[24] **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

[23] **viXra:1906.0318 [pdf]**
*submitted on 2019-06-17 18:09:21*

**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 of non-trivial zeros for the Riemann Zeta Function.

**Category:** Number Theory

[22] **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

[21] **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

[20] **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

[19] **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

[18] **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

[17] **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

[16] **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

[15] **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

[14] **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

[13] **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

[12] **viXra:1906.0114 [pdf]**
*submitted on 2019-06-07 09:50:33*

**Authors:** Igor Hrnčić

**Comments:** 4 Pages.

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

[11] **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

[10] **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

[9] **viXra:1906.0091 [pdf]**
*replaced on 2019-06-15 03:54:25*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

In the previous paper “Consideration of the Riemann hypothesis” c = 0.5 and x is non-trivial zero value, and it was described that it converges to 0, but a serious proof in mathematical expression could not be obtained.
In this paper, we will give a proof of mathematical expression.
”the non-trivial zero values of all positive infinity and negative infinity lie on the real value 0.5” I am here explained.

**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]**
*submitted on 2019-06-06 03:14:54*

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

**Comments:** 9 Pages.

A 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 disproof 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