[38] **viXra:1906.0570 [pdf]**
*submitted on 2019-06-30 18:22:13*

**Authors:** Akalabu, Emmanuel Chukwuemeka

**Comments:** 8 Pages.

--

**Category:** Number Theory

[37] **viXra:1906.0544 [pdf]**
*submitted on 2019-06-28 11:10:36*

**Authors:** Simon Plouffe

**Comments:** 8 Pages.

Un nouveau modèle est proposé pour représenter ces quantités. En premier lieu, 4 formules sont données qui sont déduites des résultats classiques, ensuite un principe est appliqué, appelé matriochkas ou des poupées russes qui permet de trouver des développements asymptotiques remarquablement simples et élégants. De plus, les développements obtenus sont tous très similaires.
A new model is proposed to represent these quantities. In the first place, 4 formulas are given which are deduced from the classical results, then a principle is applied, called matriochkas or Russian dolls which allows to find remarkably simple and elegant asymptotic expansions. Moreover, the developments obtained are all very similar.

**Category:** Number Theory

[36] **viXra:1906.0508 [pdf]**
*submitted on 2019-06-27 04:33:42*

**Authors:** Oksana Vozniuk, Bogdana Oliynyk, Roman Yavorskyi

**Comments:** 5 Pages. Text in Ukrainian. Mohyla Mathematical Journal, Vol 1 (2018) http://mmj.ukma.edu.ua/article/view/152597

iotope spaces were introduced by Marchevsky-Steinhaus in for the needs of mathematical biology, namely the study of ecosystems. Biotope distance is defined on the set of all subsets of some finite set X. The distance between any subsets A1 and A2 of X is calculated by the rule: d(A1, A2) = (0, if A1 = A2 = ∅; |A1⊕A2| |A1∪A2| , if A1, A2 ∈ B(X)).We introduce a new generalization of a biotope metric to the infinite case using supernatural or Steinitz numbers. A supernatural number (or Steinitz number) is an infinite formal product of the form Y p∈P p kp where P is the set of all primes and kp ∈ N ∪ {0, ∞}. On the set of all periodic {0, 1}-sequences with the period that is a divisor of some supernatural u; we define the metric dB for any infinite periodic sequences x¯ and y¯ by the rule: dB(¯x, y¯) = dBn (¯xn, y¯n) where n is a common period of periodic sequences x¯ and y¯, and the formula dB(¯xn, y¯n) denotes the biotope distance between the first n coordinates of sequences x¯ and y¯ in the finite biotope metric space Bn. We denote the periodic biotope space that is defined by some Steinitz number u as B(u). If u is a finite Steinitz number, i.e. u is a positive integer, then B(u) is isometric finite biotope space Bu. We also prove that the introduced metric between such two periodic sequences does not depend on a choice of a common period.
A family of such introduced periodic biotope spaces is naturally parametrized by supernatural numbers. More precisely, the family of these spaces forms a lattice that is isomorphic to the lattice of supernatural numbers. Moreover, each of these spaces B(u) is invariant with respect to the shift.
We prove that the diametr of any periodic biotope space equals 1. We also show that any finite subset of a countable biotope space introduced in is isometric embedding in the periodic biotope space B(u) for any u.

**Category:** Number Theory

[35] **viXra:1906.0498 [pdf]**
*submitted on 2019-06-27 08:44:40*

**Authors:** Nurlan Qasimli

**Comments:** 6 Pages.

History of conjecture

**Category:** Number Theory

[34] **viXra:1906.0488 [pdf]**
*submitted on 2019-06-25 08:29:56*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents a simple formula for Pi.

**Category:** Number Theory

[33] **viXra:1906.0463 [pdf]**
*replaced on 2019-06-25 11:55:41*

**Authors:** Hung Tran

**Comments:** 5 Pages. Proof of the Riemann hypothesis using a Hamiltonian and a self-adjoint operator

We first find a Hamiltonian H that has the Hurwitz zeta functions ζ(s,x) as eigenfunctions. Then we continue constructing an operator G that is self-adjoint, with appropriate boundary conditions. We will find that the ζ(s,x)-functions do not meet these boundary conditions, except for the ones where s is a nontrivial zero of the Riemann zeta, with the real part of s being greater than 1/2. Finally, we find that these exceptional functions cannot exist, proving the Riemann hypothesis, that all nontrivial zeros have real part equal to 1/2.

**Category:** Number Theory

[32] **viXra:1906.0423 [pdf]**
*submitted on 2019-06-22 16:50:25*

**Authors:** Israel Meireles Chrisostomo

**Comments:** 2 Pages.

Mostre que o seno de um arco na forma 1/p, com p inteiro, resulta em um
irracional.
Observe que

**Category:** Number Theory

[31] **viXra:1906.0422 [pdf]**
*submitted on 2019-06-22 20:43:47*

**Authors:** Israel Meireles Chrisostomo

**Comments:** 2 Pages.

Title, authors and abstract should also be included in the PdF file. These should be in English. If the submission is not in English please translate the title and abstract here.

**Category:** Number Theory

[30] **viXra:1906.0421 [pdf]**
*submitted on 2019-06-22 20:58:02*

**Authors:** Xuan Zhong Ni

**Comments:** 2 Pages.

In this article, we use method of sieve of Eratosthenes to prove that there is a larger prime gap near any primorial number.

**Category:** Number Theory

[29] **viXra:1906.0420 [pdf]**
*submitted on 2019-06-22 21:11:07*

**Authors:** Israel Meireles Chrisostomo

**Comments:** 2 Pages. irrationality and pi other transformation

irrationality and pi other transformationirrationality and pi other transformationirrationality and pi other transformationirrationality and pi other transformation

**Category:** Number Theory

[28] **viXra:1906.0408 [pdf]**
*submitted on 2019-06-20 13:40:49*

**Authors:** James Edwin Rock

**Comments:** 1 Page.

We show that attempting to map the set of real numbers to the natural numbers by listing them as infinite decimal fractions is futile. The real numbers are represented as the limit of partial decimal sums. This allows them to be explicitly referenced and makes them into a countable set. We conjecture that the Pi, i, and e generate the Real Numbers.

**Category:** Number Theory

[27] **viXra:1906.0391 [pdf]**
*replaced on 2019-08-05 10:36:01*

**Authors:** Ralf Wüsthofen

**Comments:** 2 Pages. Proof of the Goldbach conjecture on http://vixra.org/abs/1702.0300

Based on a strengthened form of the strong Goldbach conjecture, this paper presents an antinomy within the Peano arithmetic (PA). We derive two contradictory statements by using the same main instrument as in the proof of the conjecture, that is, a structuring of the natural numbers starting from 3.

**Category:** Number Theory

[26] **viXra:1906.0377 [pdf]**
*submitted on 2019-06-21 22:02:41*

**Authors:** Xuan Zhong Ni

**Comments:** 4 Pages.

In this article, we assume that the Riemann Zeta Function equals to the Euler product at the non zero points of the Riemann Zeta function. From this assumption we can prove that there are no zero points of Riemann Zeta function, ς(s) in Re(s) > 1/2. We applied proof by contradiction.

**Category:** Number Theory

[25] **viXra:1906.0374 [pdf]**
*submitted on 2019-06-22 06:25:55*

**Authors:** Julian TP Beauchamp

**Comments:** 6 Pages.

Catalan's Conjecture was first made by Belgian mathematician Eugène Charles Catalan in 1844, and states that 8 and 9 (2^3 and 3^2) are the only consecutive powers, excluding 0 and 1. That is to say, that the only solution in the natural numbers of a^x - b^y=1 for a,b,x,y > 1 is a=3, x=2, b=2, y=3. In other words, Catalan conjectured that 3^2-2^3=1 is the only nontrivial solution. It was finally proved in 2002 by number theorist Preda Mihailescu making extensive use of the theory of cyclotomic fields and Galois modules.

**Category:** Number Theory

[24] **viXra:1906.0373 [pdf]**
*submitted on 2019-06-19 07:35:33*

**Authors:** Méhdi Pascal

**Comments:** 20 Pages.

The bute of this algebra is to give a tool which makes it possible to find new formulas for the sequences of the numbers, for example, I take the numbers of Bernoulli (Bn), and the numbers of Fibonacci (Fn), and this algebra allows us the following formula:
n*F(n)=sum(binomial(n,j)*(F(2n-2j+1)-F(n-j+1))*B(j)), From j=0 to j=n.

**Category:** Number Theory

[23] **viXra:1906.0322 [pdf]**
*submitted on 2019-06-17 08:54:10*

**Authors:** James Edwin Rock

**Comments:** 1 Page.

We exploit some rudimentary facts about the number one: (-1)(-1) = 1, 1 = sqrt(1 squared), and 1 squared = 1 to show an anomaly in the set of Complex Numbers.

**Category:** Number Theory

[22] **viXra:1906.0318 [pdf]**
*replaced on 2019-06-18 12:20:34*

**Authors:** Alan M. Gómez

**Comments:** 2 Pages.

Assuming the Riemann Hypothesis to be true, we propose an asymptotic and closed-form formula to find the imaginary part for non-trivial zeros of the Riemann Zeta Function.

**Category:** Number Theory

[21] **viXra:1906.0315 [pdf]**
*submitted on 2019-06-17 22:43:25*

**Authors:** Pedro Hugo García Peláez

**Comments:** 6 Pages.

All prime numbers are represented as factors of Fibonacci numbers, following a relationship with the corresponding Fibonacci number index.

**Category:** Number Theory

[20] **viXra:1906.0282 [pdf]**
*submitted on 2019-06-15 15:16:11*

**Authors:** Sally Myers Moite

**Comments:** 6 Pages.

For a fixed last prime, sieve the positive integers as follows. For every prime up to and including that last prime, choose one arbitrary remainder and its negative. Sieve the positive integers by eliminating all numbers congruent to the chosen remainders modulo their prime. Consider the maximum of the first open numbers left by all such sieves for a particular last prime. Computations for small last primes support a conjecture that the maximum first open number is less than (last prime)^1.75. If this conjecture could be proved, it would imply Goldbach’s Theorem is true.

**Category:** Number Theory

[19] **viXra:1906.0273 [pdf]**
*submitted on 2019-06-16 04:31:04*

**Authors:** Silvio Gabbianelli

**Comments:** 14 Pages.

By arranging the prime numbers on four columns ten-to-ten (columns of one, three, seven, nine) and establishing a suitable correspondence between the quadruples obtained and the numbers between zero and fifteen, we obtain a synthetic representation of them which allows to establish that the order in the distribution of prime numbers among positive natural numbers is not random.

**Category:** Number Theory

[18] **viXra:1906.0243 [pdf]**
*submitted on 2019-06-13 11:24:19*

**Authors:** David Rudisill

**Comments:** 14 Pages.

We prove an important new result on this problem: Given any epsilon > 0 and k >= 5, and given any set of speeds s_1 < s_2 < ... < s_k, there is a set of speeds v_1 < v_2 < ... < v_k for which the lonely runner conjecture is true and for which |s_i - v_i| < epsilon. We also prove some measure theorems.

**Category:** Number Theory

[17] **viXra:1906.0242 [pdf]**
*submitted on 2019-06-13 11:35:10*

**Authors:** David Rudisill

**Comments:** 10 Pages.

We prove that the lonely runner conjecture is equivalent to a set of Diophantine approximation problems.

**Category:** Number Theory

[16] **viXra:1906.0241 [pdf]**
*submitted on 2019-06-13 11:51:58*

**Authors:** David v. Rudisill

**Comments:** 8 Pages.

We prove some measure and covering problems related to the lonely runner conjecture.

**Category:** Number Theory

[15] **viXra:1906.0199 [pdf]**
*submitted on 2019-06-13 05:44:33*

**Authors:** Julian TP Beauchamp

**Comments:** 4 Pages.

In this paper, we show how a^x - b^y can be expressed as a binomial expansion (to an indeterminate power, z, and use it as the basis for a proof for the Beal Conjecture.

**Category:** Number Theory

[14] **viXra:1906.0195 [pdf]**
*replaced on 2019-06-13 07:12:16*

**Authors:** Timothy W. Jones

**Comments:** 4 Pages. Additional comments and examples added.

The rational root test gives a means for determining if a root of a polynomial is rational. If none of the possible rational roots are roots, then if the roots are real, they must be irrational. Combining this observation with Taylor polynomials and the Taylor series for $\sin (x)$ gives intimations that $\pi$, and $e$, are likely irrational.

**Category:** Number Theory

[13] **viXra:1906.0131 [pdf]**
*submitted on 2019-06-08 13:38:48*

**Authors:** Robert C. Hall

**Comments:** 46 Pages.

The concept and application of Benford's Law have been examined a lot in the last 10 years or so, especially with regard to accounting forensics. There have been many papers written as to why Benford's Law is so prevalent and the concomitant reasons why(proofs). There are, unfortunately, many misconceptions such as the newly coined phrase "the Summation theorem", which states that if a data set conforms to Benford's Law then the sum of all numbers that begin with a particular digit (1,2,3,4,5,6,7,8,9) should be equal. Such is usually not the case. For exponential functions (y=aexp(x) it is but not for most other functions. I will show as to why this is the case. The distribution tends to be a Benford instead of a Uniform distribution.
Also, I will show that if the probability density function (pdf) of the logarithm of a data set begins and ends on the x axis and if the the values of the pdf between all integral powers of ten can be approximated with a straight line then the data set will tend to conform to Benford's Law.

**Category:** Number Theory

[12] **viXra:1906.0121 [pdf]**
*submitted on 2019-06-07 08:28:35*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We recall a simple representation for Pi.

**Category:** Number Theory

[11] **viXra:1906.0114 [pdf]**
*replaced on 2019-07-16 14:40:47*

**Authors:** Igor Hrnčić

**Comments:** 4 Pages. Rectified an obvious small error, sigma>1 instead of sigma>1/2, in the section Disproof of RH.

This paper disproves the Riemann hypothesis by generalizing the results from Titchmarsh's book The Theory of the Riemann Zeta-Function to rearrangements of conditionally convergent series that represent the reciprocal function of zeta. When one replaces the conditionally convergent series in Titchmarsh's theorems and consequent proofs by its rearrangements, the left hand sides of equations change, but the right hand sides remain invariant. This contradiction disproves the Riemann hypothesis.

**Category:** Number Theory

[10] **viXra:1906.0111 [pdf]**
*submitted on 2019-06-07 11:38:42*

**Authors:** Arthur Shevenyonov

**Comments:** 8 Pages. bridging

Some testing criteria or decision-procedures, notably when deployed as part of automated proving vehicles, might pose more of an AI threat than they do in terms of an opportunity leverage. In particular, tautology, unless rethought, will likely prove just that--irrelevant and inefficient. Mochizuki's IUT, referred to for benchmarking and illustration purposes, may well bear fruit beyond ABC if shown to be Teichmueller legacy-invariant.

**Category:** Number Theory

[9] **viXra:1906.0103 [pdf]**
*submitted on 2019-06-07 23:49:45*

**Authors:** Franco Sabino Stoianoff Lindstron

**Comments:** 4 Pages.

The method used in this article is based on analytical geometry, abstract algebra and number theory.

**Category:** Number Theory

[8] **viXra:1906.0069 [pdf]**
*submitted on 2019-06-05 23:59:34*

**Authors:** Sally Myers Moite

**Comments:** 8 Pages.

For the n-th prime P, P# or P primorial is the product of all the primes up to and including P. Let (c, d) be a pair of integers that represents a point in the primorial square, 1 < c, d < P#. For each prime p, 2 < p < P, the remainders of c and d mod p may be the same, opposite (sum to a multiple of p) or neither. Count the number of remainders of (c, d) which have same, opposite or either agreement for any such P. This gives three partitions of the primorial square, by counts for same, opposite and either agreement. Polynomial multiplication is used to find the number of points in each part of these partitions.

**Category:** Number Theory

[7] **viXra:1906.0066 [pdf]**
*replaced on 2020-01-17 01:34:11*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 14 Pages.

Twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research
paper, my attempt is to provide a valid proof for twin prime conjecture.

**Category:** Number Theory

[6] **viXra:1906.0044 [pdf]**
*submitted on 2019-06-05 00:21:36*

**Authors:** Pedro Hugo García Peláez

**Comments:** 6 Pages.

Con este algoritmo podrás encontrar el máximo común divisor de dos polinomios o de números complejos y por supuesto también de números naturales de una manera fácil.

**Category:** Number Theory

[5] **viXra:1906.0042 [pdf]**
*submitted on 2019-06-05 01:16:23*

**Authors:** Pedro Hugo García Peláez

**Comments:** 6 Pages.

With this algorithm you can find the greatest common divisor of two polynomials or complex numbers and of course also natural numbers in an easy way.

**Category:** Number Theory

[4] **viXra:1906.0028 [pdf]**
*submitted on 2019-06-03 18:13:17*

**Authors:** Bertrand Wong

**Comments:** 20 Pages.

This paper explicates the Riemann hypothesis and proves its validity. [The paper is published in a journal of number theory.]

**Category:** Number Theory

[3] **viXra:1906.0025 [pdf]**
*submitted on 2019-06-04 03:57:36*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 7 Pages. Submitted to the Ramanujan Journal. Comments welcome.

In this paper, using the recent result that $c<rad(abc)^2$, we will give the proof of the $abc$ conjecture for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=e^{\frac{1}{\epsilon^2} $. Some numerical examples are presented.

**Category:** Number Theory

[2] **viXra:1906.0018 [pdf]**
*submitted on 2019-06-02 15:45:53*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The six seminal equations evaluated are not tautologous, refuting the subsequent claimed proof of the ABC conjecture, and forming a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[1] **viXra:1906.0010 [pdf]**
*submitted on 2019-06-01 14:44:08*

**Authors:** Arthur Shevenyonov

**Comments:** 5 Pages. pre-ordual

While seeking to bypass the complex matching/ordering/comparability issue, the paper appears to have straddled areas seemingly as diverse as RH, Mikusinski operators, Euler equation for variations, and Veblen ordinals.

**Category:** Number Theory