[22] **viXra:1905.0614 [pdf]**
*replaced on 2019-06-04 08:31:22*

**Authors:** Surajit Ghosh

**Comments:** 36 Pages.

Based on Eulers formula a concept of dually unit or d-unit circle is discovered. Continuing with, Riemann hypothesis is proved from diﬀerent angles, Zeta values are renormalised to remove the poles of Zeta function and relationships between numbers and primes is discovered. Other unsolved prime conjectures are also proved with the help of theorems of numbers and number theory. Imaginary number i can be deﬁned such a way that it eases the complex logarithm without needing branch cuts. Pi can also be a base to natural logarithm and complement complex logarithm.Grand integrated scale is discovered which can reconcile the scale diﬀerence between very big and very small. Complex constants derived from complex logarithm following Goldbach partition theorem and Eulers Sum to product and product to unity can explain lot of mysteries in the universe.

**Category:** Number Theory

[21] **viXra:1905.0584 [pdf]**
*submitted on 2019-05-29 09:05:21*

**Authors:** Henry Wong

**Comments:** 2 Pages.

An addendum to elementary number theory.

**Category:** Number Theory

[20] **viXra:1905.0574 [pdf]**
*submitted on 2019-05-29 17:57:53*

**Authors:** Colin James III

**Comments:** 1 Page. © 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 root in partition Jensen polynomials for hyperbolicity is not tautologous. Hence its use to prove the Riemann hypothesis is denied. These conjectures form a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[19] **viXra:1905.0571 [pdf]**
*submitted on 2019-05-29 20:34:27*

**Authors:** Pedro Hugo García Peláez

**Comments:** Pages.

With this algorithm you can easily find the greatest common divisor of two numbers even with large numbers of figures and the same can be done if you want to find the greatest common divisor of polynomials easily and also complex numbers.

**Category:** Number Theory

[18] **viXra:1905.0570 [pdf]**
*submitted on 2019-05-29 20:36:13*

**Authors:** Pedro Hugo García Peláez

**Comments:** 5 Pages.

Con este algoritmo podrás hallar fácilmente el máximo comun divisor de dos números incluso con gran cantidad de cifras y lo mismo podrás hacer si quieres hallar el máximo común divisor de polinomios fácilmente.

**Category:** Number Theory

[17] **viXra:1905.0565 [pdf]**
*submitted on 2019-05-30 02:35:37*

**Authors:** Aryan Phadke

**Comments:** 12 Pages. email :- phadkearyan007@gmail.com

Set of Pythagorean triple consists of three values such that they comprise the three sides of a right angled triangle. Euclid gave a formula to find Pythagorean Triples for any given number. Motive of this paper is to find number of possible Pythagorean Triples for a given number. I have been able to provide a different proof for Euclid’s formula, as well as find the number of triples for any given number. Euclid’s formula is altered a little and is expanded with a variable ‘x’. When ‘x’ follows the conditions mentioned the result is always a Pythagorean Triple.

**Category:** Number Theory

[16] **viXra:1905.0560 [pdf]**
*submitted on 2019-05-28 08:35:26*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We give some infinite series for Pi.

**Category:** Number Theory

[15] **viXra:1905.0559 [pdf]**
*submitted on 2019-05-28 08:38:47*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

This note presents some remarks on the equation: x^x-x-1=0,x>0

**Category:** Number Theory

[14] **viXra:1905.0546 [pdf]**
*replaced on 2019-07-30 20:11:59*

**Authors:** Toshiro Takami

**Comments:** 68 Pages.

I considered Riemann’s hypothesis. At first, the purpose was to prove, but can not to prove.
It is written in the middle of the proof, but it can not been proved at all.
(The calculation formula is also written, but the real value 0.5 was not shown at all) The non-trivial zero values match perfectly in the formula of this paper.
However, the formula did not reach the real value 0.5.
In this case, it only reaches the pole near the real value 0.5.

**Category:** Number Theory

[13] **viXra:1905.0502 [pdf]**
*replaced on 2019-05-30 09:52:05*

**Authors:** Gang tae geuk

**Comments:** 1 Page.

Riemann hypothesis means that satisfying ζ(s)=0(ζ(s) means Riemann Zeta function) unselfevidenceable root's part of true numbers are 1/2.
Dennis Hejhal, and John Dubisher explained this hypothesis to :
"Choosed Any natural numbers(exclude 1 and constructed with two or higher powered prime numbers) then the probability of numbers that choosed number's forming prime factor become an even number is 1/2."
I'll prove this explain to prove Riemann hypothesis indirectly.
In binomial coefficient, C(n,0)+C(n+1)+...+C(n,n)=2^n. And C(n,1)+C(n,3)+C(n,5)+...+C(n,n) and C(n,0)+C(n,2)+C(n,4)+...+C(n,n) is 2^(n-1).
If you pick up 8 prime numbers, then you can make numbers that exclude 1 and constructed with two or higher powered prime numbers, and the total amount of numbers that you made is 2^8.
Same principle, if you pick the numbers in k times(k is a variable), the total amount of numbers you made is C(8,k).
If k is an even number, the total amount of numbers you can make is C(8,0)+C(8,2)+...+C(8,8)-1(because we must exclude 1,same for C(8,0)), and as what i said, it equals to 2^(8-1)-1.
So, the probability of the numbers that forming prime factor's numbers is an even number is 2^(8-1)-1/2^8
If there are amount of prime numbers exist, and we say that amount to n(n is a variable, as the k so), and sequence of upper works sameas we did, so the probability is 2^(n-1)/2^n.
If you limits n to inf, then probability convergents to 1/2.
This answer coincident with the explain above, so explain is established, same as the Riemann hypothesis is.

**Category:** Number Theory

[12] **viXra:1905.0498 [pdf]**
*replaced on 2019-07-24 13:26:15*

**Authors:** Esteve J., Martinez J.E.

**Comments:** 7 Pages. Minor corrections were made on the latest version.

We proof Goldbach's Conjecture. We use results obtained by Srinivasa A. Ramanujan (specifically in his paper A Proof of Bertrand's Postulate). A generalization of the conjecture is also proven for every natural not coprime with a natural m > 1 and greater or equal than 2m.

**Category:** Number Theory

[11] **viXra:1905.0485 [pdf]**
*replaced on 2019-11-13 03:34:03*

**Authors:** Aryan Phadke

**Comments:** 11 Pages.

Background : Sum of Harmonic Progression is an old problem. While a few complex approximations have surfaced, a simple and efficient formula hasn't. Most of the previous formulas use summation as a tool for the approximation. Also, most notable formulas are only applicable to the harmonic progression where the first term and common difference of the corresponding Arithmetic progression is one.
Aim of this paper is to create a formula without using 'summation' as a tool. On the contrary to previous formulas, the resultant formula is applicable to every harmonic progression regardless of the first term and common difference of its corresponding arithmetic progression.
The resultant formula is applicable for numerical values in a harmonic progression. Algebraic terms in a harmonic progression do not value to any numeric terms and for that reason cannot be accounted for the approximation. The resultant formula has a variability in terms of accuracy and complexity to suit different users with different priorities.
The Formula was obtained by equating the area enclosed in the graph of harmonic progression and area under the curve of equation (y=1/x). The logic being that the harmonic progression is completely inclusive in the function (y=1/x) is used to formulate the answer.

**Category:** Number Theory

[10] **viXra:1905.0468 [pdf]**
*replaced on 2019-05-28 18:53:14*

**Authors:** Bambore Dawit Geinamo

**Comments:** 9 Pages.

This paper magically shows very interesting and simple proof of Fermat’s Last Theorem. The proof identifies sufficient derivations of equations that holds the statement true and describes contradictions on them
to satisfy the theorem. If Fermat had proof, his proof is most probably
similar to this one. The proof does not require any higher field of mathematics and it can be understood in high school level of mathematics. It uses only modular arithmetic, factorization and some logical statements.

**Category:** Number Theory

[9] **viXra:1905.0365 [pdf]**
*replaced on 2019-08-02 13:21:54*

**Authors:** Emmanuil Manousos

**Comments:** 22 Pages.

“Every natural number, with the exception of 0 and 1, can be written in a unique way as a linear combination of consecutive powers of 2, with the coefficients of the linear combination being -1 or +1�?. According to this theorem we define the L/R symmetry of the natural numbers. The L/R symmetry gives the factors which determine the internal structure of natural numbers. As a consequence of this structure, an algorithm for the factorization of Fermat numbers is derived. Also, we determine a sequence of prime numbers, and we prove an essential corollary for the composite Mersenn numbers.

**Category:** Number Theory

[8] **viXra:1905.0269 [pdf]**
*submitted on 2019-05-17 15:12:11*

**Authors:** Wilson Torres Ovejero

**Comments:** 16 Pages.

160 years ago that in the complex analysis a hypothesis was raised, which was used in principle
to demonstrate a theory about prime numbers, but, without any proof; with the passing Over the years, this
hypothesis has become very important, since it has multiple applications to physics, to number theory, statistics,
among others In this article I present a demonstration that I consider is the one that has been dodging all this
time.

**Category:** Number Theory

[7] **viXra:1905.0250 [pdf]**
*submitted on 2019-05-16 16:10:59*

**Authors:** Yuly Shipilevsky

**Comments:** 5 Pages.

We consider a new conjecture regarding powers of integer numbers and
more specifically, we are interesting in existence and finding pairs of integers:
n ≥ 2 and m ≥ 2, such that nm
= mn. We conjecture that n = 2, m = 4
and n = 4, m = 2 are the only integral solutions.
Next, we consider the corresponding generalizations for Hypercomplex
Integers: Gaussian and Lipschitz Integers.

**Category:** Number Theory

[6] **viXra:1905.0210 [pdf]**
*submitted on 2019-05-14 15:29:38*

**Authors:** Arthur Shevenyonov

**Comments:** 6 Pages. trilinear

A set of minimalist demonstrations suggest how the key premises of RH may have been inspired and could be qualified, by proposing a linkage between the critical strip (0..n) and Re(s)=x-1/2 interior of candidate solutions. The solution density may be concentrated around the focal areas amid the lower and upper bound revealing rarefied or latent representations. The RH might overlook some of the ontological structure while confining search to phenomena while failing to distinguish between apparently concentrated versus seemingly non-distinct candidates.

**Category:** Number Theory

[5] **viXra:1905.0137 [pdf]**
*replaced on 2019-06-29 04:06:52*

**Authors:** Marko V. Jankovic

**Comments:** 20 Pages.

In this paper a proof of the existence of an infinite number of Sophie Germain primes, is going to be presented. In order to do that, we analyse the basic formula for prime numbers and decide when this formula would produce a Sophie Germain prime, and when not. Originally very difficult problem (in observational space) has been transformed into a simpler one (in generative space) that can be solved by elementary math.

**Category:** Number Theory

[4] **viXra:1905.0098 [pdf]**
*submitted on 2019-05-06 16:48:48*

**Authors:** Zeolla Gabriel Martín

**Comments:** 7 Pages.

This document develops and demonstrates the discovery of a new cubic potentiation algorithm that works absolutely with all the numbers using the formula of the cubic of a binomial.

**Category:** Number Theory

[3] **viXra:1905.0041 [pdf]**
*submitted on 2019-05-02 12:38:19*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 9 Pages. Submitted to the journal Monatshefte für Mathematik. Comments welcome.

In this paper, we consider the abc conjecture in the case c=a+1. Firstly, we give the proof of the first conjecture that c

**Category:** Number Theory

[2] **viXra:1905.0021 [pdf]**
*submitted on 2019-05-01 08:54:31*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages.

This is an easy approach to proving zeta(2) is irrational. The reasoning is by analogy with gym weights that are rational proportions of a unit. Sometimes the sum of such weights is expressible as a multiple of a single term in the sum and sometimes it isn't. The partials of zeta(2) are of the latter type. We use a result of real analysis and this fact to show the infinite sum has this same property and hence is irrational.

**Category:** Number Theory

[1] **viXra:1905.0010 [pdf]**
*submitted on 2019-05-01 18:09:11*

**Authors:** Yuly Shipilevsky

**Comments:** 7 Pages.

We introduce a special class of complex numbers, wherein their
absolute values and arguments given in a polar coordinate system are integers,
which when considered within the complex plane, constitute Unicentered
Radial Lattice and similarly for quaternions.

**Category:** Number Theory