Number Theory

Syracuse-Collatz Conjecture: Exploration, Analysis and Demonstration of a Mathematical Enigma

Authors: Mostafa Senhaji
Comments: 12 Pages. In French

In the infinite universe of numbers, the Syracuse conjecture emerges as a captivating enigma, defying mathematical conventions and arousing the curiosity of the most daring minds.
Category: Number Theory

On the Convergence of the Generalized Collatz Function

Authors: Oussama Basta
Comments: 6 Pages.

The Collatz conjecture, which states that repeated application of the function ( f(n) = begin{cases} n/2, & text{if } n equiv 0 pmod{2} 3n+1, & text{if } n equiv 1 pmod{2} end{cases} ) to any positive integer ( n ) will eventually reach the number 1, has been a long-standing open problem in mathematics. In this paper, we investigate a generalized version of the Collatz function, denoted as ( f(E, T) ), where ( E ) is a positive integer and ( T ) is a fixed positive integer. We prove that for any positive integer ( E ), repeated application of ( f(E, T) ) will eventually lead to an even number. Furthermore, we show that any even number will eventually reach a power of 2 under repeated application of ( f(E, T) ), and once a power of 2 is reached, the sequence will enter the cycle ( 1 ightarrow 4 ightarrow 2 ightarrow 1 ). These results provide new insights into the behavior of the generalized Collatz function and its convergence properties.
Category: Number Theory

Attempt at a Proof of Collatz Conjecture

Authors: Ryujin Choi
Comments: 2 Pages. (Correction made by viXra Admin to conform with the requirements of viXra.org - Future non-compliant submission will not be accepted!)

This is an attempt of a proof of Collatz Conjecture.
Category: Number Theory

Contributions to the Langlands Program

Authors: Romain Viguier
Comments: 7 Pages. (Note by viXra Admin: Please cite and list scientific references as reminded previously - Future non-compliant submission/repalcement will not be accepted!)

This article is a contribution to the Langlands Program.
Category: Number Theory

Simple Arrangement in Which it is Possible to Arrange the Prime Numbers in Order to Obtain a Range of Occupied Consecutive Positions Greater Than Those Defined by the Legendre Conjecture

Authors: Andrea Berdondini
Comments: 4 Pages.

The Legendre conjecture tells us that there always exists a prime number between ��^2 and (�� + 1)^2. In this article, we will see that it is possible to arrange prime numbers less than or equal to N in order to obtain a consecutive number of occupied positions greater than the interval from ��^2to (�� + 1)^2. It is important to note that this result does not define a counterexample to the Legendre conjecture, but it represents an interesting theoretical result that can help us resolve many of the conjectures regarding the gap between two prime numbers
Category: Number Theory

Proving & Teaching Beal Conjecture

Authors: A. A. Frempong
Comments: 8 Pages. Copyright © by A. A. Frempong

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. More formally, the conjectured equality, r^x = t^x would be true if and only if (r^x /t^x) =1; and the conjectured equality s^y = t^y would be true if and only if (s^y/ t^y) = 1. These conjectures would be proved in the Beal conjecture proof. 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

Elementary Matrix Algorithm of Order 3

Authors: Ruslan Pozinkevych
Comments: 3 Pages. The author analyzes algorithm for 3 component vector bases

The aim of our research is to establish relationship between linearly independent vectors and the R^3 Space The means for that are going to be taken from Balanced Ternary analysis e.g triplets with entries (-1,0,1) This is not going to be any specific vector, for instance (-1,0,1) It’s going to be rather a Coordinate system with 8 quadrants and directional vectors The novelty of the proposed research lies in the fact that unlike traditional Cartesian system-vector basis our system uses 8 vectors which are at the same time a polyvector
Category: Number Theory

Collatz Conjecture Integer Series Has no Looping Except One

Authors: Tsuneaki Takahashi
Comments: 2 Pages. (Note by viXra Admin: Further repetition/regurgitation will not be accepted!)

If the series of Collatz Conjecture integer has looping in it, it is sure the members of the looping cannot reach to value 1. Here it is proven that the possibility of looping is zero except one case.
Category: Number Theory

Merging the Goldbach and the Bunyakovsky Conjecture into a Unified Prime Axiom of Second-Order Logic and Investigating Much Beyond the Goldbach's Conjecture and the Prime Number Theorem

Authors: Alexis Zaganidis
Comments: 31 Pages.

Merging the Goldbach and the Bunyakovsky conjecture into a Unified Prime Axiom of second-order logic and investigating much beyond the Goldbach's conjecture and the prime number theorem.
Category: Number Theory

Kochanski's Approximation of pi

Authors: Edgar Valdebenito
Comments: 2 Pages.

The problem of the exact rectification of a circle cannot by solved by classical geometry. Many approximate methods have been developed. Such an elegant one is Kochanski's construction.
Category: Number Theory

Division by Zero is Incoherent and Contradictory

Authors: Paul Ernest
Comments: 3 Pages.

A number of authors have claimed that Division by Zero and in particular the Division of Zero by Zero (0/0) can be computed and has a definite value (Mwang 2018, Saitoh & Saitoh 2024). I refute these claims. This is trivial, but despite its elementary standing, some peripheral or recreational mathematicians make claims about 0/0 or k/0 having some value, or in some cases, several values in different contexts, according to the author’s whim. Division by zero is undefined and attempts to define it lead to contradiction.
Category: Number Theory

The Collatz Conjecture, Pythagorean Triples, and the Riemann Hypothesis: Unveiling a Novel Connection Through Dropping Times

Authors: Darcy Thomas
Comments: 15 Pages.

In the landscape of mathematical inquiry, where the ancient and the modern intertwine, few problems captivate the imagination as profoundly as the Collatz conjecture and the quest for Pythagorean triples. The former, a puzzle that has defied solution since its inception in the 1930s by Lothar Collatz, asks us to consider a simple iterative process: for any positive integer, if it is even, divide it by two; if it is odd, triple it and add one. Despite its apparent simplicity, the conjecture leads us into a labyrinth of diverse complexity, where patterns emerge and dissolve in an unpredictable dance. On the other hand, Pythagorean triples, sets of three integers that satisfy the ancient Pythagorean theorem, have been a cornerstone of geometry since the time of the ancient Greeks, embodying the harmony of numbers and the elegance of spatial relationships. This exploratory paper embarks on an unprecedented journey to bridge these seemingly disparatedomains of mathematics. At the heart of this exploration is the discovery of a novel connection between Collatz dropping times and Pythagorean triples. I will demonstrate how the dropping time of each odd number can be uniquely associated with a Pythagorean triple. As you will see, the triples seem to be encoding spatial information about Collatz trajectories. As we begin to work with triples, we’ll be motivated to move from the number line to the complex plane where we find structure andbehavior resembling that of the Riemann Zeta function and it’s zeros.
Category: Number Theory

The Symmetry of D2n+2n 、D2n×2n 、D1/2×1/2、D∞+i and Numbers Conjectures

Authors: Yajun Liu
Comments: 12 Pages. (Auther name re-ordered by viXra Admin - Future non-compliant submission/repalcement will not be accepted)

In this paper, we discuss the symmetry of D2n+2n 、D2n×2n 、D1/2×1/2、D∞+i and we find that using the symmetry characters of Natural Numbers we can give proofs of the Prime Conjectures: Goldbach Conjecture、Polignac’s conjecture (Twins Prime Conjecture) and Riemann Hypothesis. . We also gave a concise proofs of Collatz Conjecture in this paper. And we found that if the Goldbach Conjecture、Polignac’s conjecture (Twins Prime Conjecture) were proofed, we also can get a concise proof of Fermat Last Theorem and get an Unified Field Theory for physic.
Category: Number Theory

Syracuse Conjecture

Authors: Mostafa Senhaji
Comments: 7 Pages. In French

IL s'agit d'une séquence très simple d'opérations sur les nombres qui ramène toujours au même endroit, le nombre 1. D'abord un amusement, cette étonnante suite est devenue troublante pour les mathématiciens qui ne se lassent pas de l'explorer sans avoir encore réussi à la domestique.

This is a very simple sequence of number operations that always returns to the same place, the number 1. At first an amusement, this astonishing sequence has become disturbing for mathematicians who never tire of exploring it without having yet succeeded in domesticating it.
Category: Number Theory

A Machine Learning Guided Proof of Beal's Conjecture

Authors: Jonathan Wilson
Comments: 15 Pages.

This paper presents a proof of Beal's conjecture, a long-standing open problem in number theory, guided by insights from machine learning. The proof leverages a novel combination of techniques from modular arithmetic, prime factorization, and the theory of Diophantine equations. Key lemmas, including an expanded version of a modular constraint and a pairwise coprimality condition, are derived with the help of patterns discovered through computational experiments. These lemmas, together with a refined conjecture based on the distribution of prime factors in the dataset, are used to derive a contradiction, proving that any solution to Beal's equation must have a common prime factor among its bases. The proof demonstrates the potential of machine learning in guiding the discovery of mathematical proofs and opens up new avenues for research at the intersection of artificial intelligence and number theory.
Category: Number Theory

Geometric Interpretations of Riemann Hypothesis and the Proof

Authors: Tae Beom Lee
Comments: 5 Pages.

The Riemann zeta function(RZF), ζ(s), is a function of a complex variable s=x+iy. Riemann hypothesis(RH) states that all the non-trivial zeros of RZF lie on the critical line, x=1/2. The symmetricity of RZF zeros implies that if ζ(α+ iβ)=0, then ζ(1-α+ iβ)=0, too. In geometric view, if RH is false, two trajectories ζ(α+ iy) and ζ(1-α+ iy) must intersect at the origin when y=β. But, according to the functional equations of RZF, two trajectories ζ(α+ iy) and ζ(1-α+ iy) can’t intersect except when α=1/2. So, they can’t intersect at the origin, too, proving RH is true.
Category: Number Theory

Set Theory Can’t be Directly Representative of Algebraic Constructions in Goldbach Conjecture and Other NT Problems

Authors: Juan Elias Millas Vera
Comments: 3 Pages.

In this paper I want to express my thoughts on the non possible link between set theory arguments in Number Theory. It is maybe good a first approximation to a problem to think in relation to sets, but my actual thinking is that you can’t solve an algebraic construction with a direct implication between the set and the algebraic variable. As example I will analyze my past trying to prove strong Goldbach conjecture. Finally I explain other version of this topic also with Goldbach conjectures as examples.
Category: Number Theory