Number Theory

Why All Perfect Numbers Are Even

Authors: Derek Tucker
Comments: 1 Page.

We prove there are no odd perfect numbers by the properties of the sum of divisors function forcing the existence of a number with its sum of divisors equal to two, which is absurd.
Category: Number Theory

A Relationship Between Pythagorean Triples and Triangular Numbers

Authors: Julian Beauchamp
Comments: 1 Page.

In this paper, I share a discovery of a subset of Pythagorean Triples (X,Y,Z=Y+1), whose hypotenuses end in the digits 13 and begin in the digits of Triangular Numbers ($T_{n}$). Although this is a slightly curious relationship, it is of little mathematical significance, and I believe it to be previously unknown.
Category: Number Theory

On Distribution of Some Prime Numbers

Authors: Alazar I. Ali
Comments: 9 Pages.

Abstract. This paper focuses on one of the most studied areas of number theory — the distribution of prime numbers. Particularly, the last section of this paper comes up with a conjecture that could lead to a new way to look the distribution of prime numbers.
Category: Number Theory

Fermat's Last Theorem: the Numbers a, B, C Are Infinite, or What P. Fermat Did not Write in the Margins

Authors: Victor Sorokine
Comments: 6 Pages.

For the values of the numbers A, B, C (with the last digits a, b, c) equal to a^(n^k), b^(n^k), c^(n^k), equality (1*) is satisfied by (k+1)-digit endings and is not performed by (k+2)-th digits (which follows from the sum of n-1 equalities)..To turn this inequality into equality, it is necessary to increase the value of the exponent k by 1, but after that, the equality is not satisfied for the (k+3)-digits. And so endlessly.
Category: Number Theory

A Refined Pothole Method and the Scholz Conjecture on Addition Chains

Authors: Theophilus Agama
Comments: 6 Pages.

Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove the inequality $$iota(2^n-1)leq frac{3}{2}n-left lfloor frac{n-2}{2^{lfloor frac{log n}{log 2}-1floor+1}}ight floor-lfloor frac{log n}{log 2}-1floor +frac{1}{4}(1-(-1)^n)+iota(n)$$ where $lfloor cdot floor$ denotes the floor function and $iota(n)$ the shortest addition chain producing $n$.
Category: Number Theory

Another Conjecture for the Sum of Divisors

Authors: Julian Beauchamp
Comments: 2 Pages.

In this paper, we conjecture that no sum of divisors exists of the form x^2+1. We offer a proof only for prime numbers, showing that prime numbers cannot have a sum of divisors that is of the form x^2+1.
Category: Number Theory

Disjointed Equivalence of Gravitational and Inertial Mass

Authors: D. J. Pons
Comments: 41 Pages. Preprint of peer reviewed and published version at https://doi.org/10.4236/jmp.2023.143017

Problem- Contemporary physics offers no underlying reason for the equivalence of inertial and gravitational mass. Approach- The equivalence is examined from the new physics provided by the cordus theory, being a non-local hidden-variable (NLHV) theory. Mathematical formalisms are derived for masses and observers in different fabric densities. Findings- A disjointed equivalence is predicted, whereby inertial and gravitational mass are equivalent in any one situation, but a different equivalence holds when the fabric densities change. Consequently this theory predicts that the gravitational constant G varies with fabric density, and hence would be different across the universe and across time. Not only is the gravitational constant non-constant, but the formulation of gravitation changes with fabric density. Specifically, the theory predicts gravity is stronger at genesis (and the end of the universe) such that orbit velocity v_B∝√(r_B ) (where r_Bis orbit radius), compared to weaker gravitation at middle life epochs with v_B∝1/√(r_B ). The current Earth location and epoch corresponds to the latter case, i.e. Newtonian gravitation is recovered. The findings disfavour the existence of both dark energy and dark matter, and instead attribute these effects to differences in the fabric density. Originality — The work makes the contribution of deriving a mass equivalence relationship that includes fabric density, identifying a disjointed mass equivalence, and showing that the gravitation formulation itself changes with relative fabric densities.
Category: Number Theory

Euler's Constant

Authors: Edgar Valdebenito
Comments: 5 Pages.

The constant gamma=0.5772157...is the limit as n tends to infinity of the difference between the sum of n terms of the harmonic series and the natural logarithm of n. It is not know whether this number is algebraic or transcendental, nor even whether it is rational or irrational.
Category: Number Theory

On the Irrationality of Riemann Zeta Functional Value at Odd Integers

Authors: Shubhayan Ghosal
Comments: 4 Pages.

In this article we provide a proof of the irrationality of ζ(2n+1) ∀n ∈ N.Also,in our attempt, we construct an upper bound to the Zeta values at odd integers.It is interesting to see how the irrationality of Zeta values at even positive integers mixed up with Dirichletsirrationality criterion and this bound accelerates our proof further,case by case.
Category: Number Theory

The Proof of the Collatz Conjecture

Authors: Kwangsun Song
Comments: 3 Pages.

Prove the Collatz conjecture and summarize it as an formula.
Category: Number Theory

Proof of the Riemann Hypothesis

Authors: Hyunho Song
Comments: 6 Pages.

I decided to use the Fourier series to solve the Riemann Hypothesis. This article approached the Riemann Hypothesis as a periodic function with a Fourier series.To solve the Riemann Hypothesis, I approached the main content of the Riemann Hypothesis, the function of the Riemann Hypothesis. Here, I did not write the calculation process in this article, so if you want to check the authenticity, do the calculation yourself.To understand this article, you must be familiar with the Fourier series. Because the important point in this proof is that the Fourier series is important. Here, the Fourier series is a Fourier series, a Fourier transform, and an inverse Fourier transform.In this proof I converted the Riemann zeta function into a periodic function and then approached the solution of the Riemann zeta function as a Fourier series. The way I approached it is simple. After converting the solution into a periodic function, the Fourier series was used to confirm its authenticity. Here, the proof was attempted with the real part of the solution period function of the Riemann zeta function fixed as a critical line. And in the main body of this article, we will show the detailed proof process.
Category: Number Theory

La Función Zeta Como un Caso Particular de la Fórmula de Euler-Maclaurin
The Zeta Function as a Particular Case of the Euler-Maclaurin Formula

Authors: Marco Burgos
Comments: 14 Pages.

Una de las funciones más famosas y llenas de enigmas es la función zeta de Riemann, ya que es la base de una de las hipótesis más sorprendentes por su relación con la función contadora de primos. Pero en este trabajo, revelaremos algunos de esos enigmas, utilizando la fórmula de Euler-Maclaurin.

One of the most famous and enigmatic functions is the Riemann zeta function, since it is the basis of one of the most surprising hypotheses due to its relationship with the prime counting function. But in this paper, we will reveal some of those enigmas, using the Euler-Maclaurin formula.
Category: Number Theory

On the Sequence:{0 Larger Than xn Smaller Than 1 and 3*(xn^n)+3*xn=1, N=1,2,3,...}

Authors: Edgar Valdebenito
Comments: Pages.

We study the sequence: {0<xn<1 and 3*(xn^n)+3*xn=1, n=1,2,3,...}.
Category: Number Theory

A Formula of the Dirichlet Character Sum

Authors: JinHua Fei
Comments: 12 Pages.

In this paper, We use the Fourier series expansion of realvariables function, We give a formula to calculate the Dirichlet charactersum, and four special examples are given
Category: Number Theory

A Recurrence Formula for the Sum of Divisors

Authors: Julian Beauchamp
Comments: 2 Pages.

In this paper, we describe a simple recurrence formula for the sum of divisors.
Category: Number Theory

The Determination of Integer Coordinates of Elliptic Curves

Authors: Abdelmajid Ben Hadj Salem
Comments: 14 Pages. Submitted to Ramanujan Journal. Comments welcome.

In this paper, we give an elliptic curve $(E)$ given by the equation: $$y^2=varphi(x)=x^3+px+q $$ with $p,qin Z$ not null simultaneous. We study the conditions verified by $(p,q)$ so that $exists ,(x,y) in Z^2$ the coordinates of a point of the elliptic curve $(E)$ given by the equation above.
Category: Number Theory

On a Conjecture of Erdos on Additive Basis of Large Orders

Authors: Theophilus Agama
Comments: 6 Pages.

Using the methods of multivariate circles of partition, we prove that for any additive base A [...] holds for sufficiently large values of k provided the counting function [...] is an increasing function for all k sufficiently large.
Category: Number Theory

Fermat's Last Theorem as a Consequence of the Theorem on Second Digits of Degree N

Authors: Victor Sorokine
Comments: 2 Pages.

In a base number system with prime base n>2 and natural numbers A, B, C [with last digits a, b, c, where a>0 and b>0] the equality A^n+B^n-C^n=0 is impossible.In each of the sets V and W, respectively, of the numbers d^n (d=1, u2026, n-1) and (ad)^n+(bd)^n-(cd)^n [where a, b, c are single-valued, a+b-c=0 (mod n), a>0 and b>0] with equal sums S of all n-1 two-digit endings of numbers in V and W [with the ending of S equal to 00] are numbers with the second (from the end) digits not equal to zero.Otherwise, the second digit in the number S is not 0 (see the arithmetic progression formula).
Category: Number Theory

Prime Numbers Estimation in A Special Palindromic Number

Authors: Zhi Li, Hua Li
Comments: 3 Pages.

The repunit number refers to the natural number composed of all the numbers 1, which is a special palindromic number. Whether the number of prime numbers in the repunit number is infinite, and how to calculate the number of prime numbers less than a certain integer have not been determined. This paper proves that the number of prime numbers of this kind is infinite by using the method of probability and statistics, and derives the calculation formula of it. On this basis, two conjectures are further proposed, that is, the ratio of the number of prime numbers in this palindromic number to the number of prime numbers in Mersenne number is 16ln (2)/(15ln (10)), and the number of prime numbers less than 24862048 digits is close to 15.
Category: Number Theory

Possible Proof of the Riemann Hypothesis Resulting From Prediction of Exact Values of the Landau Constants

Authors: Miroslav Sukenik
Comments: 5 Pages.

Based on the predicted value of the Landau constants L = 0.5331588u2026.and A = 0.7301772..., it is possible to identify the critical line x=0.5 with the horizon line. It is known that the size of entropy is proportional to the size of the surface of the horizon, and according to the holographic principle, all information must lie on this surface. So also the zero points of the Riemann Zeta function, which carry information about the distribution of primenumbers.
Category: Number Theory

Finite Dimensional Series and Its Applications in Number Theory and Mathematical Analysis

Authors: Shazly Abdullah
Comments: 23 Pages.

In this study we used an algebraic method that uses elementary algebra and binomial theorem to create finite binomial series. This is a type of series that has several properties in variables. We used these series to prove results in congruences and primitive roots, Diophantine equation. We also obtained several results in finite series.
Category: Number Theory

Using Finite Binomial Series to Prove Results in Prime Numbers and Congruences

Authors: Shazly Abdullah
Comments: 8 Pages.

In this work used an algebraic method that uses elementary algebra and binomial theorem to create finite binomial series.This is a type of series that has several properties in variables. We used these series to prove results in congruences and prime numbers.
Category: Number Theory