numbers from natural to complex. A close relationship between these numbers and three direct operations ([1] – addition, [2] – multiplication, [3] – exponentiation) is revealed. A method of constructing new direct operations is found. Two new direct operations ([4] and [0]) are constructed. Two inverse operations are constructed for each of them. If equations that are unsolvable in complex numbers are found for them, new numbers can be constructed on the basis of these unsolvabilities.
So far in this way in this work the question numbers are constructed (?- numbers) for [3] direct operation.
Along the way, numbers-strings and N-numbers are described. The topology of the numerical line is traced on the basis of N-numbers.
In this paper we find the relation of N-numbers and ?-numbers'.

**Category:** Number Theory

[7] **viXra:1810.0255 [pdf]**
*submitted on 2018-10-16 17:40:35*

**Authors:** Zeolla Gabriel Martín

**Comments:** 3 Pages.

This paper develops the construction of the Golden Pattern for different prime divisors, the discovery of patterns towards infinity. The discovery of infinite harmony represented in fractal numbers and patterns. These patterns form the sequence of prime numbers.

**Category:** Number Theory

[6] **viXra:1810.0240 [pdf]**
*submitted on 2018-10-15 14:59:35*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

I found out my original make prime number built method.
30n+7 (n is positive integer, including 0).
30n+17 (n is positive integer, including 0).
Prime has a period of 30. We focused on that point and developed it.
However, those that are not prime are also quite included.
And,
30n+11 (n is positive integer, including 0).
30n+13 (n is positive integer, including 0).
Also considered.

**Category:** Number Theory

[5] **viXra:1810.0223 [pdf]**
*replaced on 2018-10-23 16:14:29*

**Authors:** Marco Ripà

**Comments:** 4 Pages.

In 2011, in his book “La strana coda della serie n^n^...^n", M. Ripà analyzed some properties involving the rightmost figures of integer tetration, the iterated exponentiation a^^b, characterized by an increasing number of stable digits for any base a > 1.
A few conjectures arose about how many new stable digits are generated by any unitary increment of the hyperexponent b, and Ripà indicated this value as V(a) or “convergence speed” of a. In fact, when b is large enough, V(a) seems to not depend from b, taking on a (strictly positive) unique value, and many observations supported this claim. Moreover, we claim that V(a) = 1 for any a(mod 25) congruent to {2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23} and V(a)>=2 otherwise.

**Category:** Number Theory

[4] **viXra:1810.0175 [pdf]**
*submitted on 2018-10-12 02:45:55*

**Authors:** Anna Povazanova, Ivo Povazan

**Comments:** 13 Pages.

p is prime.The article describes the new Abelian groups of type
p=4k+1 and p = 4k-1, for which a theorem similar to the Fermat's
little theorem applies. The multiplicative group (Z/pZ)* in some sense
similar to the Abelian group of type p = 4k+1. Abelian group of type
p = 4k-1 is a different structure compared to group (Z/pZ)*. This
fact is used for the primality test of integer N = 4k-1. The primality
test was veried up to N = 2^(64).

**Category:** Number Theory

[3] **viXra:1810.0153 [pdf]**
*submitted on 2018-10-09 07:38:30*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents an integral for Catalan's constant: G=0.915965...

**Category:** Number Theory

[2] **viXra:1810.0141 [pdf]**
*submitted on 2018-10-09 16:57:44*

**Authors:** Jonathan Trousdale

**Comments:** 6 Pages.

This paper sets forth a representation of the hyperbolic substratum that defines order on $\mathbb{N}$. Degeneracy of $\mathbb{N}$ at points of intersection with the substratum is observed as violations of the fundamental theorem of arithmetic in the form of mutable prime factorization. At a point of maximum symmetry on the representation manifold, an exact expression of $\pi$ is available as a combination of three integers.

**Category:** Number Theory

[1] **viXra:1810.0046 [pdf]**
*submitted on 2018-10-05 01:33:59*

**Authors:** Idriss Olivier Bado

**Comments:** 5 Pages.

The main contribution of this paper is to achieve the proof of Riemann hypothesis. The key idea is based on new formulation of the problem
$$\zeta(s)=\zeta(1-s) \Leftrightarrow re(s)=\frac{1}{2}$$. This proof is considered as a great discovery in mathematic.

**Category:** Number Theory