# Number Theory

## 1811 Submissions

 viXra:1811.0503 [pdf] submitted on 2018-11-29 17:48:12

### Classify Positive Integers to Prove Collatz Conjecture by the Mathematical Induction

Authors: Zhang Tianshu

In this article, the author uses the mathematical induction, classifies positive integers gradually, and passes necessary operations by the operational rule to achieve finally the purpose proving Collatz conjecture.
Category: Number Theory

 viXra:1811.0471 [pdf] submitted on 2018-11-29 00:56:11

### A Proof of Infinite Primes

In this small paper , I have given a simple proof of already well established fact that there are infinitely many prime numbers.
Category: Number Theory

 viXra:1811.0459 [pdf] submitted on 2018-11-27 10:05:16

### Fermat’s Last Theorem First and Second Cases

Authors: Victor Sorokine

he essence of the proof of the FLT: The first case (ABC is not a multiple of n): In one of the equivalent Fermat equations, the 3rd digit of the sum of the powers of the last digits of the bases greater than 1, which cannot be zeroed using the second digits with the sum of the latter equal to 0 or n-1. +++ The second case (A or B or C is a multiple of n): (k+2)-th digit in the number D=(A+B)^n-(C-B)^n-(C-A)^n, where the number A+B-C ends by k zeros, is not zero, but after adding to the number D zero as 0=A^n+B^n-C^n (k+2)-th digit is zero.
Category: Number Theory

 viXra:1811.0457 [pdf] submitted on 2018-11-27 10:06:38

### Fermat’s Last Theorem First and Second Cases (Russian)

Authors: Victor Sorokine

Суть доказательства ВТФ: Первый случай (ABC не кратно n): В одном из эквивалентных равенств Ферма 3-я цифра суммы степеней последних цифр оснований больше 1, которую невозможно обнулить с помощью вторых цифр с суммой последних, равной 0 или n-1. +++ Второй случай (A или B или C кратно n): (k+2)-я цифра в числе D=(A+B)^n-(C-B)^n-(C-A)^n, где число A+B-C оканчивается на k нулей, не равна нулю, но после прибавления к числу D нуля в виде 0=A^n+B^n-C^n (k+2)-я цифра равна нулю.
Category: Number Theory

 viXra:1811.0338 [pdf] replaced on 2018-11-23 22:07:51

### Theorem on the Congruence Speed of Tetration

Authors: Marco Ripà

We provide a preliminary proof of Ripà’s Conjecture 3 about the convergence speed of tetration, published in October 2018, which states that, for any natural number "v", exists (at least) another natural number "a", not a multiple of 10, such that V(a)=v, where V(a) represents the convergence speed of the tetration a^^b.
Category: Number Theory

 viXra:1811.0320 [pdf] submitted on 2018-11-20 09:10:58

### New Multiplication Algorithm.

Authors: Zeolla Gabriel Martín

This paper develops a new multiplication algorithm that works absolutely with all the numbers.
Category: Number Theory

 viXra:1811.0288 [pdf] submitted on 2018-11-18 11:45:11

### Proof that a Data Set that Conforms to Benford's Law is not Always Sum Invariant with Respect to the First Digits

Authors: Robert C. Hall

The summation test consists of adding all numbers that begin with a particular first digit or first two digits and determining its distribution with respect to these first or first two digits numbers. Most people familiar with this test believe that the distribution is a uniform distribution for any distribution that conforms to Benford's law i.e. the distribution of the mantissas of the logarithm of the data set is uniform U[0,1). The summation test that results in a uniform distribution is true for an exponential function (geometric progression) i.e. y = a exp(kt) but not necessarily true for other data sets that conform exactly to Benford'a law.
Category: Number Theory

 viXra:1811.0282 [pdf] submitted on 2018-11-18 21:44:13

### Beal Conjecture as a General Form of Fermat's Last Theorem

Authors: Stefan Bereza

Fermat's Last Theorem (FLT) x^p + y^p = z^p could be seen as a special case of more generalized Beal's Conjecture (BC) x^m + y^n = z^r. Those equations are impossible when x, y and z are natural numbers and coprimes and {p, m, n, r}> = 3; if m = n = r (= p), then it is FLT; if not, Beal's Conjecture. In BC, if x, y and z are integers and have a common factor, they can be measured (without rest) with this factor as a common unit - making x, y and z in the equation rational to each other. FLT can be proved with proving irrationality of triangles inscribed into an ellipse whose sides x, y and z represent the Fermat's equation x^p + y^p = z^p ; here, for x, y and z a common unit cannot be found. The BC equation x^m + y^n = z^r (without a common factor) can be simplified to the Fermat's equation x^p + y^p = z^p which - at the lacking common unit - makes x, y and z impossible to be all rational to each other.
Category: Number Theory

 viXra:1811.0263 [pdf] submitted on 2018-11-17 20:39:55

### Considérations Sur la Formule du Binome de Newton

Authors: Olivier Massot

The binomial formula, set by Isaac Newton, is of utmost importance and has been extensively used in many different fields. This study (in French) aims at coming up with alternative expressions to the Newton's formula.
Category: Number Theory

 viXra:1811.0262 [pdf] submitted on 2018-11-17 20:44:21

### Considerations on the Newton's Binomial Expansion

Authors: Olivier Massot

The binomial formula, set by Isaac Newton, is of utmost importance and has been extensively used in many different fields. This study aims at coming up with alternative expressions to the Newton's formula.
Category: Number Theory

 viXra:1811.0250 [pdf] submitted on 2018-11-16 11:08:43

### A Proof Minus Epsilon of the Abc Conjecture

In this paper, we give a proof minus $\epsilon$ of the $ABC$ conjecture, considering that Beal conjecture is true. Some conditions are proposed for the proof, perhaps it needs some justifications that is why I give the title of the paper " a proof minus $\epsilon$ of the $ABC$ conjecture".
Category: Number Theory

 viXra:1811.0211 [pdf] submitted on 2018-11-13 08:01:46

### Nuevo Algoritmo de Multiplicación.

Authors: Zeolla Gabriel Martín

Este documento desarrolla y demuestra el descubrimiento de un nuevo algoritmo de multiplicación que funciona absolutamente con todos los números.
Category: Number Theory

 viXra:1811.0179 [pdf] submitted on 2018-11-11 19:03:33

### Formula of Π

Authors: Yuji Masuda

This formula shows Π.
Category: Number Theory

 viXra:1811.0159 [pdf] submitted on 2018-11-11 04:13:06

### Another Way to Write the Riemann Zeta Function

Authors: Zach Don

In this paper, I will be presenting an alternative way of writing the Riemann zeta function in terms of Euler's constant, e.
Category: Number Theory

 viXra:1811.0158 [pdf] submitted on 2018-11-11 04:19:06

### An Alternative Way of Writing the Riemann Zeta Function

Authors: Zach Don

In this paper, I will propose a legitimate way of re-writing the Riemann zeta function in terms of Euler's constant, e.
Category: Number Theory

 viXra:1811.0119 [pdf] submitted on 2018-11-07 10:57:26

### Periodic Table of Prime Number Intervals

Authors: Viktor Strohm

In accordance with the General Theory Systems of Urmantsev (GTSU) [1, 2, 3], the set of primes is considered as a system of objects. For the relationship between objects taken the difference of prime numbers. Revealed periodicity of pairs of intervals.
Category: Number Theory

 viXra:1811.0116 [pdf] replaced on 2018-12-22 21:36:55

### Refutation of Riemann Hypothesis by Two Zeta Properties

Authors: Colin James III

Properties of the zeta function of the Riemann hypothesis are not confirmed as tautologous and hence refute it.
Category: Number Theory

 viXra:1811.0112 [pdf] replaced on 2018-11-09 19:22:45

### A New Proof of the Strong Goldbach Conjecture

Authors: Es-said En-naoui

The Goldbach conjecture dates back to 1742 ; we refer the reader to - for a history of the conjecture. Christian Goldbach stated that every odd integer greater than seven can be written as the sum of at most three prime numbers. Leonhard Euler then made a stronger conjecture that every even integer greater than four can be written as the sum of two primes. Since then, no one has been able to prove the Strong Goldbach Conjecture.\\ The only best known result so far is that of Chen , proving that every sufficiently large even integer N can be written as the sum of a prime number and the product of at most two prime numbers. Additionally, the conjecture has been verified to be true for all even integers up to $4.10^{18}$ in 2014 , J\"erg  and Tom\'as . In this paper, we prove that the conjecture is true for all even integers greater than 8.
Category: Number Theory

 viXra:1811.0080 [pdf] replaced on 2019-05-13 20:21:52

### (pk mk qk) or an Unexpected Inconsistency

Authors: Ralf Wüsthofen

Based on a strengthened form of the strong Goldbach conjecture, this paper presents an antinomy within the Peano arithmetic (PA).
Category: Number Theory

 viXra:1811.0072 [pdf] replaced on 2019-01-01 23:58:16

### Theorem for W^n and Fermat's Last Theorem

Authors: Quang Nguyen Van

We give a expression of w^n and the possible to apply for solving Fermat's Last theorem.
Category: Number Theory

 viXra:1811.0046 [pdf] submitted on 2018-11-03 21:04:42

### Refutation of Karush-Kuhn-Tucker Constraints for Linear Programming

Authors: Colin James III

Karush-Kuhn-Tucker constraints for linear programming are not tautologous.
Category: Number Theory

 viXra:1811.0032 [pdf] replaced on 2019-07-02 11:20:24

### An Optimization Approach to the Riemann Hypothesis

Authors: Hassine Saidane

A great deal of research has been and still is being devoted to the zeros of the Riemann Zeta function (RZF) that are in the critical strip and known as the nontrivial zeros of RZF. The Riemann Hypothesis (RH) states that these zeros are all located on the critical line . Although a large number of nontrivial zeros have proved to be located on the critical line through numerical computation methods, starting with Riemann’s manual computation of the first few zero, no analytical proof or disproof of RH has been found since its conjecture by Riemann in 1859. In this paper, we implement a novel analytical approach to RH based on optimization. This analysis tool proved successful in deriving some important scientific theories and laws. Such a success prompted us to use this tool to analytically derive the location of RZF nontrivial zeros in order to either prove or disprove the Riemann Hypothesis. This was achieved by formulating and solving the appropriate location optimization problem.
Category: Number Theory

 viXra:1811.0017 [pdf] submitted on 2018-11-01 12:03:50

### Irrational Numbers and Density

Authors: Salvatore Gerard Micheal