Number Theory

Remarks on Lucas—Lehmer Test

Authors: V. Barbera
Comments: 2 Pages.

This paper presents some remarks on Lucas—Lehmer test for primality test of Mersenne numbers.
Category: Number Theory

Kouider Number with Base a

Authors: Kouider Mohammed Ridha
Comments: 5 Pages.

Kouider function with basis a is a new numerical function which presented by Kouider (2021,[1]). In this paper we interesting in study this function for all natural number. In order to discover his properties. Therefore we present a new number called kouider number with bases a which we go developed in this study.
Category: Number Theory

On the General Erdh{o}s-Moser Equation Via the Notion of Olloids

Authors: Theophilus Agama
Comments: 6 Pages.

We introduce and develop the notion of the textbf{olloid}. We apply this notion to study a variant and a generalized version of the ErdH{o}s-Moser equation under some special local condition.
Category: Number Theory

The Notion of Olloids and the Erdh{o}s-Moser Equation

Authors: Theophilus Agama
Comments: 5 Pages.

We introduce and develop the notion of the textbf{olloid}. We apply this notion to study the ErdH{o}s-Moser equation.
Category: Number Theory

Fermat's Little Theorem and Composite Numbers

Authors: Héras Joël
Comments: 6 Pages.

En utilisant le Petit théorème de Fermat ap ≡ a mod p, il est possible de factoriser presque tous les nombres composés.

Using the Fermat's little theorem ap ≡ a mod p, it is possible to factorize into primes almost all composite numbers.
Category: Number Theory

A Formula of Zhi-Wei Sun

Authors: Edgar Valdebenito
Comments: 5 Pages.

In this note we discuss various equivalent formulations for the sum of an infinite series considered by Zhi-Wei Sun.
Category: Number Theory

Proofs Using my Definition Series 3

Authors: Yuji Masuda
Comments: 1 Page.

The purpose of this short paper is to provide a proof based on my definition series to determine the correctness of my previous papers No.19, No.89.
Category: Number Theory

Proofs Using My Definition Series 2

Authors: Yuji Masuda
Comments: 1 Page.

The purpose of this short paper is to provide a proof based on my definition series to determine the correctness of my previous papers No.21.
Category: Number Theory

The Distribution of Prime Numbers

Authors: Mohammed Bouras
Comments: 2 Pages. New Sequence of prime numbers

In this paper, we discovered a new sequence of prime numbers, such that every term of this sequence is either a prime number or equal to 1.
Category: Number Theory

On the Regularity of Prime Numbers

Authors: Yuji Masuda
Comments: 1 Page.

In this short paper, I will publish one prediction based on the regularity of prime numbers.
Category: Number Theory

Collatz Conjecture Part 1: Alternative Definition for Collatz's Transformations

Authors: Gaurav Sharma
Comments: 12 Pages. (Corrections made by viXra Admin - Please conform)

We consider n to have only odd values, and even values are written in the form; n.2^b. We create a predefined function r_b (n).Define, g(n)=r_b (n)+r_(b-1) (n) and prove g(n)=f(n). g(n) being an identical function to Collatz transformations, we use the properties of said function to probe the conjecture.
Category: Number Theory

Euler-Mascheroni Constant

Authors: Edgar Valdebenito
Comments: 7 Pages.

A double integral for Euler-Mascheroni Constant.
Category: Number Theory

Exploration of Expression for Pi

Authors: Edgar Valdebenito
Comments: 2 Pages. (Note by viXra Admin: Please write in complete sentences and refrain from using irrelevant title etc.)

In this paper, the author explores the expression for Pi.
Category: Number Theory

Determination of the Higgs Boson’s Mass

Authors: Gang Chen, Tianman Chen, Tianyi Chen
Comments: 5 Pages.

In our previous papers, we gave formulas of the fine-structure constant and their corresponding applications along with a mass model of the elementary particles. And in a recent paper, we redefined Hartree atomic units to Hartree-Chen atomic units. In this paper, we apply our mass model of the elementary particles and Hartree-Chen atomic units to determine the exact value of the Higgs boson’s mass. Based on our hypothetical formulas, the Higgs boson’s in Hartree-Chen atomic units should be 245280.001934, and the exact value of the Higgs boson’s mass should be 125.33782309(4) GeV. Compared to the value of 125.35 ± 0.15 GeV which was measured out by CERN in 2016, our calculated value is almost absolutely precise if it is correct. By the new value of 125.11(11) GeV reported by ATLAS Collaboration, we revised the calculated value for the Higgs boson’s mass to 125.15375720(4) GeV.
Category: Number Theory

On Conjectures About the Simultaneous Pell Equations X2−(a2−1)y2=1 and Y2−pz2=1

Authors: Michael C. I. Nwogugu
Comments: 10 Pages. The copyright license-type for this article is CC-BY-NC-ND

This article shows that the Qu (2018) conjectures, the Yang & Fu (2018) conjectures, the Jiang (2020) Conjecture-#1, the Tao (2016) Conjecture-#1, the Cipu & Mignotte (2007) Conjecture, the Ai, Chen, Zhang & Hu (2015) Conjecture, the Yuan (2004) Conjecture, the Keskin, Karaatlı, et. al. (2017) Conjecture, the Cipu (2018) Conjecture, and the Cipu (2007) Conjecture [all of which pertain to the system of Simultaneous Pell equations x^2−(a^2−1)y^2=1 and y^2−pz^2=1] are wrong or incomplete (incomplete in the sense that they didn’t provide complete solutions for the system of equations). This article also introduces simple Java codes for solutions to this class of equations for positive-integers up to (10)^2457600000 (and even greater positive-integers depending on available computing power).
Category: Number Theory

Zero-Dimensional Number Theory

Authors: Stephen H. Jarvis
Comments: 40 Pages.

Examined here is a proposed zero-dimensional number theory as the process of labelling zero-dimensional space as a point and zero-dimensional time as a moment as the different mathematical values of 0 and 1 respectively. By such it can be shown how zero-dimensional time in being mathematically labelled as a unit can form relationships between zero-dimensional spatial points labelled as 0. Here, zero-dimensional time can be demonstrated to derive a suite of mathematical operators for zero-dimensional points that then relate with each other in the form of equations for 1d, 2d, and 3d timespace. By such, time can be shown to represent the fundamental basis for all mathematical equation operators (addition, subtraction division, multiplication, equality, exponentiation, etc) for points in space. Subsequently, it is proposed that the resulting time and space (timespace) equations are synonymous with the mathematical equations that describe both the physical phenomenal field forces and their associated particle activity. In this process, solutions can be shown for Goldbach’s conjecture, the Riemann hypothesis, and Fermat’s last theorem, together with the formulation of a physical theory matching known physics theory equations and associated constants. The result of such is a zero-dimensional number theory that both prescribes the basis for a mathematical theorem together with becoming a physical theory as a process of accounting for the equations of physical phenomena.
Category: Number Theory

General Base Decimals with the P-Series of Calculus Shows All Zeta(n) Irrational

Authors: Timothy W. Jones
Comments: 10 Pages. This is a different proof that uses limits in a more conventional way.

We give a new approach to the question of whether or not all greater than one, integer arguments of Zeta are irrational. Currently only Zeta(2n) and Zeta(3) are known to be irrational. We show that using the denominators of the terms of Zeta(n)-1=z_n as decimal bases gives all rational numbers in (0,1) as single decimals, property one. We also show the partial sums of z_n are not given by such single digits so using the denominators of the partial sum's terms as number bases, property two. Next, using integrals for the p-series contracting upper and lower bounds for partial sum remainders of z_n are generated. Assuming z_n is rational, it is expressible as a single decimal using the denominator of a term of z_n (property one) and eventually these bounds will consist of infinite decimals (property two) with their first decimal equal to this single decimal. But as no single decimal can be between two infinite decimals with the same first digit a contradiction is derived and all z_n are proven irrational.
Category: Number Theory

With the Pythagorean Triples C^2 = A^2 + B^2 We Obtain that C^2 * N = A^2 * N + B^2 * N and C^2 * N is a Power of C with Exponent> 2

Authors: Giovanni Di Savino
Comments: 4 Pages.

It is possible to multiply the three powers by the same value square of a Pythagorean triple and obtain that the sum of two is equal to the power of the third.
Category: Number Theory

Fermat's Last Theorem: A Proof by Contradiction

Authors: Benson Schaeffer
Comments: 6 Pages.

In this paper I oer an algebraic proof by contradiction of Fermat's Last Theorem. Using an alternative to the standard binomial expansion, (a+b)n = an + b Pn i=1 ani(a + b)i1, a and b nonzero integers, n a positive integer, I showthat a simple rewrite of the equation stating the theorem, Ap + (A + b)p = (2A + b c)p; A; b and c positive integers, entails the contradiction of two positive integers that sum to less than zero,(2f + g)(f + g)(f + g + b) Xp2 i=1 (2f + g)p2i(3f + 2g + b)i1 + (f + b)(f + g)(3f + 2g + b)p2 + fb(3f + 2g + b)p2 < 0; f and g positive integers. This contradiction shows that the rewrite has nonon-trivial positive integer solutions and proves Fermat's Last Theorem.
Category: Number Theory

A Lower Bound for Length of Addition Chains

Authors: Theophilus Agama
Comments: 4 Pages.

In this paper we show that the shortest length $iota(n)$ of addition chains producing numbers of the form $2^n-1$ satisfies the lower bound $$iota(2^n-1)geq n+lfloor frac{log (n-1)}{log 2}floor$$ where $lfloor cdot floor$ denotes the floor function.
Category: Number Theory

A Simple Proof that Goldbach's Conjecture is True

Authors: Timothy W. Jones
Comments: 3 Pages.

A induction proof shows Goldbach's conjecture is correct. It is as simple as can be imagined. A table consisting of two rows is used. The lower row counts from 0 to any n and and the top row counts down from 2n to n. All columns will have all numbers that add to 2n. Using a sieve, all composites are crossed out and only columns with primes are left. For the base case of k=5 suppose that primes on the lower row always map to composites on the top and that this results in too many composites on the top. This is true for this base case. Suppose it is true for k=n, then the shifts and additions necessary for the k=n+1 case maintain this property of too many composites on top. The contrapositive is that there exists a prime on the bottom that maps to a prime on top and Goldbach is established: the sum of these two primes is 2(n+1).
Category: Number Theory

Beal Conjecture Proved Very Simply & Very, Very Simply

Authors: A. A. Frempong
Comments: 7 Pages. Copyright © by A. A. Frempong

Two versions of the proof are covered. The first version is very simple; and the second version is very, very simple. By applying basic mathematical principles, the author surely, and instructionally, proves, directly, the original Beal conjecture which states that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. One will let r, s, and t be prime factors of A, B and C, respectively, such that A = Dr, B = Es, C = Ft, where D, E, and F are positive integers. Then, the equation A^x + B^y = C^z becomes D^xr^x + E^ys^y = F^zt^z. The proof would be complete after showing that the equalities, r^x = t^x, s^y = t^y and r = s = t, are true. The proof of the above equalities would involve showing that the ratios, (r^x)/(t^x) = 1 and (s^y)/(t^y) =1, which would imply that r = s = t. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the power of each prime factor on the left side of the equation equals the same power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture for a bonus question on a final class exam.
Category: Number Theory

Ramanujan, Integral, Continued Fraction

Authors: Edgar Valdebenito
Comments: 7 Pages.

Ramanujan (1887-1920), notable achievements in the evaluation of integrals.
Category: Number Theory