Number Theory

A New Identity For Prime Counting Function

Authors: Amisha Oufaska
Comments: 4 Pages.

In this article, the author proves on a new identity (or equation) which asserts that for every natural number n the sum of the prime-counting function π(2n) and the con-counting function π ̅(2n) equals n , explicitly and simply ∀ n∈N^* we have π(2n)+ π ̅(2n) = n . The new identity (or equation) may have many applications in Number Theory and its related to one of the famous problems in Mathematics .
Category: Number Theory

Very Simple Proof of 3x+1

Authors: Samuel Ferrer Colas
Comments: 2 Pages.

The Collatz or 3x + 1 conjecture is perhaps the simplest stated yet unsolved problem inmathematics in the last 70 years. It was circulated orally by Lothar Collatz at the InternationalCongress of Mathematicians in Cambridge, Mass, in 1950 (Lagarias, 2010).The problem is known as the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm(after Helmut Hasse), or the Syracuse problem.In this concise paper I provide a very simple proof of this conjecture.
Category: Number Theory

On the Existence of Solutions to Erdh{o}s-Straus Type Equations

Authors: Theophilus Agama
Comments: 5 Pages.

We apply the notion of the textbf{olloid} to show that the family of ErdH{o}s-Straus type equation $$frac{4^{2^l}}{n^{2^l}}=frac{1}{x^{2^l}}+frac{1}{y^{2^l}}+frac{1}{z^{2^l}}$$ has solutions for all $lgeq 1$ provided the equation $$frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$$ has solution for a fixed $n>4$.
Category: Number Theory

Natural Number Infinite Formula and the Nexus of Fundamental Scientific Issues

Authors: Budee U. Zaman
Comments: 9 Pages.

Within this paper, we embark on a comprehensive exploration of the profound scientific issues intertwined with the concept of the infinitewithin the realm of natural numbers. Through meticulous analysis, we delve into three distinct perspectives that shed light on the nature of natural number infinity. By considering the framework of time reference, we confront and address the inherent challenges that arise when contemplating the infinite. Furthermore, we navigate the intricate relationship betweenthe infinite and fundamental scientific questions, seeking to unveil novel insights and resolutions. In a departure from conventional viewpoints,our examination of natural number infinity takes on a relativistic dimension, scrutinizing the role of time and the observer’s perspective. Strikingly, as we delve deeper into the foundational strata, we uncover the pivotal significance of relativity not only in physics but also in mathematics. This realization propels us towards a more holistic and consistentmathematical framework, underlining the inextricable link between the infinitude of natural numbers and the essential constructs of time and perspective.
Category: Number Theory

An Extension to Fermat's Pythagorean Triangle Area Proof, and Fermat's Last Theorem

Authors: Richard Kaufman
Comments: 12 Pages. This is a new paper with new results from Darmon and Merel.

Pierre de Fermat proved that the area of a Pythagorean triangle is not a square. Here we extend his result for Pythagoran triangles to consider cubic integer areas and higher power areas. We show how each such power �� immediately leads to a Fermat’s equation ��^k + ��^k = ��^k for integer �� > 2 and positive integers ��, ��, and ��. Using only elementary results, we show that a Pythagorean triangle area is not a cube. Using non-elemantary results from Darmon and Merel, we can extend Fermat’s Pythagorean triangle area result to show that these areas cannot be higher powers either. The results from Darmon and Merel are an alternative to using Andrew Wiles more complex result for Fermat Last Theorem to establish the same result - using the impossibility of the Fermat equations ��^k + ��^k = ��^k. Based on equations derived in this paper, we may wonder if some of these elementary results could have been known to Fermat himself. That is, could Fermat’s proof that the area of a Pythagorean triangle is not a square have helped him to envision what we have come to know as Fermat equations and Fermat’s Last Theorem?
Category: Number Theory

The Condition for the Real Part of Dirichlet Function to be 1/2

Authors: Xiaohui Li
Comments: 1 Page.

Find the trigonometric function sin(n/2)π that satisfies the Dirichlet feature, and then analyze the conditions for making the real part of the L-function 1/2.
Category: Number Theory

The Simple Structure of Prime Numbers

Authors: Ihsan Raja Muda Nasution
Comments: 3 Pages.

The prime numbers have a pseudo-random structure. And this structure is not simple. In this paper, we analyze the behavior of prime numbers. And we diagnose the inner body of the prime numbers.
Category: Number Theory

Proof of Collatz Conjecture Using Division Sequence Ⅲ

Authors: Masashi Furuta
Comments: 5 Pages.

This paper is positioned as an extra edition of [1]. First, as in [1], define "division sequence", "complete division sequence", and "star conversion". Next, we consider loops and divergences in the Collatz conjecture, respectively. Theorem Proving is not used in this paper.
Category: Number Theory

Any Even Number Greater Than 6 Can be Written as the Sum of Two Prime Numbers

Authors: Xiaohui Li
Comments: 3 Pages.

The so-called strong Goldbach conjecture, which means that any even number greater than 6 can be written as the sum of two prime numbers, is also known as the "strong Goldbach conjecture" or the "Goldbach conjecture about even numbers".This paper utilizes the basic principle that the number of all odd numbers in the positive integer remains constant, and the sum of all odd numbers remains constant, and the values of odd numbers are not equal to each other. It is found that there are both odd prime numbers pr1 and pr2 in the [3, n] and (n, 2n-2) intervals, respectively. Then, the equivalent transformation is performed by using the principle that the number of all odd numbers, the sum of all odd numbers remains constant, and the values of odd numbers are not equal to each other, thereby proving that 2n=pr1+pr2.
Category: Number Theory

"Eureka" Shift, Taylor Shift, Offset, Symmetry Point, and Symmetry in Polynomials

Authors: Charles Kusniec
Comments: 16 Pages.

In this study we show the existence of three types of shifts in polynomial curves that will always result in integer sequences: 1. "Eureka" shift, 2. Taylor shift, and 3. Offset. Then, we demonstrate that every polynomial equation has a reference point that we call sp - symmetry point. From the symmetry point of any polynomial sequence of integers we can define two types of symmetry and one type of asymmetry. At the end, we name and define asymmetry, and the two types of symmetries.
Category: Number Theory

Exact Sum of Prime Numbers in Matrix Form

Authors: Budee U. Zaman
Comments: 6 Pages.

This paper introduces a novel approach to represent the nth sum of prime numbers using column matrices and diagonal matrices. The proposed method provides a concise and efficient matrix form for computing and visualizing these sums, promising potential insights in number theory and matrix algebra. The innovative representation offers a new perspectiveto explore the properties of prime numbers in the context of matrix algebra.
Category: Number Theory

Connected Old and New Prime Number Theory with Upper and Lower Bounds

Authors: Budee U. Zaman
Comments: 9 Pages.

In this article, we establish a connection between classical and modern prime number theory using upper and lower bounds. Additionally, weintroduce a new technique to calculate the sum of prime numbers.
Category: Number Theory

On the Zeta Distribution and Riemann Hypothesis

Authors: Yahya Grari
Comments: 7 Pages.

we will be very optimistic and give what seems to be a probabilistic argument in favor of theRiemann hypothesis through Denjoy's version.
Category: Number Theory

An Alternative Form of Hardy-Littlewood Conjecture

Authors: Junho Choi
Comments: 5 Pages.

I found an alternative form of Hardy-Littlewood Conjecture using Mertens’ 3rd theorem. This new form has a theoretical background and coincides with prime number theorem. It is expected to provide an easier way to prove the conjecture.
Category: Number Theory

Contraction of Ramanujan Formulas in the Letter to Hardy

Authors: Juan Elias Millas Vera
Comments: 4 Pages.

In this paper we show an approach to the Ramanujan summation of series formulas, proving that it is possible a contracted version of them.
Category: Number Theory

A New Type of Approximation for the Gamma Function Based on the Windschitl’s Formula

Authors: HyonChol Kim
Comments: 6 Pages.

In this paper, we present a new approximate formula based on the Windschitl’s type formula, one of the important approximate formulas of the Gamma function.And we introduce interesting double inequality associated with our new formula.
Category: Number Theory

Ramanujan’s Infinite Summation Formula: Key Fundamental Scientific Issues

Authors: Pankaj Mani
Comments: 18 Pages. (Corrections made by viXra Admin to conform with scholarly norm)

The author tries to Look at the famous Ramanujan's Infinite Summation Result from a Relativistic Time reference frame and resolving the fundamental issues. Mathematics unlike conventional views ,when looked at Relativistic Time, Observer's perspective, many contradictions.conflicts seem to get resolved. In fact at deeper foundational level, there is critical role of relativity, time and observer even in mathematics like physics that must be incorporated to make mathematics consistent.
Category: Number Theory

Prime Numbers, Finding Them All with a Method Based on Divisible Numbers (Using Only Additions and not Divisions)

Authors: Filiberto Marra, Cristina Gabrielli
Comments: 29 Pages.

This study on prime numbers presents a method that allows us to know divisible numbers without performing complex calculations. It is based on a simple calculation system using additions of numbers instead of divisions, and it enables finding all divisible numbers. By eliminating them, we can identify all prime numbers.
Category: Number Theory

On the Numbers 3F2(1,(1-N)/2,-N/2; 3/2,1/2-N;4) , N=0,1,2,3,...

Authors: Edgar Valdebenito
Comments: 4 Pages.

In this note we give some formulas related to the numbers 3F2(1,(1-n)/2,-n/2;3/2,1/2-n;4),n=0,1,2,3,...,where 3F2 is the generalized hypergeometric function.
Category: Number Theory

A Conjecture On σ(n) Function

Authors: Sourav Mandal
Comments: 9 Pages.

We know many Arithmetical Functions [1] like ϕ(n),σ(n),τ(n) etc. In this paper we will discuss about σ(n) and will see a phenomenal observation.And later we will claim this observation as a conjecture.
Category: Number Theory

Unexpected Connection Between Triangular Numbers and the Golden Ratio

Authors: Waldemar Puszkarz
Comments: 5 Pages. Originally posted on ResearchGate in March 2023.

We find out that when a sum of five consecutive triangular numbers, $S_5(n)= T(n)+...+T(n+4)$, is also a triangular number $T(k)$, the ratios of consecutive terms of $a(i)$ that represent values of $n$ for which this happens, tend to $phi^2$ or $phi^4$ as $i$ tends to infinity, where $phi$ is the Golden Ratio. At the same time, the ratios of consecutive terms $S_5(a(i))$ tend to $phi^4$ or $phi^8$. We also note that such ratios that are the powers of $phi$ can appear in the sequences of triangular numbers that are also higher polygonal numbers, one case of which are the heptagonal triangular numbers.
Category: Number Theory

Constant C Makes the Abc Conjecture Hold

Authors: Xiaohui Li
Comments: 4 Pages.

The ABC conjecture in number theory was first proposed by Joseph Oesterl é and David Masser in 1985. Mathematicians declare this conjecture using three related positive integers a, b, and c (satisfying a+b=c). The conjecture states that if there are certain prime powers in the factors of a and b, then c is usually not divisible by the prime powers.This paper utilizes the fact that the prime factor among all factors in the root number rad (c) can only be a power of 1. Then, analyze all combinations of c that satisfy rad (c)=c, calculate the value of the combination, and find the maximum and minimum values of the root number rad, as well as the maximum exponent between them. Using this maximum index, an equivalent inequality is constructed to prove the ABC conjecture.
Category: Number Theory

An Elementary Proof of Goldbach’s Conjecture v. 3.0

Authors: Ronald Danilo Chávez
Comments: 29 Pages.

In this present paper we will show you an elementary proof of the Goldbach’s Conjecture based on probabilities.
Category: Number Theory

Analytic Proof of The Prime Number Theorem

Authors: Subham De
Comments: 24 Pages.

In this paper, we shall prove the textit{Prime Number Theorem} by providing a brief introduction about the famous textit{Riemann Zeta Function} and using its properties.
Category: Number Theory

Quasi-Perfect Numbers Have at Least 8 Prime Divisors

Authors: B. Zemann
Comments: 17 Pages.

Quasi-perfect numbers satisfy the equation sigma(N) = 2*N+1, where sigma is the divisor summatory function. By computation, it is shown that no quasi-perfect number has less than 8 prime divisors. For testing purposes, quasi-multiperfect numbers are examined also.
Category: Number Theory

Combination Rule and Last Fermat's Theorem

Authors: Carlos Alejandro Chiappini
Comments: 6 Pages. carloschiappini@gmail.com

The central objective of this document is to reason regarding the combined use of two or more methods to solve a mathematical problem.To facilitate understanding I present the topic with the help of a known problem. I have chosen Fermat's Last Theorem because the polynomial that expresses it has few monomials and few variables.
Category: Number Theory

Regla de Combinación y Último Teorema de Fermat (Combination Rule and Last Fermat's Theorem)

Authors: Carlos Alejandro Chiappini
Comments: 8 Pages. In Spanish (email: carloschiappini@gmail.com)

El objetivo central de este documento es razonar respecto al uso combinado de dos o más metodos para resolver un problema matemático. Para facilitar la comprensión presento el tema con ayuda de un problema conocido. He escogido el último teorema de Fermat porque el polinomio que lo expresa posee pocos monomios y pocas variables.

The central objective of this document is to reason regarding the combined use of two or more methods to solve a mathematical problem. To facilitate understanding I present the topic with the help of a known problem. I have chosen Fermat's Last Theorem because the polynomial that expresses it has few monomials and few variables.
Category: Number Theory

Proof that the Real Part of All Non Trivial Zeros of Riemann Zeta Functions is 1/2

Authors: Xiaohui Li
Comments: 4 Pages.

Riemann hypothesis is that the real part of all nontrivial zeros of Riemann zeta functions is 1/2.Mr. Riemann formed Riemann zeta function by Analytic continuation of Euler zeta function,There are trivial and non trivial zeros in the Riemannian zeta function that make its value zero.Standard By analyzing the Trigonometric functions relationship in the equivalent Algebraic expression of Riemannian zeta function, it is concluded that the real part of all nontrivial zeros is 1/2.
Category: Number Theory