Number Theory

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Recent submissions

Any replacements are listed farther down

[3162] viXra:2403.0084 [pdf] submitted on 2024-03-19 02:49:58

On Almost Trivial Zeros of the Function ζ

Authors: Berkouk Mohamed
Comments: 14 Pages. In French (Correction made by viXra Admin - Future nonconforming submission or replacement will not be accepted!)

I don't know to what extent the demonstration of the Riemann hypothesis could help to find an explicit and non-recurring formula, which generates the prime numbers, I don't think because the prime numbers were designed so that we do not discover their secrets, they are too clever to divulge it to the mind.We will see how from Euler's formula we constructed numbers of the form ZPT = 0.000(k times zeros)000c1c2c3u2026cm, depending on the order of the imaginary parts of the non-trivial numbers Zo such that Zo =1 /2+bi and ζ(1/2+bi)=0, that the more we advance in the successive order of b, the more we come across a ZPT almost equal to zero; ζ(ZPT) ≈ 0
Category: Number Theory

[3161] viXra:2403.0082 [pdf] submitted on 2024-03-18 22:37:29

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

[3160] viXra:2403.0079 [pdf] submitted on 2024-03-18 00:10:12

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

[3159] viXra:2403.0077 [pdf] submitted on 2024-03-18 00:22:05

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

Authors: Darcy Thomas
Comments: 14 Pages. (Note by viXra Admin: Please cite and list scientific references in the future))

In the landscape of mathematical inquiry, where the ancient and the modern intertwine, few problemscaptivate 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 ofdiverse 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 connectionbetween Collatz dropping times and Pythagorean triples. I will demonstrate how the dropping timeof each odd number can be uniquely associated with a Pythagorean triple. As you will see, the triplesseem 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 and behavior resembling that of the Riemann Zeta function and it’s zeros.
Category: Number Theory

[3158] viXra:2403.0071 [pdf] submitted on 2024-03-15 19:05:49

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

[3157] viXra:2403.0065 [pdf] submitted on 2024-03-16 03:03:22

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

[3156] viXra:2403.0042 [pdf] submitted on 2024-03-10 20:53:03

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

[3155] viXra:2403.0029 [pdf] submitted on 2024-03-07 07:52:34

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

[3154] viXra:2403.0024 [pdf] submitted on 2024-03-06 02:39:57

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

[3153] viXra:2402.0163 [pdf] submitted on 2024-02-29 23:10:30

Isolating the Prime Numbers

Authors: Adrian M. Stokes
Comments: 4 Pages.

Prime numbers greater than 3 belong to the number sequences 6n ± 1 wheren ≥ 1. These sequences also include the composites that are not divisible by 2 and/or3 and therefore their factors must also be of the form 6n ± 1. This allows all of the6n±1 composites to be equivalently written in the form of factors (6n1±1)(6n2±1),where n1 and n2 ≥ 1, creating three sub-sequences that exclude prime numbers.Finding and isolating the prime numbers can be achieved by selecting a numberrange and creating a set of 6n ± 1 numbers for that range before subtracting thesubsets (6n1 ± 1)(6n2 ± 1) to isolate and identify all the primes in the set.
Category: Number Theory

[3152] viXra:2402.0157 [pdf] submitted on 2024-02-28 22:07:11

The Zeta Function and the Euler-Maclaurin Formula

Authors: Marco Burgos
Comments: 15 Pages. (Note by viXra Admin: Author name is required on the article in pdf)

The Riemann Zeta function is very famous because hidden within it lies the much-desired prime counting function. In this paper, we will unlock the door using the Euler-Maclaurin formula and present the proof of the Riemann Hypothesis.
Category: Number Theory

[3151] viXra:2402.0115 [pdf] submitted on 2024-02-21 20:37:00

The First Counterexample of Riemann Hypothesis Found Through Computer Calculation

Authors: Zhiyang Zhang
Comments: 7 Pages.

The counterexample of the Riemann hypothesis causes a significant change in the image of the Riemann Zeta function, which can be distinguished using mathematical judgment equations. The first counterexample can be found through this equation.
Category: Number Theory

[3150] viXra:2402.0113 [pdf] submitted on 2024-02-21 18:17:43

Incomplete Gamma Function and pi

Authors: Edgar Valdebenito
Comments: 3 Pages.

In this note we give three double series for Pi.
Category: Number Theory

[3149] viXra:2402.0111 [pdf] submitted on 2024-02-20 21:25:04

Collatz Conjecture [Being] Truly a Definite Conclusion is Drawn by Using the Principle of Net Induction Rate and Net Reduction Rate

Authors: Chandan Chattopadhyay
Comments: 10 Pages.

This research work establishes a theory for concluding an affirmative answer to the famous, long-standing unresolved problem "TheCollatz Conjecture".
Category: Number Theory

[3148] viXra:2402.0110 [pdf] submitted on 2024-02-20 14:03:36

Further Investigations on Euler's Odd Perfect Numbers

Authors: Chandan Chattopadhyay
Comments: 11 Pages.

It is a long-standing question whether there exists an odd perfect number. This article establishes a complete theory in order to prove that if an oddperfect number n exists then n = pm^2 with p prime and p is congruent to 1 (mod 4), andgcd (p, m) = 1.
Category: Number Theory

[3147] viXra:2402.0109 [pdf] submitted on 2024-02-20 21:22:36

Diophantine Nth-tuples

Authors: Claude Michael Cassano
Comments: 2 Pages. (Note by viXra Admin: Please use complete sentences in the abstract)

Diophantine equations of sums of terms of various degrees [are explored.]
Category: Number Theory

[3146] viXra:2402.0097 [pdf] submitted on 2024-02-18 20:09:49

Riemann Hypothesis is [Claimed to Be] Proven

Authors: Dmitri Martila
Comments: 2 Pages. (Note by viXra Admin: Please use complete sentences and do not use grandiose title - Future non-compliant submission will be rejected))

A short research about the Riemann Hypothesis [is given].
Category: Number Theory

[3145] viXra:2402.0094 [pdf] submitted on 2024-02-18 20:04:47

Euler-Mascheroni Constant is Irrrational

Authors: Dmitri Martila
Comments: 2 Pages.

[The attempted proof that the Euler-Mascheroni constant is irrrational is given.]
Category: Number Theory

[3144] viXra:2402.0092 [pdf] submitted on 2024-02-18 11:11:18

Proof of Riemann Hypothesis via Robin's Theorem

Authors: Dmitri Martila
Comments: 3 Pages.

I show that the minimum of the function F=e^gamma*ln(ln n)-sigma(n)/n isfound to be positive. Therefore, F>0 holds for any n>5040.
Category: Number Theory

[3143] viXra:2402.0091 [pdf] submitted on 2024-02-18 11:13:59

The Signature of Abc Conjecture is Proven

Authors: Dmitri Martila
Comments: 4 Pages.

Equivalent view of abc conjecture is proven. Some crucial properties of the abc conjecture are presented and proven. For example, there exist three numbers (a, b, c) that satisfy the abc conjecture for an arbitrary value c.
Category: Number Theory

[3142] viXra:2402.0090 [pdf] submitted on 2024-02-18 12:04:49

An Attempt to Prove the Riemann Hypothesis Simply

Authors: Dmitri Martila
Comments: 1 Page.

This work says that Riemann Hypothesis is true.
Category: Number Theory

[3141] viXra:2402.0089 [pdf] submitted on 2024-02-18 12:07:06

Proof of Some Conjectures and Riemann Hypothesis

Authors: Dmitri Martila
Comments: 5 Pages.

A simple proof confirms Riemann, Generalized Riemann, Collatz, Swinnerton-Dyer conjectures and Fermat's Last Theorem.
Category: Number Theory

[3140] viXra:2402.0087 [pdf] submitted on 2024-02-18 19:16:34

Proof of Strong Golbach Conjecture

Authors: Dmitri Martila
Comments: 2 Pages. (Note by viXra Admin: Please use complete sentences in the abstract)

Proof of Strong Golbach Conjecture [is explored in this article].
Category: Number Theory

[3139] viXra:2402.0081 [pdf] submitted on 2024-02-17 22:17:47

A New Attempt to Check Whether Ramanujan's Formula Pi^4 Approx 97.5-1/11 is a Part of Some Completely Accurate Formula

Authors: Janko Kokošar
Comments: 7 Pages.

Intuitively, it seems that Ramanujan's formula $pi^4approx 97.5-1/11$ is an approximation for some perfectly accurate formula for $pi$. Here is one attempt to prove this. The principle of proof, however, is based on closeness of the every rest term to the inverse of integers. Although it is indeed somewhat closer to integers than it is on average, this proof is not complete. So we cannot say for sure whether this proves or disproves that this Ramanujan's formula has higher approximations; however, it gives hints and opens up space for further research.Moreover, this attempted proof is quite original. Also, such a method could also help in physics.
Category: Number Theory

[3138] viXra:2402.0070 [pdf] submitted on 2024-02-14 16:46:34

A Curious Family of Integrals that Give pi

Authors: Edgar Valdebenito
Comments: 2 Pages.

In this note we give a set of integrals for Pi.
Category: Number Theory

[3137] viXra:2402.0058 [pdf] submitted on 2024-02-12 23:03:16

Validating Collatz Conjecture Through Binary Representation and Probabilistic Path Analysis

Authors: Budee U. Zaman
Comments: 15 Pages.

The Collatz conjecture, a longstanding mathematical puzzle, posits that, regardless of the starting integer, iteratively applying a specific formula will eventually lead to the value 1. This paper introduces a novelapproach to validate the Collatz conjecture by leveraging the binary representation of generated numbers. Each transition in the sequence is predetermined using the Collatz conjecture formula, yet the path of transitionsis revealed to be intricate, involving alternating increases and decreases for each initial value. The study delves into the global flow of the sequence, investigating thebehavior of the generated numbers as they progress toward the termination value of 1. The analysis utilizes the concept of probability to shed light on the complex dynamics of the Collatz conjecture. By incorporatingprobabilistic methods, this research aims to unravel the underlying patterns and tendencies that govern the convergence of the sequence.The findings contribute to a deeper understanding of the Collatz conjecture,offering insights into the inherent complexities of its trajectories. This work not only validates the conjecture through binary representation but also provides a probabilistic framework to elucidate the global flow ofthe sequence, enriching our comprehension of this enduring mathematical mystery.
Category: Number Theory

[3136] viXra:2402.0032 [pdf] submitted on 2024-02-06 20:57:56

Derivation/Correction of Hardy-Littlewood Twin Prime Constant using Prime Generator Theory (PGT)

Authors: Jabari Zakiya
Comments: 7 Pages.

The Hardy-Littlewood twin prime constant is a metric to compute the distribution of twin primes. Using Prime Generator Theory (PGT), it is shown it is more easily mathematically and conceptually derived, and the correct value is a factor of 2 larger.
Category: Number Theory

[3135] viXra:2402.0024 [pdf] submitted on 2024-02-05 22:47:02

Adaptive Polynomial Factorization (APF) Method: Enhanced Factorization using Modified Pollard Rho Algorithm

Authors: Anil Sharma
Comments: 3 Pages. (Note by viXra Admin: Please list scientific references in future submissions)

This research paper introduces the Adaptive Polynomial Factorization (APF) Method, an enhanced factorization technique based on the Modified Pollard Rho Algorithm. The method incorporates adaptive polynomial evaluation, providing efficiency in factorization tasks. The paper presents a mathematical representation, performance analysis, and examples showcasing the APF Method’s versatility and superiority over the original Pollard Rho algorithm.
Category: Number Theory

[3134] viXra:2402.0023 [pdf] submitted on 2024-02-05 22:44:40

The Riemann Hypothesis Assumes that the First Counterexample is Located Near S=0.383+(1.578 * 10 ^ 16) I

Authors: Zhiyang Zhang
Comments: 10 Pages.

We already know the distribution of non trivial zeros in the Riemann hypothesis, and there is a formula for calculating counterexamples. The first counterexample can be obtained using a computer, and its value is s=0.383+15786867949799975i
Category: Number Theory

[3133] viXra:2402.0022 [pdf] submitted on 2024-02-05 22:46:45

On the Generation of Odd Prime Numbers Through a Modified Composite Expression

Authors: Anil Sharma
Comments: 2 Pages. (Note by viXra Admin: Please list scientific references in future submissions)

This research paper investigates a distinctive mathematical expression involving natural numbers, unveiling its remarkable property of generating odd prime numbers. The expression, given by N+1 / N × (N! mod PN k=1 k) for natural positive integers N ranging from 2 to infinity, serves as the focal point of our exploration. The paper formulates a formal conjecture, provides a comprehensive proof, and elucidates the claim through stepwise examples.
Category: Number Theory

[3132] viXra:2402.0020 [pdf] submitted on 2024-02-05 22:20:54

Beyond Fibonacci: the Sequence of Integer Powers of Numbers

Authors: Jean-Philippe Vassan
Comments: 15 Pages. In French

Neighboring triangles of Pascal's triangle, cousin numbers of the golden ratio, a simplified formula giving the numbers of generalized Fibonacci sequences, associated generating function, chaos theory and tent function equation.

Triangles voisins du triangle de Pascal, nombres cousins du nombre d'or, une formule simplifiée donnant les nombres des suites de Fibonacci généralisées, fonction génératrice associée, théorie du chaos et équation de la fonction tente.
Category: Number Theory

[3131] viXra:2402.0016 [pdf] submitted on 2024-02-04 22:36:44

The Symmetry of S∞+i and Number Conjectures

Authors: Yajun Liu
Comments: 4 Pages. (Author name reversed by viXra Admin - Future non-compliant submission will not be accepted!)

In this paper, we discuss the symmetry of S∞+i and we find that using the symmetry characters of S∞+i , we can give proofs of the Hodge Conjecture and the Prime Conjectures: Goldbach Conjecture、Polignac’s conjecture and Twins Prime Conjecture. And we also give a proof of Collatz conjecture.
Category: Number Theory

[3130] viXra:2402.0010 [pdf] submitted on 2024-02-03 22:20:19

Function for Prime Numbers

Authors: Massimo Russo
Comments: 66 Pages. In Italian (Correction made by viXra Admin to conform with the requirements of viXra.org)

The Function 5*(1+1/x) + 1 for every value of x determined by Sequence A x = (5^2)+5*2*(n(n+1)/2) where n ≥ 0 determines an infinite series of fractional numbers N/d: 5*(1+1/x) + 1 = N/d such that N and d are prime. numbers.
Category: Number Theory

[3129] viXra:2402.0003 [pdf] submitted on 2024-02-01 23:02:33

Analyzing the Connection from (H(z)) to the Riemann Zeta Function

Authors: Oussama Basta
Comments: 6 Pages.

This paper explores the intriguing connection between the function (H(z) = ln(|sec(pi z/log(z))|)) and the Riemann Zeta Function (zeta(s)). The journey begins by investigating the zeros of (H(z)) and employing advanced mathematical tools such as the Taylor series expansion, the argument principle, and the inverse Mellin transform. Through this exploration, we establish a relationship that leads to a complex integral representation connecting (H(z)) to the Riemann Zeta Function (zeta(s)).
Category: Number Theory

[3128] viXra:2402.0002 [pdf] submitted on 2024-02-01 23:01:56

A Mathematical Criterion for the Validity of the Riemann Hypothesis

Authors: Zhiyang Zhang
Comments: 7 Pages.

We already know in what situations there will be counterexamples for the Riemann hypothesis, but simply increasing Im (s) to find counterexamples for the Riemann hypothesis is still very slow. If there is only a counterexample when Im (s)=10 ^ 1000, or even 10 ^ 10000, then the performance requirements for the computer are very demanding. So, we must create a numerical order determinant to determine whether the Riemann hypothesis holds.
Category: Number Theory

[3127] viXra:2401.0153 [pdf] submitted on 2024-01-31 08:25:27

Pythagorean Triples and Fermat's Theorem N = 4

Authors: Rolando Zucchini
Comments: 7 Pages.

This article contains a theorem to build the Primitive Pythagorean triples and the proof of the last Fermat’s Theorem for n = 4.
Category: Number Theory

[3126] viXra:2401.0132 [pdf] submitted on 2024-01-26 18:00:12

Periodic Function Approach to Prime Number Analysis with Graphical Illustrations

Authors: Budee U. Zaman
Comments: 15 Pages.

This paper introduces a novel approach employing periodic functions for the comprehensive analysis of prime numbers. The method ncompassesprimality testing, factor counting and listing, prime distribution calculation, and the determination of the Nth prime. The exposition of the technique is presented in a clear and sequential manner, guiding the reader through each step with explicit equations. Graphs are strategically incorporated between crucial stages to facilitate a rapid and intuitive visualization of the rationale and outcomes of each maneuver. The paper concludes with concise reflections and ongoing inquiries into the potential applications and refinements of the proposed method.
Category: Number Theory

[3125] viXra:2401.0122 [pdf] submitted on 2024-01-24 20:12:56

Rsa 2048 Factoring NPQ

Authors: Ricardo Gil
Comments: 5 Pages.

The purpose of this paper is to provide an algorithm that has 5 lines of code and that finds P & Q when N is given. It will work for RSA 2048 & RSA-617.
Category: Number Theory

[3124] viXra:2401.0104 [pdf] submitted on 2024-01-21 22:03:34

The Riemann Hypothesis Has no Counterexamples When Imaginary Part Below One Million Billion

Authors: Zhiyang Zhang
Comments: 2 Pages.

This article aims to identify counterexamples of the Riemann hypothesis. Although no upper bound was found for the counterexample, at least it was proven that there were no counterexamples when imaginary part below one million billion, significantly increasing the lower bound of the counterexample.
Category: Number Theory

[3123] viXra:2401.0091 [pdf] submitted on 2024-01-21 01:01:32

Continued Fraction Generalization [Part 3]

Authors: Isaac Mor
Comments: 18 Pages.

This is a list of ten types of continued fraction generalization. (This is [Part 3] , every volume contains 10 formulas) I am using Euler's continued fraction formula in order to find some nice continued fraction generalization.
Category: Number Theory

[3122] viXra:2401.0087 [pdf] submitted on 2024-01-20 00:56:23

A Simple Proof of The ABC Conjecture

Authors: Oussama Basta
Comments: 3 Pages.

This work analyzes the ABC conjecture, which states that for any positive real number ε, there exists a constant Kε such that for all coprime positive integer triples (a, b, c) with a + b = c, c < Kε * rad(abc)^(1 + ε). We focus on the case where a > F, F > ε, and b = (a + F - ε), c = (a + F + ε), where F and ε are positive real numbers with F > ε.
Category: Number Theory

[3121] viXra:2401.0084 [pdf] submitted on 2024-01-18 01:20:00

Riemann Hypothesis: Direct Demonstration Proposal

Authors: Vincent Koch
Comments: 3 Pages.

In his 1859 article "On the number of prime numbers less than a given quantity", Bernhard Riemann formulated the hypothesis that all non-trivial zeros of the Zeta function have the real part 1/2.This assertion, known as the "Riemann Hypothesis", remains unproven to this day. The present paper is an attempt at a direct demonstration.
Category: Number Theory

[3120] viXra:2401.0081 [pdf] submitted on 2024-01-16 23:55:05

Hypothesis: Distribution of Primes and the Logarithmic Expression

Authors: Anil Sharma
Comments: 4 Pages.

This research explores the distribution of prime numbers using a novel ljogarithmic expression. The hypothesis suggests that an expression, partitions natural numbers into groups, revealing a systematic distribution of primes. Experimental results demonstrate an intriguing pattern as the range of N increases, with the average number of primes in each group stabilizing around 15. The paper discusses thebackground, mathematical expression, experimental results, and potentialavenues for future research.
Category: Number Theory

[3119] viXra:2401.0072 [pdf] submitted on 2024-01-16 01:10:42

Revolutionizing Prime Factorization: A Time Complexity-Optimized Approach for Efficient Composite Number Analysis

Authors: Anil Sharma
Comments: 2 Pages.

This research investigates patterns in prime number distributions and proposes an optimized factorization method. A novel approach is introduced to explore the position of the first prime factor in composite numbers, focusing on a specific range for potential computational time savings.
Category: Number Theory

[3118] viXra:2401.0064 [pdf] submitted on 2024-01-13 21:07:52

An Efficient Method to Prove that the Riemann Hypothesis Is Not Valid

Authors: Zhiyang Zhang
Comments: 12 Pages. (Name added to article by viXra Admin - Please conform!)

Analytical number theory is a combination of trigonometric functions and polynomial symbols, which can be solved no matter how difficult it is. Therefore, I believe that the Riemann hypothesis is not unsolvable. In the field of number theory, the mathematical community tends to seek a maximum number to overturn the conclusion. Whether it is the Riemann hypothesis or the Goldbach conjecture, this should be the solution.
Category: Number Theory

[3117] viXra:2401.0054 [pdf] submitted on 2024-01-13 04:07:31

On the Sum of Reciprocals of Primes

Authors: Young Deuk Kim
Comments: 5 Pages.

Suppose that $y>0$, $0leqalpha<2pi$ and $0K$ and $P^-$ the set of primes $p$ such that $cos(yln p+alpha)<-K$ . In this paper we prove $sum_{pin P^+}frac{1}{p}=infty$ and $sum_{pin P^-}frac{1}{p}=infty$.
Category: Number Theory

[3116] viXra:2401.0046 [pdf] submitted on 2024-01-09 21:34:50

Redefining Mathematical Structure: From the Real Number Non-Field to the Energy Number Field

Authors: Parker Emmerson
Comments: 8 Pages.

The traditional classification of real numbers (R) as a complete ordered field is contested throughcritical examination of the field axioms, with a focus on the absence of a multiplicative inverse for zero. We propose an alternative mathematical structure based on Energy Numbers (E), deriving from quantum mechanics, which addresses the classical anomalies and fulfills field properties universally, including an element structurally analogous but functionally distinctive from the zero in R.
Category: Number Theory

[3115] viXra:2401.0018 [pdf] submitted on 2024-01-04 20:18:27

Charles Hutton's Formula

Authors: Edgar Valdebenito
Comments: 3 Pages.

Some remarks about a formula of Charles Hutton.
Category: Number Theory

[3114] viXra:2401.0009 [pdf] submitted on 2024-01-02 04:44:33

Contribution to Goldbach's Conjectures

Authors: Radomir Majkic
Comments: 6 Pages.

The internal structure of the natural numbers reveals the relation between the weak and strong Goldbach's conjectures. Explicitly, if the weak Goldbach's conjecture is true, the strong Goldbach's conjecture is, and Goldbach's conjectures are true.
Category: Number Theory

[3113] viXra:2401.0008 [pdf] submitted on 2024-01-02 05:10:07

Goldbach's Number Construction

Authors: Radomir Majkic
Comments: 6 Pages.

Goldbach’s numbers, all-natural integers which satisfy Goldbach’s conjectures are all odd integers and a subset of the even integers. Naturally, they appear in the proof of Goldbach’s conjectures. In this paper, the construction of Goldbach’s numbers approach is used to prove Goldbach’s conjectures, hopefully, it will bring a happy end.
Category: Number Theory

[3112] viXra:2312.0164 [pdf] submitted on 2023-12-30 16:19:53

Introduction to the Intersection_p Operator

Authors: C. Pokorski
Comments: 13 Pages. in French

This article introduces a new mathematical operator, Intersection_p, designed to establish connections between numbers based on their prime factors, which can also be seen as an extension of GCD.
Category: Number Theory

[3111] viXra:2312.0163 [pdf] submitted on 2023-12-30 22:15:38

Multivariate Circle of Partitions and the Squeeze Principle

Authors: Theophilus Agama
Comments: 5 Pages.

The goal of this paper is to extend the squeeze principle to circle of partitions with at least two resident points on their axes.
Category: Number Theory

[3110] viXra:2312.0158 [pdf] submitted on 2023-12-29 22:42:36

Introduction to the Union_p Operator

Authors: C. Pokorski
Comments: 15 Pages. In French

This article introduces a new mathematical operator, the Union_p, designed to explore relationships between numbers through their prime factors, which can also be seen as an extension of the LCM.
Category: Number Theory

[3109] viXra:2312.0157 [pdf] submitted on 2023-12-29 13:48:28

Investigation on Brocard-Ramanujan Problem

Authors: Akash Shivaji Pawar
Comments: 7 Pages. 6 Lemma,2 Theorem

Exploring n! + 1 = m^2 for natural number solutions beyond n = 4, 5, 7 confirms no further solutions exist,validated by using GCD Linear Combination Theorem
Category: Number Theory

[3108] viXra:2312.0143 [pdf] submitted on 2023-12-27 02:32:49

On the Nonexistence of Solutions to a Diophantine Equation Involving Prime Powers

Authors: Budee U. Zaman
Comments: 7 Pages.

This paper investigates the Diophantine equation pr + (p + 1)s = z2 Where p > 3, s ≥ 3 , z is an even integer. The focus of the study is to establish rigorous results concerning the existence of solutions within this specific parameter space. The main result presented in this paper demonstrates the absence of solutions under the stated conditions. The proof employs mathematical techniques to systematically address the case when the prime p exceeds 3, and the exponent s is equal to or greater than2, while requiring the solution to conform to the constraint of an even z. This work contributes to the understanding of the solvability of the given Diophantine equation and provides valuable insights into the interplay between prime powers and the resulting solutions.
Category: Number Theory

[3107] viXra:2312.0140 [pdf] submitted on 2023-12-27 02:26:49

Euler’s Totient Function, Sum of Divisors and Primes

Authors: Rédoane Daoudi
Comments: (Note by viXra Admin: Future stub page/paper will not be accepted!)

Here I present a conjecture about Euler’s totient function, sum of divisors andprimes.
Category: Number Theory

[3106] viXra:2312.0135 [pdf] submitted on 2023-12-25 22:25:51

On the Notion of Carries of Numbers 2^n-1 and Scholz Conjecture

Authors: Theophilus Agama
Comments: 16 Pages.

Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-1$ has carries of degree at most $$kappa(2^n-1)=frac{1}{2(1+c)}lfloor frac{log n}{log 2}floor-1$$ for $c>0$ fixed, then the inequality $$iota(2^n-1)leq n-1+(1+frac{1}{1+c})lfloorfrac{log n}{log 2}floor$$ holds for all $nin mathbb{N}$ with $ngeq 4$, where $iota(cdot)$ denotes the length of the shortest addition chain producing $cdot$. In general, we show that all numbers of the form $2^n-1$ with carries of degree $$kappa(2^n-1):=(frac{1}{1+f(n)})lfloor frac{log n}{log 2}floor-1$$ with $f(n)=o(log n)$ and $f(n)longrightarrow infty$ as $nlongrightarrow infty$ for $ngeq 4$ then the inequality $$iota(2^n-1)leq n-1+(1+frac{2}{1+f(n)})lfloorfrac{log n}{log 2}floor$$ holds.
Category: Number Theory

[3105] viXra:2312.0134 [pdf] submitted on 2023-12-25 22:02:28

A Proof of the Wen-Yao Conjecture

Authors: David Adam
Comments: 24 Pages. In French

In this article, we characterize monomials in de facto values.Carlitz-Goss rielle defined on the complement of Fq (T) in a finite place which arealgebraic on Fq (T ). In particular, this confirms Wen-Yao's conjecturestated in 2003. This gives a necessary and sufficient condition on an en-p-adic tier so that the value of the Carlitz-Goss factorial in it is algebraic on Fq (T ). When restricted to rational arguments, we determinenot all algebraic relations between the values u200bu200btaken by this function, this which gives the counterpart for finite places of a result of Chang, Papanikolas, Thakur and Yu obtained in the case of infinite place.

Dans cette article, nous caractérisons les monômes en les valeurs de la facto-rielle de Carlitz-Goss définie sur le complété de Fq (T ) en une place finie qui sont algébriques sur Fq (T ). En particulier, cela confirme la conjecture de Wen-Yaoénoncée en 2003 . Celle-ci donne une condition necessaire et suffisante sur un en-tier p-adique pour que la valeur de la factorielle de Carlitz-Goss en celui-ci soit algébrique sur Fq (T ). Lorsque restreint aux arguments rationnels, nous détermi-nons toutes les relations algébriques entre les valeurs prises par cette fonction, cequi donne le pendant pour les places finies d’un résultat de Chang, Papanikolas, Thakur et Yu obtenu dans le cas de la place infinie.
Category: Number Theory

[3104] viXra:2312.0108 [pdf] submitted on 2023-12-20 06:32:37

Complete Operations

Authors: Pith Peishu Xie
Comments: 26 Pages.

The Operator axioms have produced complete operations with real operators. Numerical computations have been constructed for complete operations. The classic calculator could only execute 7 operator operations: + operator operation(addition), - operator operation(subtraction), $times$ operator operation(multiplication), $div$ operator operation(division), ^{} operator operation(exponentiation), $surd$ operator operation(root extraction), log operator operation(logarithm). In this paper, we invent a complete calculator as a software calculator to execute complete operations. The experiments on the complete calculator could directly prove such a corollary: Operator axioms are consistent.
Category: Number Theory

[3103] viXra:2312.0099 [pdf] submitted on 2023-12-19 23:00:40

A Proof of a Result of James Stirling

Authors: Hervé Gandran-Tomeng
Comments: 2 Pages.

A recent paper contains a proof of a result of James Stirling,$sum_{n=1}^inftyfrac{1}{n^2binom{2n}{n}}=frac{1}{3}sum_{n=1}^infty frac{1}{n^2}$What is following is another proof of this equality.
Category: Number Theory

[3102] viXra:2312.0081 [pdf] submitted on 2023-12-15 17:48:12

Rediscovering Ramanujan

Authors: Edgar Valdebenito
Comments: 14 Pages.

In this note, we revisit Ramanujan-type series for 1/pi .
Category: Number Theory

[3101] viXra:2312.0076 [pdf] submitted on 2023-12-15 01:17:12

Collatz Conjecture: a Countably Infinite Sequence

Authors: Vishesh Mangla
Comments: 2 Pages. CC BY-SA (Note by viXra admin: The article needs an abstract and scientific references are required

In this paper, I have explored the Collatz conjecture and presented a new result regarding the behavior of the sequence. The proof demonstrated that for natural numbers n subjected to the Collatz Algorithm, the sequence can potentially havea countably infinite number of terms.
Category: Number Theory

[3100] viXra:2312.0054 [pdf] submitted on 2023-12-10 20:35:22

The Twin Prime Conjecture - An Analytical Approach

Authors: Patrick DiCarlo
Comments: 37 Pages.

This paper offers a proof of the twin prime conjecture. The basic strategy is to first establish that there is no highest prime number by calculating the rates at which the multiples of each successive prime preclude higher numbers from being prime, and then proving that this rate (in the aggregate) can never reach 100%. The same basic methodology is then used to show that there can also be no highest twin prime.
Category: Number Theory

[3099] viXra:2312.0053 [pdf] submitted on 2023-12-10 22:28:10

One Theorem Complementary to the Fundamental Theorem of Arithmetic

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

In this paper I want to show a complementary theorem of the Fundamental Theorem of Arithmetic. Using the delta notation (Δ) I was able to deduce a generic formula involving prime numbers and natural numbers.
Category: Number Theory

[3098] viXra:2312.0036 [pdf] submitted on 2023-12-07 21:09:02

New Equivalent of the Riemann Hypothesis

Authors: Leonardo de Lima
Comments: 10 Pages.

In this article, it is demonstrated that if the zeta function does not have a sequence of zeros whose real part converges to 1, then it cannot have any zeros in the critical strip, showing that the Riemann Hypothesis is false.
Category: Number Theory

[3097] viXra:2312.0023 [pdf] submitted on 2023-12-05 21:32:41

Cracking the Collatz Code: a Journey from Conjecture to Certainty Through Infinity

Authors: Eric Lough
Comments: 7 Pages.

The Collatz conjecture, a puzzle that has intrigued mathematicians for decades, has met its resolution through a simple combination of trial and error combined with a spreadsheet. While the primary goal was to definitively prove the conjecture, the exploration revealed unexpected insights into the underlying patterns of the Collatz sequence. This study introduces a novel approach to organizing and analyzing Collatz data, laying bare the formulas governing the sequence's behavior. The results present a conclusive proof of the Collatz conjecture and unveil a spectrum of sets and formulas reaching to infinity, each contributing to a deeper understanding of its workings. Comprehensive instructions are provided, encouraging collaboration and furthering the collective understanding of mathematical sequences.
Category: Number Theory

[3096] viXra:2312.0021 [pdf] submitted on 2023-12-04 21:59:36

The Magic of Mirror Composite Numbers

Authors: Emilio Sánchez, Óscar E. Chamizo
Comments: 6 Pages.

In this paper, continuation and completion of some previous researches, we fully develop the new concept of mirror composite numbers. Mirror composite numbers are composite numbers of the form 2n-p for some n natural number and p prime. We shall show that the factorization of these numbers have interesting properties in order to face the Goldbach conjecture [2][3] by the divide et impera method.
Category: Number Theory

[3095] viXra:2312.0016 [pdf] submitted on 2023-12-03 23:47:17

Proof for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 7 Pages.

I am going to use telescoping series and then a proof by induction. I am using Lambert's continued fraction for the base case.
Category: Number Theory

[3094] viXra:2312.0014 [pdf] submitted on 2023-12-03 23:49:08

Elliptic Curve

Authors: Ian Connell
Comments: 327 Pages.

The first version of this handbook was a set of notes of about 100 pages handed out to the class of an introductory course on elliptic curves given in the 1990 fall semester at McGill University in Montreal. Since then I haveadded to the notes, holding to the principle: If I look up a certain topic a year from now I want all the details right at hand, not in an "exercise", so if I’ve forgotten something I won’t waste time. Thus there is much that anordinary text would either condense, or relegate to an exercise. But at the same time I have maintained a solid mathematical style with the thought of sharing the handbook.
Category: Number Theory

[3093] viXra:2312.0005 [pdf] submitted on 2023-12-01 07:01:52

Fermat's Last Theorem for Odd Primes

Authors: Minho Baek
Comments: 26 Pages.

It was already proved right that xn+yn=zn, (n>2) has no solutions in positive integers which we called Fermat’s Last Theorem (FLT) by Andrew Wiles. But his proof would be impossible in the 17th century. Since Fermat showed he proved n=even by leaving proof for n=4, many people have tried to prove odd primes. I took the idea from Euler proof and proved in case of n=odd primes by simple method.
Category: Number Theory

[3092] viXra:2312.0004 [pdf] submitted on 2023-12-01 21:33:10

A Proof of Riemann Hypothesis by Symmetry and Circular Properties of Riemann Zeta Function

Authors: Tae Beom Lee
Comments: 9 Pages.

Riemann zeta function(RZF) z(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, 0.5+iy. The symmetricity of RZF zeros implies that if z(a+ib)=0,0<a<0.5, then z(1-a+ib)=0, too. The graphs of RZF are similar to the graphs of circles with non-uniform radius and argument. These two, symmetry and circular properties of RZF, are the basis of our proof.
Category: Number Theory

[3091] viXra:2311.0152 [pdf] submitted on 2023-11-29 21:52:02

Complete Collatz Directed Graph

Authors: Wiroj Homsup, Nathawut Homsup
Comments: 8 Pages.

The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even, or (3n+1)/2 , if n is odd. The conjecture states that for all starting values n the sequence eventually reaches a trivial cycle 1, 2, 1, 2u2026u2026The Collatz sequences can be represented as a directed graph. If the Collatz conjecture is false, then either there is a nontrivial cycle, or one sequence goes to infinity. In this paper, we construct a Collatz directed graph by connecting infinite number of basic directed graphs. Each basic directed graph relates to each natural number. We prove that the Collatz directed graph covers all positive integers and there is only a trivial cycle and no sequence goes to infinity.
Category: Number Theory

[3090] viXra:2311.0139 [pdf] submitted on 2023-11-27 21:46:55

Solution of a Five Degree Equation

Authors: Ait saadi Ahcene
Comments: 4 Pages. (Correction made by viXra Admin to conform with scholarly norm)

In this article, I solve the general equation of degree 5 of the a particular form. For this I used Mathematics that I invented. The method I invented allows me to solve all equations of degrees greater than 4, as well as equations of another form: The principle, is to find for all these equations, an equation of degree three (3) which has at least one solution in common with those of degree greater than 4. With my method of course this is possible.
Category: Number Theory

[3089] viXra:2311.0137 [pdf] submitted on 2023-11-27 15:31:50

New Bounds on Mertens Function

Authors: Juan Moreno Borrallo
Comments: 7 Pages.

In this brief paper we study and bound Mertens function. The main breakthrough is theobtention of a Möbius-invertible formulation of Mertens function, which with some transformationsand the application of the generalization of Möbius inversion formula, allows us to reach anasymptotic equivalence of the absolute value of Mertens function that proves the Riemann Hypothesis.
Category: Number Theory

[3088] viXra:2311.0126 [pdf] submitted on 2023-11-25 21:34:20

Generalization for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 2 Pages. (Correction made by viXra Admin to conform with scholarly norm - Future non-comforming submission will not be accepted.))

I came across "The Ramanujan Machine" on the Internet and, using my intuition on those kind of stuff, I found some interesting results.
Category: Number Theory

[3087] viXra:2311.0125 [pdf] submitted on 2023-11-24 14:19:12

No Collatz Conjecture Integer Series Have Looping

Authors: Tsuneaki Takahashi
Comments: 3 Pages. Note by viXra Admin: Future repetition/regurgitation will not be accepted.

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

[3086] viXra:2311.0119 [pdf] submitted on 2023-11-24 23:54:05

Zeta Function

Authors: Leonardo de Lima
Comments: 8 Pages.

This article delves into the properties of the Riemann zeta function, providing a demonstration of the existence of a sequence of zeros ${z_k}$, where $lim operatorname{Re}(z_k) = 1$. The exploration of these mathematical phenomena contributes to our understanding of complex analysis and the behavior of the zeta function on the critical line.
Category: Number Theory

[3085] viXra:2311.0118 [pdf] submitted on 2023-11-25 04:56:20

Riemann Hypothesis

Authors: Bertrand Wong
Comments: 14 Pages.

This paper discusses the distribution of the non-trivial zeros of the Riemann zeta function ζ. It looks into the question of whether any non-trivial zeros would ever possibly be found off the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1, e.g., at Re(s) = 1/4, 1/3, 3/4, 4/5, etc., and why all the non-trivial zeros are always found at the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1 and not anywhere else on this critical strip, with the first 1013 non-trivial zeros having been found only at the critical line Re(s) = 1/2. It should be noted that a conjecture, or, hypothesis could possibly be proved by comparing it with a theorem that has been proven, which is one of the several deductions utilized in this paper. Through these several deductions presented, the paper shows how the Riemann hypothesis may be approached to arrive at a solution. In the paper, instead of merely using estimates of integrals and sums (which are imprecise and may therefore be of little or no reliability) in the support of arguments, where feasible actual computations and precise numerical facts are used to support arguments, for precision, for more sharpness in the arguments, and for "checkability" or ascertaining of the conclusions. This paper is the revised and expanded version of a paper [5] published in 2022.
Category: Number Theory

[3084] viXra:2311.0105 [pdf] submitted on 2023-11-22 21:58:46

Proof of the Collatz Conjecture

Authors: Wiroj Homsup, Nathawut Homsup
Comments: 8 Pages. (Note by vXra Admin: Future repetition/regurgitation will not be accepted!)

The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even, or (3n+1)/2 , if n is odd. The conjecture states that for all starting values n the sequence eventually reaches the trivial cycle 1, 2, 1, 2u2026u2026The inverted Collatz sequences can be represented as a tree with 1 as its root node. In order to prove the Collatz conjecture, one must demonstrate that the tree covers all natural numbers. In this paper, we construct a Collatz tree with 1 as its root node by connecting infinite number of basic trees. Each basic tree relates to each natural number. We prove that a Collatz tree is a connected binary tree and covers all natural numbers.
Category: Number Theory

[3083] viXra:2311.0094 [pdf] submitted on 2023-11-20 20:05:24

Zero Times Zero Equals Nonzero

Authors: Michael Graham
Comments: 3 Pages.

The current Multiplication and Division Properties of Zero are flawed and illogical. This paper illustrates why and presents logical solutions that resolve the issue of dividing by zero.
Category: Number Theory

[3082] viXra:2311.0086 [pdf] submitted on 2023-11-19 02:46:21

On the Largest Prime Factor of the K-Generalized Lucas Numbers

Authors: Herbert Batte, Florian Luca
Comments: 14 Pages.

Let $(L_n^{(k)})_{ngeq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $kge 2$ whose first $k$ terms are $0,ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. For an integer $m$, let $P(m)$ denote the largest prime factor of $m$, with $P(0)=P(pm 1)=1$. We show that if $n ge k + 1$, then $P (L_n^{(k)} ) > (1/86) log log n$. Furthermore, we determine all the $k$--generalized Lucas numbers $L_n^{(k)}$ whose largest prime factor is at most $ 7$.
Category: Number Theory

[3081] viXra:2311.0070 [pdf] submitted on 2023-11-12 21:44:27

3n+1 Conjecture: A Proof or Almost

Authors: A. Makarenko
Comments: 4 Pages.

The Collatz algorithm is rewritten to remove divisions by two and to transform it from a hailstone to a steadily growing value. In contrast with the original problem this new sequence becomes reversible and it is reverted in combinatorial way to find all integers leading to the sequence end. Computer programs are available for demonstrations and experimenting.
Category: Number Theory

[3080] viXra:2311.0063 [pdf] submitted on 2023-11-10 23:22:13

A Proof of Fermat’s Last Theorem by Relating to Monic Polynomial Properties

Authors: Tae Beom Lee
Comments: 5 Pages.

Fermat's Last Theorem(FLT) states that there is no natural number set {a,b,c,n} which satisfies a^n+b^n=c^n or a^n=c^n-b^n when n≥3. In this thesis, we related LHS and RHS of a^n=c^n-b^n to the constant terms of two monic polymials x^n-a^n and x^n-(c^n-b^n). By doing so, we could inspect whether these two polynomials can be identical when n≥3, i.e., x^n-a^n=x^n-(c^n-b^n), which satisfies a^n=c^n-b^n. By inspecting the properties of two polynomials such as factoring, root structures and graphs, we found that x^n-a^n and x^n-(c^n-b^n) can’t be identical when n≥3, except when trivial cases.
Category: Number Theory

[3079] viXra:2311.0059 [pdf] submitted on 2023-11-10 23:14:38

Divisible Cyclic Numbers

Authors: Julian Beauchamp
Comments: 5 Pages. (Author name added to the article by viXra Admin - Please conform!)

There are known to exist a number of (multiplicative) cyclic numbers (CNs), but in this paper I introduce what appears to be a new kind of number, which we call divisible cyclic numbers (DCNs) and wonder what properties they may possess. Strangely, I can find no reference to them anywhere. Given that they are simple to understand and quite commonplace, it would be remarkable if they were hitherto unknown to the mathematical world.
Category: Number Theory

[3078] viXra:2311.0056 [pdf] submitted on 2023-11-10 01:13:32

For a Number k, Can (2[k]m)+1 Always be Prime for All Number m?

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

This paper is about hyperoperators. In this paper I ask myself and the mathematical community if there is possible that a k-ation of the number 2 will be always a number prime for any number m if we add the number one to the result.
Category: Number Theory

[3077] viXra:2311.0052 [pdf] submitted on 2023-11-10 01:07:57

On the Incompletely Predictable Problems of Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 72 Pages.

We validly ignore even prime number 2. Based on all arbitrarily large number of even prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with Prime number theorem for Arithmetic Progressions. With this condition being satisfied by all Odd Primes, we argue that Polignac's and Twin prime conjectures are proven to be true with these conjectures treated as Incompletely Predictable Problems. In so doing [and with the famous Riemann hypothesis being a special case], the generalized Riemann hypothesis formulated for Dirichlet L-function is also supported. Riemann hypothesis is separately proven to be true with this hypothesis treated as an Incompletely Predictable Problem.
Category: Number Theory

[3076] viXra:2311.0050 [pdf] submitted on 2023-11-08 21:38:34

Mathematics for Incompletely Predictable Problems Required to Prove Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 69 Pages.

As two different but related infinite-length equations through analytic continuation, Hasse principle is satisfied by Riemann zeta function as a certain type of equation that generates all infinitely-many trivial zeros but this principle is not satisfied by its proxy Dirichlet eta function as a dissimilar type of equation that generates all infinitely-many nontrivial zeros. Based on two seemingly different location that are in fact identical, all nontrivial zeros are mathematically located on critical line or geometrically located on Origin point. Thus we prove location for complete Set nontrivial zeros to be critical line confirming Riemann hypothesis to be true. Sieve of Eratosthenes as a certain type of infinite-length algorithm is exactly constituted by an Arbitrarily Large Number of (self-)similar infinite-length sub-algorithms that are specified by every even Prime gaps. Modified Hasse principle is satisfied by this algorithm and its sub-algorithms that perpetually generate the Arbitrarily Large Number of all Odd Primes. Thus we prove Set even Prime gaps with corresponding Subsets Odd Primes all have cardinality Arbitrarily Large in Number confirming Modified Polignac's and Twin prime conjectures to be true.
Category: Number Theory

[3075] viXra:2311.0049 [pdf] submitted on 2023-11-08 07:36:01

Collatz Conjecture Proved Ingeniously & Very Simply

Authors: A. A. Frempong
Comments: 12 Pages. Copyright © by A. A. Frempong

Collatz conjecture states that beginning with a positive integer, if one repeatedly performs the following operations to form a sequence of integers, the sequence will eventually reach the integer one; the operations being that if the integer is even, divide it by 2, but if the integer is odd, multiply it by 3 and add one; and also, use the result of each step as the input for the next step. One would note the patterns of the sequence terms as the Collatz process reaches the equivalent powers, 2^(2k) (k = 2, 3, . . . ) and continues as 2^(2k-1), 2^(2k-2), 2^(2k-3), . . ., 2^(2k-2k). Two main cases are covered. In Case 1, the integer can be readily written as a power of 2 as 2^(k) (k=1,2,3, . . . ), and the sequence would reach the integer one by following 2^(k-1), 2^(k-2), 2^(k-3,), . . . , 2^(k-k). In Case 2, the integer cannot be readily written as a power of 2, but the sequence terms reach the equivalent power, 2^(2k) (k = 2, 3, . . . ) which will continue as 2^(2k-1), 2^(2k-2), 2^(2k-3), . . . , 2^(2k-2k). In Case 2, when the sequence terms reach some particular integers such as 5, 21 and 85, the application of 3n + 1 to these integers will result in the powers, 2^(2k). One would call these integers, the 2k-power converters. There are infinitely many 2k-power converters as there are 2^(2k) powers. A term of the sequence must be converted to 2^(2k). There are infinitely many paths for converting integers to 2^(2k) powers. Of these paths, the integer 5-path, is the nearest 2^(2k) converter path to the integer 1 on the 2^(2k)-route. For the 5-path, when a sequence terms reach the integer, 5, the next term would be 16. Other integers can follow the integer 5-path to 16 as follows: Let n be an integer whose sequence terms would reach 16, and let n ± r = 5, where r is the net change in the sequence terms before the integer 5; and one uses the positive sign if n < 5, but the negative sign if n > 5. One will call the following, the 5-path 2k-converter template: 3(n ± r) + 1 = 16. By the substitution axiom, using this template, the sequence of every positive integer would reach 16; and applying repeated division by 2, the sequence will reach the integer 1.
Category: Number Theory

[3074] viXra:2311.0047 [pdf] submitted on 2023-11-08 21:32:28

Mirror Composite Numbers: Their Factorization and Their Relationship with Goldbag Conjecture.

Authors: Ángeles Jimeno Yubero, Óscar E. Chamizo Sánchez
Comments: 5 Pages.

Mirror composite numbers are composite numbers of the form 2n-p for some n positive natural number and p prime. We shall show that the factorization of these numbers have interesting properties in order to face the Goldbach conjecture by the divide et impera method.
Category: Number Theory

[3073] viXra:2311.0040 [pdf] submitted on 2023-11-08 04:08:29

A Simple Proof that E^(p/q) is Irrational

Authors: Timothy W. Jones
Comments: 3 Pages.

Using a simple application of the mean value theorem, we show that rational powers of e are irrational.
Category: Number Theory

[3072] viXra:2311.0030 [pdf] submitted on 2023-11-06 14:27:29

Euler's Identity, Leibniz Tables, and the Irrationality of Pi

Authors: Timothy W. Jones
Comments: 6 Pages.

Using techniques that show show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's formula. The proof involves evaluation of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.
Category: Number Theory

[3071] viXra:2311.0026 [pdf] submitted on 2023-11-07 01:27:25

Convergence and Computation of Sum of a Series on the Riemann Zeta Function

Authors: HaeRyong Kim, HyonChol Kim, YongHun Jo
Comments: 13 Pages.

In this paper, we present a new method of evaluating the convergence and sum of a series with the Riemann zeta function in its general term.We consider the convergence and sum of a series by means of difference other than previous methods.
Category: Number Theory

[3070] viXra:2311.0025 [pdf] submitted on 2023-11-07 02:14:45

Optimal FractionalPIβ(t)Dα(t) Controllers and Numerical Simulation for DC Motor Speed Control

Authors: Ji-Song Ro, Myong-Hyok Sin, Yong-Ho Kim, Sung-Il Gang
Comments: 12 Pages.

We model the rotation process of the motor for variable-order fractional control, which has been active in recent research, and perform numerical simulation of its optimal control and automatic control process. In this paper, we verify numerical method and error estimation of variable order fractional linear dynamic system with time-varying coefficients, a variable-order fractional PID controller design method where the integral of the absolute error with time weight is minimized is proposed using particle swarm optimization algorithm and demonstrate its effectiveness through numerical simulation for DC motor speed control. Numerical experiments show that the performance of the VFPID controller is superior to PID and FPID, especially VFPIDB (B-type variable order FPID) controller has the best performance. Finally, when the differential order varies, the subtypes of variable-order fractional derivatives are analyzed for the effects on the control objective, its effectiveness is newly clarified, and their research and practice is highlighted. 
Category: Number Theory

[3069] viXra:2311.0015 [pdf] submitted on 2023-11-03 09:25:52

Reverse Chebyshev Bias in the Distribution of Superprimes

Authors: Waldemar Puszkarz
Comments: 12 pages. Originally posted on ResearchGate in September 2023.

We study the distribution of superprimes, a subsequence of prime numbers with prime indices, mod 4. Rather unexpectedly, this subsequenceexhibits a reverse Chebyshev bias: terms of the form 4k + 1 are more common than those of the form 4k + 3, whereas the opposite is the case in the sequence of all primes. The effect, while initially weak and easy to overlook, tends tobe several times larger than the Chebyshev bias for all primes for samples of comparable size, at least, by one simple measure. By two other measures, it can be seen as fairly strong; by the same measures the ordinary Chebyshev effectis very strong. Both of these measures also imply that the reverse Chebyshev bias for superprimes is more volatile than the ordinary Chebyshev bias.
Category: Number Theory

[3068] viXra:2311.0010 [pdf] submitted on 2023-11-04 00:20:41

On the Binomial and Fermat's Last Theorem

Authors: Carlos Villacres
Comments: 9 Pages.

An approach to the classic problem of Fermat's last theorem. Using the binomial theorem and the cases where n is even or odd, we find a solution as well as a Pythagorean triple generator.
Category: Number Theory

[3067] viXra:2311.0006 [pdf] submitted on 2023-11-03 03:13:33

The Symmetry of N-domain and Numbers Conjuctures

Authors: Yajun Liu
Comments: 11 Pages.

In this paper, we discuss the symmetry of N-domain 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.
Category: Number Theory

[3066] viXra:2311.0003 [pdf] submitted on 2023-11-01 21:26:30

Proving the Goldbach Conjecture

Authors: Jim Rock
Comments: 2 Pages.

In 1742 Christian Goldbach suggested that any even number four or greater is the sum of two primes. The Goldbach Conjecture remains unproven to the present day though it has been verified for all even numbers up to 4 x 1018. This paper suggests an algorithm for checking the Goldbach conjecture for individual even numbers and a generalization that could be used to prove the Goldbach conjecture.
Category: Number Theory

[3065] viXra:2310.0145 [pdf] submitted on 2023-10-30 18:26:40

An Truly Easy Proof: Pi is Irrational

Authors: Timothy W. Jones
Comments: 1 Page.

Using the derivative of an integer polynomial composed with Euler's formula we prove that pi is irrational.
Category: Number Theory

[3064] viXra:2310.0133 [pdf] submitted on 2023-10-28 18:58:39

Guessing that the Riemann Hypothesis Is Unprovable

Authors: T. Nakashima
Comments: 3 Pages.

Riemann Hypothesis has been the unsolved conjecture for 164 years. This conjecture is the last one of conjectures without proof in "Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse"(B.Riemann). The statement is the real part of the non-trivial zero points of the Riemann Zeta function is 1/2. Very famous and difficult this conjecture has not been solved by many mathematicians for many years. In this paper, I guess the independence (unprovability) of the Riemann Hypothesis.
Category: Number Theory

[3063] viXra:2310.0117 [pdf] submitted on 2023-10-24 19:43:17

Proof of Collatz Conjecture Using Division Sequence Ⅳ

Authors: Masashi Furuta
Comments: 8 Pages.

This paper is positioned as a sequel edition of [1]. First, as in [1], define "division sequence", "complete division sequence", "star conversion", and "extended star conversion". Next, we use Well-Founded Induction and Peirce's law to prove the Collatz conjecture. This proof uses the theorem proving system Idris.
Category: Number Theory

[3062] viXra:2310.0115 [pdf] submitted on 2023-10-24 19:40:31

Proof of ABC Conjecture

Authors: Xiaohui Li
Comments: 4 Pages.

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 exponent then an equivalent inequality is constructed to prove the ABC conjecture.
Category: Number Theory

[3061] viXra:2310.0113 [pdf] submitted on 2023-10-25 00:17:20

New Maximum Interval Between Any Number and the Nearest Prime Number and Related Conjectures

Authors: Juan Moreno Borrallo
Comments: 5 Pages.

In this short paper we prove that for n ≥ 2953652287 it exists some prime number between nand n + log(n), improving the best known proved bounds for the maximum interval between anynumber and the nearest prime number, as well as the maximum difference between two consecutiveprime numbers (prime gap). We note that this result proves some open conjectures on prime gapsand maximum intervals between any number and the nearest prime number.
Category: Number Theory

[3060] viXra:2310.0111 [pdf] submitted on 2023-10-24 01:26:55

Sufficiently Large Number N Makes ∑_(k=n)^(2n-1) C/(ak+b) = C/a Lnu20612

Authors: Tai-Choon Yoon, Yina Yoon
Comments: 5 Pages.

According to the Riemann rearrangement theorem, when a sequence converges, the sum can be changed by rearranging the order of the sequence. However, the result cannot be changed simply by rearranging the order of any sequences. In the case of the alternating harmonic series exemplified by Riemann, even if the result was the same by chance, the sum of the series was obtained by ignoring the sum oflim┬( n→∞)u2061∑_(k=n)^(2n-1) 1/(k+1)=lnu20612.
Category: Number Theory

[3059] viXra:2310.0110 [pdf] submitted on 2023-10-24 01:22:55

Les Preuves de Syracuse (Evidence from Syracuse)

Authors: Pierre Lamothe
Comments: 15 Pages. In French

The monoid algebra of transition functions between numbers of generalized Collatzsequences has revealed the universal cause of cycles and were used to demonstrate, both :— The absolute truth of the Syracuse conjecture cannot be verified mathematically, as a random cycle is always possible when the ratio (d/m) of d divisions by 2 to m multiplications by 3 is a rational approximation of log 3/ log 2.— The Syracuse conjecture remains true with a zero probability status in practice dueto the exponential decay of its probability as a function of cycle length.
Category: Number Theory

[3058] viXra:2310.0083 [pdf] submitted on 2023-10-17 18:31:42

An Algorithm for Finding the Factors of Fermat Numbers

Authors: Emmanuil Manousos
Comments: 3 Pages.

In this article we present an algorithm for finding the factors Q of composite Fermat numbers. The algorithm finds the Q factors with less tests than required through the equation Q=2n K+1.
Category: Number Theory

[3057] viXra:2310.0046 [pdf] submitted on 2023-10-10 21:48:11

The Philosophical and Mathematical Implications of Division by 0/0 = 1 in Light of Einstein’s Theory of Special Relativity

Authors: Budee U. Zaman
Comments: 9 Pages.

The enigma of dividing zero by zero 0 0 has perplexed scholars across philosophy, mathematics, and physics, remaining devoid of a clear-cut solution. This lingering conundrum leaves us in an unsatisfactory position,as there emerges a genuine necessity for such divisions, particularly in scenarios involving tensor components that are both set at zero. This article endeavors to grapple with this profound issue by leveraging the insights of Einstein’s theory of special relativity. Surprisingly, when we wholeheartedly embrace the ramifications of this theory, it becomes evidentthat zero divided by zero must equate to one 00 = 1. Essentially, we are confronted with a pivotal decision: either embrace the feasibility and definition of dividing zero by zero, in accordance with Einstein’s theory of special relativity, or reevaluate the integrity of this fundamental theory itself. This exploration delves into the profound consequences arisingfrom this critical choice.
Category: Number Theory

[3056] viXra:2310.0041 [pdf] submitted on 2023-10-10 01:15:19

Exist[ence Of] a Prime in Interval N^2 and "N^2+epsilon n"

Authors: Hashem Sazegar
Comments: 5 Pages.

Oppermance’ conjecture states that there is a prime number between n^2 and n^2 + n for every positive integer n,first we show that , All integer numbers between x^2 and x^2 + ϵx can be written as x^2 + i > 4p that 1 ≤ i ≤ ϵx andp = (x − m − 2)2 + j in which j is a number in intervals 1 ≤ j ≤ ϵ(x − m − 2),and then we prove generalization of Oppermance’ conjecture i.e there is a prime number in interval n^2 and n^2 + ϵn such that 0 < ϵ ≤ 1.
Category: Number Theory

[3055] viXra:2310.0020 [pdf] submitted on 2023-10-04 07:27:08

A Theorem on the Golden Section and Fibonacci Numbers

Authors: Rolando Zucchini
Comments: 13 Pages.

In the chapter 12° of his most significant book LIBER ABACI, Leonardo Pisano known as Fibonacci (Pisa 1170-1240 (?)) proposed a problem on the reproduction of rabbits [*]. So many scholars deduced that he arrives to his famous numerical sequence starting from this problem. In this article is explained a new hypothesis. The Fibonacci sequence was generated by the iteration of a theorem on the Golden Section, and it is presumably that was the great Italian mathematician to state and demonstrate it. The theorem allow us to proof a lot of properties of Fibonacci numbers. [*] A man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced by the initial couple in a year supposing every couple each month produces a new pair that can reproduce itself from the second month?
Category: Number Theory

[3054] viXra:2310.0019 [pdf] submitted on 2023-10-04 08:06:50

A New Formulation of Mertens Function

Authors: Juan Moreno Borrallo
Comments: 2 Pages.

In this brief note there are showed original formulations for the reciprocal of the Riemann zeta function evaluated at 1, and Mertens function.
Category: Number Theory

[3053] viXra:2310.0012 [pdf] submitted on 2023-10-03 23:46:39

Infinity Tensors, the Strange Attractor and the Riemann Hypothesis

Authors: Parker Emmerson
Comments: 6 Pages.

The Riemann Hypothesis can be reworded to indicate that the real part of one half always balanced at the infinity tensor by stating that the Riemann zeta function has no more than an infinity tensor's worth of zeros on the critical line.
Category: Number Theory

[3052] viXra:2310.0002 [pdf] submitted on 2023-10-01 22:26:18

A New Closed Formula for the Riemann Zeta Function at Prime Numbers

Authors: Oussama Basta
Comments: 2 Pages. (Note by viXra Admin: Please only submit complete/finalized paper)

The Riemann zeta function is one of the most important functions in mathematics, but it is also one of the most difficult to compute. In this paper, we present a new closed formula for the Riemannzeta function at prime numbers. Our formula is based on a new function.
Category: Number Theory

[3051] viXra:2309.0142 [pdf] submitted on 2023-09-28 20:15:53

Flows of the Riemann Hypothesis

Authors: Tai-Choon Yoon, Yina Yoon
Comments: 4 Pages.

The Riemann hypothesis is a mathematical conjecture that relates to the calculation of prime numbers through the Riemann product formula, which represents the product of Riemann zeta function and factorial. There were flows in deriving ∫x^(s-1)/(e^x-1) dx from Riemann product formula and, in attempting to represent the negative region by substituting x with —x. Furthermore, asserting that the Riemann zeta function, in the absence of a definition for negative factorial, obtains trivial zeros for negative even numbers through the Bernoulli exponential generating formula in the negative domain is also incorrect.
Category: Number Theory

[3050] viXra:2309.0135 [pdf] submitted on 2023-09-27 10:48:11

About an Integral in Valean's Book

Authors: Edgar Valdebenito
Comments: 3 Pages.

The evaluation of integrals is an important subject in mathematics, physics and applied sciences. In this note we give some integrals fot pi^3 .
Category: Number Theory

[3049] viXra:2309.0106 [pdf] submitted on 2023-09-22 00:35:38

Sum of Three Cubes Explored - Proof

Authors: James DeCoste
Comments: 15 Pages. Contact: jbdecoste@eastlink.ca

Using already known techniques along with some not so obvious innovations on my part, I was able to show (prove) that there are solutions for all K (except those of the form 9m+/-4 and 9m+/-5 which are impossible) for +/-K = +/- (x^3) +/- (y^3) +/- (z^3). A further stipulation is that x, y and z must be whole numbers that can be a combination of positives and negatives. This is achieved through simple subtraction. Setting up a table showing that all K can be represented using a multiple of 27 plus a mask lends validity to a portion of the proof. These representations may and often do contain many more than the required number of cubes summed up. I side step that problem by showing that no matter the K picked and how ever many cubes are required to create it in my representations, they can all be reduced to a maximum of cubes summed. Exactly what we require for the proof. Having done that we are complete. The three new cubes we have just reduced to are already included in table. They are items I have already represented in the above format.
Category: Number Theory

[3048] viXra:2309.0095 [pdf] submitted on 2023-09-19 22:21:42

A Direct, Simple, and Basic Computation of a Difference of Two Dilogarithms

Authors: Hervé Gandran-Tomeng
Comments: 2 Pages.

The computation of dilog(sqrt(2)-1)-dilog(1-sqrt(2)) is performed.
Category: Number Theory

[3047] viXra:2309.0088 [pdf] submitted on 2023-09-18 01:20:31

Collatz Conjecture Proof for Special Integer Subsets and a Unified Criterion for Twin Prime Identification

Authors: Budee U. Zaman
Comments: 5 Pages.

This paper presents a proof of the Collatz conjecture for a specific subset of positive integers, those formed by multiplying a prime number "p" greater than three with an odd integer "u" derived using Fermat’s little theorem. Additionally, we introduce a novel screening criterion for identifying candidate twin primes, extending our previous work linking twin primes (p and p+2) with the equation 2(p−2) = pu+v, where unique solutions for u and v are required. This unified criterion offers a promising approach to twin prime identification within a wider range of integers, further advancing research in this mathematical domain.
Category: Number Theory

[3046] viXra:2309.0080 [pdf] submitted on 2023-09-16 16:05:33

The Floor and Ceiling Functions

Authors: Edgar Valdebenito
Comments: 4 Pages.

In this note we give some properties of the Floor and Ceiling functions.
Category: Number Theory

[3045] viXra:2309.0079 [pdf] submitted on 2023-09-17 00:12:32

Proof of the Collatz Conjecture 3x+1 by Adriano Bertaggia

Authors: Adriano Bertaggia
Comments: 52 Pages.

I will demonstrate that the conjecture 3x+1 is true using a new approach based on an ancient symbol the ENNEAGRAM, and how "numerical gravity" arises from the deterministic divisibility that combinations of integers allow. I will use modular arithmetic. With the help of flow and block diagrams I will find an equation that, applying the 2 conditions, binds all odd numbers and consequently positive numbers to powers of 2.I will find the analytical expression of the function. I will go up the Collatz graph represented by the inverse function which forms a tree with the exception of the cycle 1-4-2-1...I will show how all positive integers are present in the tree, that is connected to the number 1, making extensive use of tables, drawings and colors in order to represent the beauty of mathematics. I will show that for every integer n, n ≡ 1 (mod 2) if and only if 3n+1 ≡ 4 (mod 6).I will follow the exact chronology of the insights. Careful observation of the numbers will return an elementary (-a)rithmetic (double logical negation equals affirmation). I will not omit passages that are obvious,for these are the substrate on which the approach is grounded. I hope you can appreciate the extreme simplicity, harmony and rhythm that the numbers manifest.
Category: Number Theory

[3044] viXra:2309.0066 [pdf] submitted on 2023-09-13 21:59:25

Authors: G. Hervé
Comments: 2 Pages. (Abstract added by viXra Admin)

[This note is about a proof of a conjecture on the Riemann zeta-function at even integers]
Category: Number Theory

[3043] viXra:2309.0062 [pdf] submitted on 2023-09-13 00:17:56

Expressing Even Numbers Beyond 6 as Sums of Two Primes

Authors: Budee U. Zaman
Comments: 3 Pages.

The "strong Goldbach conjecture" posits that any even number exceeding 6 can be represented as the sum of two prime numbers. This study explores this hypothesis, leveraging the constancy of odd integer quantities and cumulative sums within positive integers. By identifying odd prime numbers, pα1and pα2, within [3, n] and (n, 2n-2) intervals, we demonstrate a transformative process grounded in the unchanging nature of odd number counts and their cumulative sums. Through this process, we establish the equation 2n =pα1 + pα2, offering a significant stride in unraveling the enigmatic core of the strong Goldbach conjecture.
Category: Number Theory

[3042] viXra:2309.0049 [pdf] submitted on 2023-09-08 20:33:59

The Sum of Positive and Negative Prime Numbers are Equal

Authors: Budee U. Zaman
Comments: 8 Pages.

This paper unveils a profound equation that harnesses the power of natural numbers to establish a captivating theorem: the balance between positive and negative prime numbers’ summation, intricately linked through the medium of natural numbers. As a corollary, the essence ofnatural numbers emerges as a testament to the harmonious interplay between even and odd elements. Notably, we expose the remarkable revelation that odd numbers find expression as both the aggregate of prime divisors and the sum of prime numbers, fusing diverse mathematical concepts into an elegant unity. This work reshapes the landscape of number theory, illuminating the hidden connections between primes, naturals, andtheir arithmetical compositions.
Category: Number Theory

[3041] viXra:2309.0043 [pdf] submitted on 2023-09-07 20:43:08

On Bifurcations and Beauty

Authors: Matthew Russell Downey
Comments: 71 Pages. Comments welcome!

This paper focuses on two ideas: the beginning focuses on standard and chaotic bifurcations, and the end focuses on beauty through mathematical coincidences. The scope of the bifurcation side is ambitious: relating bifurcation theory not only to the logistic map but also to prime spirals, the Riemann hypothesis, the Lambert W function, the Collatz conjecture, the Mandelbrot set, and music theory. The scope of the beauty side is similar: a proposed sequence that is opposite to the primes in some sense, finite sequences with peculiar properties, Fibonacci-like sequences, trees of primitive Pythagorean triples, Babylonian math, Grimm's conjecture, and Shell sort. Rather than providing rigorous analysis, my goal is to revitalize qualitative mathematics.
Category: Number Theory

[3040] viXra:2309.0038 [pdf] submitted on 2023-09-06 23:54:12

Double Inequalities Related to Approximate Formulas of Euler-Mascheroni Constant with Continuedfraction

Authors: JiSong Ro, SongIl Kang, JinSong Yu, HyonChol Kim
Comments: 6 Pages.

In this paper, we present some new double inequalities starting from the approximate formula for Euler-Mascheroni constant the newly obtained by us.
Category: Number Theory

[3039] viXra:2309.0037 [pdf] submitted on 2023-09-06 06:29:21

New Approaches to Riemann Hypothesis Solution

Authors: Miroslav Sukenik, Magdaléna Súkeníková
Comments: 3 Pages.

In the article, we assume that the Golden Ratio plays a fundamental role in alocalosation of non-trivial zero points of the Riemann Zeta function on the critical line s = 0.5.
Category: Number Theory

[3038] viXra:2309.0022 [pdf] submitted on 2023-09-04 00:22:20

Proof by Analysis-Synthesis and by the Absurd of Goldbach's Conjecture

Authors: Yves Désiré Ipolo
Comments: 8 Pages. In French

This article proposes an original approach never before addressed to demonstrate the Goldbach conjecture which uses reasoning by analysis-synthesis followed by reasoning by the absurd. I would like to put forward the key to the demonstration in the form of a revisited Goldbach Conjecture 1: "For every natural integer n strictly greater than 3, there exists at least one prime natural integer p which is prime with n such that 2n-p is prime and prime with n." The "Goldbach solutions", if they exist, are necessarily prime with 2n.
Category: Number Theory

[3037] viXra:2309.0020 [pdf] submitted on 2023-09-04 00:24:27

Dynamical System, Prime Numbers, Black Holes, Quantum Mechanics, and the Riemann Hypothesis

Authors: M. J. Sghiar
Comments: 11 Pages.

In mathematics, the search for exact formulas giving all the prime numbers, certain families of prime numbers or the n-th prime number has generally proved to be vain, which has led to contenting oneself with approximate formulas [6]. The purpose of this article is to give a simplefunction to produce the list of all prime numbers.Finally we give a generalization of this result and we show a link with the quantum mechanics and the attraction of black Holes. And I give a newproof of lemma 1 which gave a proof of the Riemann hypothesis [4]. Another excellent new proof is given.
Category: Number Theory

[3036] viXra:2309.0015 [pdf] submitted on 2023-09-03 18:48:27

A Formula for Mertens' Function and Its Applications

Authors: Roy L. Lewis Jr.
Comments: 12 Pages.

In this article, we prove the limit formula lim M(x) / &pi(x) = lim h / log(x) = 0, h = a constant for Mertens' function M(x) using arithmetic and analytic arguments based on theorems for the prime counting function &pi(x) and the series &sum &mu(k)/k. The formula is evaluated using limit theorems to give: an alternative proof of lim M(x)/ x = 0, a new disproof of Mertens' conjecture, proof of the Odlyzko--te Riele conjecture and a disproof of the Riemann hypothesis based on Littlewood's equivalence theorem.
Category: Number Theory

[3035] viXra:2309.0003 [pdf] submitted on 2023-09-01 00:05:56

Proof of the Odd Goldbach's Conjecture

Authors: Samuel Ferrer
Comments: 2 Pages. (Abstract added to Article by viXra Admin - Please conform!)

In this work a proof is presented provided that the original Goldbach’s conjecture(Goldbach’s) has been verified.
Category: Number Theory

[3034] viXra:2308.0208 [pdf] submitted on 2023-08-31 10:02:45

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

[3033] viXra:2308.0197 [pdf] submitted on 2023-08-30 20:32:05

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

[3032] viXra:2308.0177 [pdf] submitted on 2023-08-26 23:24:33

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

[3031] viXra:2308.0173 [pdf] submitted on 2023-08-26 20:34:01

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

[3030] viXra:2308.0166 [pdf] submitted on 2023-08-25 21:42:48

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

[3029] viXra:2308.0164 [pdf] submitted on 2023-08-24 13:00:10

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

[3028] viXra:2308.0162 [pdf] submitted on 2023-08-24 14:59:25

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

[3027] viXra:2308.0156 [pdf] submitted on 2023-08-24 01:12:59

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

[3026] viXra:2308.0145 [pdf] submitted on 2023-08-23 00:15:19

"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

[3025] viXra:2308.0131 [pdf] submitted on 2023-08-21 00:51:45

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

[3024] viXra:2308.0130 [pdf] submitted on 2023-08-21 00:51:01

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

[3023] viXra:2308.0122 [pdf] submitted on 2023-08-18 21:19:16

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

[3022] viXra:2308.0110 [pdf] submitted on 2023-08-16 06:07:26

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

[3021] viXra:2308.0106 [pdf] submitted on 2023-08-15 20:11:15

Riemann Hypothesis: the Abstract Physics Behind Mathematical Equations

Authors: Pankaj Mani
Comments: 21 Pages.

In this paper, I'm looking unconventionally at the underlying Physics behind the Mathematical Equation in context of Riemann Hypothesis to show that Riemann Hypothesis would be true ! It's about the new way of looking at Mathematics where one has to imagine and look at the governing abstract physicalities e.g. symmetry behind the arithmetical operations like addition, multiplication,0,complex number etc.Mathematics has its own abstract Physics behind it. Riemann Hypothesis is about imagining that rather trying to solve the equations endlessly in my humble view.
Category: Number Theory

[3020] viXra:2308.0103 [pdf] submitted on 2023-08-15 19:09:48

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

[3019] viXra:2308.0101 [pdf] submitted on 2023-08-14 23:50:15

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

[3018] viXra:2308.0096 [pdf] submitted on 2023-08-14 20:53:32

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

[3017] viXra:2308.0078 [pdf] submitted on 2023-08-12 11:02:00

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

[3016] viXra:2308.0073 [pdf] submitted on 2023-08-12 14:26:28

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

[3015] viXra:2308.0063 [pdf] submitted on 2023-08-11 16:24:51

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

[3014] viXra:2308.0056 [pdf] submitted on 2023-08-10 23:45:22

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

[3013] viXra:2308.0040 [pdf] submitted on 2023-08-08 20:33:54

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

[3012] viXra:2308.0037 [pdf] submitted on 2023-08-08 23:26:06

An Elementary Proof of Goldbach's Conjecture

Authors: Ronald Danilo Chávez Calderón
Comments: 27 Pages.

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

[3011] viXra:2308.0025 [pdf] submitted on 2023-08-04 20:56:43

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

[3010] viXra:2308.0024 [pdf] submitted on 2023-08-04 12:06:47

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

[3009] viXra:2308.0021 [pdf] submitted on 2023-08-04 21:18:34

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

[3008] viXra:2308.0020 [pdf] submitted on 2023-08-04 21:21:23

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

[3007] viXra:2308.0002 [pdf] submitted on 2023-08-01 02:36:54

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

[3006] viXra:2307.0158 [pdf] submitted on 2023-07-29 21:35:56

Proof of the Goldbach's Conjecture

Authors: Xiaohui Li
Comments: 2 Pages.

[A]ny large even number can be written as the sum of two prime numbers, which is also called "strong Goldbach's conjecture" or "Goldbach's conjecture about even numbers". Based on the equality of the sum of all odd numbers and the equality of odd numbers, the values and numbers of prime numbers pr1 and pr2 are separately screened out from the odd combination. By using the equality of the sum of all odd numbers and the equality of all odd numbers, an identity is constructed to obtain 2n=pr1+pr2
Category: Number Theory

[3005] viXra:2307.0149 [pdf] submitted on 2023-07-27 07:28:28

Two Proofs of Fermat’s Last Theorem by Relating to Monic Polynomial Properties

Authors: Tae Beom Lee
Comments: 5 Pages. (Correction on author name made by viXra Admin - Please conform!)

Fermat's Last Theorem(FLT) states that there is no natural number set {a,b,c,n} which satisfies a^n+b^n=c^n when n≥3. In this thesis, we related the LHS of a^n=c^n-b^n to x^n-a^n and the RHS to x^n-(c^n-b^n). By doing so, we could analyse FLT in view of properties of monic polynomials such as factoring, root structure and graphs.The polynomial properties narrowed the vast possible approaches to FLT to elementary level mathematics. We relied on factoring, rational root theorem and parallel movement of graphs. And we succeeded to find simple proofs of FLT, which many people waited for so long time.
Category: Number Theory

[3004] viXra:2307.0137 [pdf] submitted on 2023-07-25 04:11:56

Beyond the Riemann Hypothesis

Authors: Hideharu Maki
Comments: 173 Pages.

The functional equation of real variable that Riemann used in his paper was subjected to elementary operations. And I obtained a lot of complex functional equations that the Riemann zeta function follows respectively. Here, functional equation transformations were the main methods for obtaining the complex functional equations. Half of those are equivalent to the complete symmetric functional equation that the Riemann Xi function follows, and one of those has an origin symmetry with correction terms. From the origin symmetric functional equation including correction terms, the representation containing the leading term of the zeta function for any complex number was obtained. And the Riemann hypothesis was proved by applying reduction to absurdity. Moreover the general representation containing the leading term of the zeta function for any odd number of 3 or more was also obtained. By suitably combining those functional equations, I observed a new explicit formula for the zeta function. The Riemann hypothesis was again proven using the deductive method. And two types of general representations for the zeta function for any odd number of either 3 or 7, or more, were also obtained from the explicit formula. In total, three types of general representations for the zeta function for any odd number of either 3 or 7, or more, were discovered. Conversely, I defined a new function, named the Chi function, for the left side of the origin symmetric functional equation that includes corrective terms. The Chi function is similar to the Riemann Xi function and exhibits origin symmetry. Furthermore, I defined a new function, the eta function, which is similar to the zeta function. The eta function's pole and trivial zeros are the same as those of the zeta function. Furthermore, the Chi and the eta functions have the same non-trivial zeros on the imaginary axis. Here, the imaginary axis corresponds to the critical line of the eta function. And I proposed a generalized Riemann hypothesis for the eta function that states that all non-trivial zeros lie on the imaginary axis. Since I was able to discover the explicit formula for the eta function, the deductive method was used to prove the generalized Riemann hypothesis for the eta function.
Category: Number Theory

[3003] viXra:2307.0136 [pdf] submitted on 2023-07-25 21:25:17

A Proof of Fermat’s Conjecture

Authors: Olivier Massot
Comments: 32 Pages.

This study (in English) leads to a different formulation of the Newton's Binomial expansion. From there, a proof of Fermat's conjecture seems possible.
Category: Number Theory

[3002] viXra:2307.0133 [pdf] submitted on 2023-07-25 21:20:01

Analytical Proof of Collatz Conjecture

Authors: Ahmed Idrissi Bouyahyaoui
Comments: 6 Pages. In English and French

Let xi = 2^αi*yi and vi = 2^βi*zi , x0, yi and zi are odd integers. The sequence {xi + vi} built by Collatz algorithm is a Collatz sequence if it exists n such that xn + vn = 1. By hypothesis S(y0) is a Collatz sequence, then it exists at least one i such that yi = 1, xi = 2^αi*yi = 2^αi and vi = zi (because vi < xi and xi + vi > 0). As for every k ≥ i yk Є [1, 4, 2], xk + vk is of form : xk + vk = 2^αk + zk. For every optimal point (k, xk + vk), continuous and differentiable function f(α) = x + v = 2^α + z has a zero derivative and the primitive function z = - 2^α + c, c is an arbitrary integer constant.For every optimum we have : f(α) = c. At the optimum minimum = 1, it exists at least one n such that, yn Є [1, 4, 2], f(αn) = xn + vn = 2^αn + zn = 2^αn - 2^αn + c = c. For the minimum f(αn) = 1, it suffices to set c = 1 and so we have :f(αn) = xn + vn = 1, xn = 2^αn and vn = — (2^αn — 1). Conclusion : The sequence S(x0 +2) ends in 1 and has the only cycle [1, 4, 2, 1].So by recurrence, every positive integer number gives a Collatz sequence.
Category: Number Theory

[3001] viXra:2307.0129 [pdf] submitted on 2023-07-24 00:20:07

A Proof of the Legendre Conjecture

Authors: Radomir Majkic
Comments: 11 Pages.

If Legendre conjecture does not hold all integers in the interior of BT (n^{2},(n+1)^{2})ET are composed numbers. The composite integers counting shown that the rate of the number of the odd composites to the number of odd integers in theinterior of BT (n^{2},(n+1)^{2})ET is smaller than one. Consequently, the Legendre conjecture holds.
Category: Number Theory

[3000] viXra:2307.0127 [pdf] submitted on 2023-07-24 05:28:17

Reformulation of Syracuse Function and Its Convergence

Authors: Rakesh Timsina
Comments: 9 Pages.

This paper presents a geometrical approach to tackle the infamous Collatz conjecture. In this approach, we represent odd natural numbers as points in 2-D space. We then define a iterative geometrical algorithm and prove that this algorithm is equivalent to the Collatz function (more precisely, Syracuse function). Using the monotone convergence theorem, we prove the sequence generated by this algorithm always converges to 1. Since, this is same as saying Collatz (Syracuse) sequence converges to 1, we prove that the Collatz conjecture is true.
Category: Number Theory

[2999] viXra:2307.0126 [pdf] submitted on 2023-07-24 21:47:50

Une Démonstation de la Conjecture de Fermat (a Proof of Fermat's Conjecture)

Authors: Olivier Massot
Comments: 32 Pages. In French

This study leads to a different formulation of the Newton's Binomial expansion. From there, a proof of Fermat's conjecture seems possible.
Category: Number Theory

[2998] viXra:2307.0122 [pdf] submitted on 2023-07-23 22:43:53

Solved the Mathematical Problem Called the Oldest of Mathematical Problems

Authors: Giovanni Di Savino
Comments: 1 Page.

It is mathematical that: there are more planes in the ocean than submarines in the sky; there are more unaccepted essays than published essays; referring to the correspondence used to control the herd 3500 years ago, the mathematical problem defined as the oldest of mathematical problems "Why cannot exist perfect odd numbers" is mathematically demonstrated, but it is with mathematics, applied to the herd, that it is demonstrated that there are even perfect numbers and there are no perfect odd solutions.
Category: Number Theory

[2997] viXra:2307.0116 [pdf] submitted on 2023-07-22 08:36:18

Undecidability of the Riemann Hypothesis

Authors: Yinghao Luo
Comments: 3 Pages.

The Riemann zeta function in the Riemann hypothesis equals zero for both all negative even integers and an infinite number of complex numbers with real part 1/2. We can conclude that Riemann hypothesis is undecidable.
Category: Number Theory

[2996] viXra:2307.0092 [pdf] submitted on 2023-07-17 06:56:01

Proof of Erdos-Straus Conjecture

Authors: Oussama Basta
Comments: 3 Pages.

This paper presents a proof of the Erdős-Straus conjecture, which asserts that for every positive integer n greater than or equal to 2, there exist positive integers x, y, and z such that 4/n = 1/x + 1/y + 1/z. The proof utilizes a specific equation derived from the original conjecture and employs algebraic manipulations to establish its validity. By demonstrating that the equation holds for all applicable values of n, this proof conclusively confirms the Erdős-Straus conjecture.
Category: Number Theory

[2995] viXra:2307.0086 [pdf] submitted on 2023-07-17 23:26:05

Rieman Hypothesis Proof

Authors: Oussama Basta
Comments: 4 Pages.

The Rieman Hypothesis, a famous unsolved problem in mathematics, posits a deep connection between the distribution of prime numbers and the nontrivial zeros of the Riemann zeta function. In this study, we investigate the presence of zeros at prime numbers in a specific mathematical expression, ln (sec (π.nlog(n))), and its implications for the Riemann hypothesis. By employing rigorous mathematical analysis, we establish a clear connection between prime numbers, trigonometric functions, and the behavior of the Riemann zeta function. Our findings contribute to the body of knowledge surrounding the Riemann hypothesis and its potential proof, shedding light on the intricate nature of prime numbers and their relationship to fundamental mathematical functions.
Category: Number Theory

[2994] viXra:2307.0035 [pdf] submitted on 2023-07-06 13:06:04

Bertrand's Postulate and the Sum of Primes

Authors: Yung Zhao
Comments: 3 Pages.

It is deduced from Bertrand's Postulate that every even integer greater than 4 is the sum of two primes.
Category: Number Theory

[2993] viXra:2307.0018 [pdf] submitted on 2023-07-04 17:22:29

[T]here Can Be No Perfect Odd Numbers

Authors: Giovanni Di Savino
Comments: 2 Pages.

A perfect number is a natural number which is equal to the sum of its integer divisors including 1 and excluding itself, but a number n is also perfect in which the sum of its divisors including 1 and itself is equal to 2n. The natural numbers are infinite, for each of them there is a successor number and if it will never be possible to know how many, among the natural numbers, there can be perfect numbers, it is possible to know why there are even perfect numbers and there cannot be odd perfect numbers .The perfect number equal to 2n recalls a measurement technique, used 35,000 years ago when numbers were not known and which is similar to today's one-to-one correspondence. The correspondence of years ago consisted in associating each element of a set A with an element of set B; a concrete correspondence today is: "in a shirt the A.soles can be associated with the B.brass". Years ago, not knowing how to count, any set A was made to correspond to a set B in order to obtain that any difference between the two sets was the confirmation or not that the two sets were equal.
Category: Number Theory

[2992] viXra:2306.0160 [pdf] submitted on 2023-06-28 21:05:35

The Perfect Cuboid Is Nothing More Than a Myth

Authors: Zakhar Pekhterev
Comments: 2 Pages. (Correction made by viXra Admin - Please conform!)

An impeccable proof of the impossibility of the existence of a perfect cuboid based on the parametrization of Leonhard Euler.
Category: Number Theory

[2991] viXra:2306.0152 [pdf] submitted on 2023-06-25 07:35:23

Possible Solutions of the Rational Cuboid Problem

Authors: B. S. Safin
Comments: 5 Pages.

This article covers a number of forms for elliptic equations that were derived from the simultaneous equations describing a rational cuboid. The analysis of these elliptic equations shows that some rational points on the elliptic curves exist, but they are not the points of infinite order, accordingly they do not belong to any of the right triangles.
Category: Number Theory

[2990] viXra:2306.0151 [pdf] submitted on 2023-06-26 01:45:14

Search for Congruent Numbers

Authors: B. S. Safin
Comments: 3 Pages.

By parametrizing the Pythagorean equation with hyperbolic functions you can obtain an algebraic equation of the 3rd degree that describes congruent numbers. In some cases this equation may facilitate the search for these numbers.
Category: Number Theory

[2989] viXra:2306.0135 [pdf] submitted on 2023-06-23 21:32:48

3

Authors: Yung Zhao
Comments: 3 Pages.

The final solution to the problem about small gaps between primes lies in Bertrand-Chebyshev Theorem. we construct a pair of intervals [3n, 6n], [6n, 12n], and a set: {P_(6n-), 6n, P_(6n+)}, where P_(6n-) denotes the largest prime in [3n, 6n] and P_(6n+) denotes the smallest prime in [6n, 12n]. Analyzing and dealing with them by the combination of Bertrand-Chebyshev Theorem, T. Tao's result and the elemental property of primes reveal that the Twin Prime Conjecture holds.
Category: Number Theory

[2988] viXra:2306.0129 [pdf] submitted on 2023-06-22 17:23:47

On Non-Trivial Zeros and Riemann Zeta Function

Authors: Bertrand Wong
Comments: 5 Pages.

This paper examines the mysterious non-trivial zeros of the Riemann zeta function ζ and explains their role, e.g., in the computation of the error term in Riemann’s J function for estimating the quantity of primes less than a given number. The paper also explains the close connection between the Riemann zeta function ζ and the prime numbers. [Published in international mathematics journal.]
Category: Number Theory

[2987] viXra:2306.0128 [pdf] submitted on 2023-06-22 17:29:39

On Energy Numbers

Authors: Ryan J. Buchanan
Comments: 7 Pages.

The "energy numbers" of Parker Emmerson are critically examined, and an application of these exotic numbers to particle physics is attempted. Along the way, we establish the quaternion field identity, which is an isomorphism between a certain characterization of the abstract structure of a Hermitian space, and the complex Borel algebra of its generators.
Category: Number Theory

[2986] viXra:2306.0110 [pdf] submitted on 2023-06-19 19:10:22

On the Set of Prime Numbers

Authors: Emmanuil Manousos
Comments: 4 Pages.

"The octets of the odd numbers" theory categorizes the odd numbers into four categories D1, Q1, D2, Q2. From this categorization we get an algorithm for finding the set of prime numbers of the form D2 and Q2. The algorithm sequentially finds all prime numbers of the form D2 and Q2 in ascending order.
Category: Number Theory

[2985] viXra:2306.0105 [pdf] submitted on 2023-06-18 00:07:17

P=NP? Proving the Problem

Authors: Mesut Kavak
Comments: 3 Pages.

The question about the problem is pretty clear:"Can any question whose solution can be quickly verified, also quickly solved?"
Category: Number Theory

[2984] viXra:2306.0095 [pdf] submitted on 2023-06-16 22:44:45

Number System in Base One Hundred

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

In this paper I show my work on the possibility of make a number system in base one hundred, showing the table of possible assignations for every symbol and showing examples and conclusions.
Category: Number Theory

[2983] viXra:2306.0093 [pdf] submitted on 2023-06-15 07:47:45

Analytical Proof of 3x + 1

Authors: Samuel Ferrer Colas
Comments: 5 Pages.

The Collatz or 3x + 1 conjecture is perhaps the simplest stated yet unsolved problem in mathematics in the last 70 years. It was circulated orally by Lothar Collatz at the International Congress 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 proof of this conjecture, by finding an upper bound to the Collatz sequence and, as a consequence, a contradiction.
Category: Number Theory

[2982] viXra:2306.0091 [pdf] submitted on 2023-06-15 13:54:44

Proof of the Riemann Hypothesis

Authors: Marcello Colozzo
Comments: 28 Pages.

We prove the Riemann Hypothesis by studying the behavior of a holomorphic function which has the same non-trivial zeros as the Riemann zeta function.
Category: Number Theory

[2981] viXra:2306.0089 [pdf] submitted on 2023-06-16 00:48:10

Generalisation of the Fibonacci Sequence

Authors: Gaspar Daguet
Comments: 18 Pages. In French (Correction made by viXra Admin - Please conform!)

This article deals with a generalisation of the Fibonacci sequence and various facts about this generalisation.
Category: Number Theory

[2980] viXra:2306.0084 [pdf] submitted on 2023-06-14 19:43:27

The Semiprime Equivalent Proof of the Goldbach Conjecture

Authors: Stephen Marshall
Comments: 6 Pages.

In number theory, for very difficult Number theory problems that have been open and unsolved for long periods of time it can often be wise to take alternative approaches to the problem. There more old unsolved Number Theory problems than most would think. The Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics, it has been unsolved for over 281 years. On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture:Every even integer which is ≥ 4 can be written as the sum of two primes. It also states that every even natural number greater than 2 is the sum of two prime numbers. Or more specifically, that the "strong" Goldbach Conjecture asserts that all positive even integers ≥ 4 can be expressed as the sum of two primes. Two primes (p,q) such that p + q = 2n for n a positive integer ≥ 2.The conjecture has been shown via computer to hold for all integers less than 4×1018, but remains unproven despite enormous effort by many mathematicians over hundreds of years. Even the author has spent much effort attempting to solve this conjecture using several different direct methods and have come very close but was not able to prove the Goldbach Conjecture using any of these direct approaches. All of this effort made the author realize how difficult the Goldbach Conjecture is to solve using direct approaches, so this made him consider looking for a back door approach, or a work around the direct approaches. Any such approach could be different than the Goldbach Conjecture, but if it is a different Conjecture must be the equivalent of the Goldbach Conjecture, conjecture otherwise it would not solve the Goldbach Conjecture. This is exactly what the author has done, an equivalent conjecture has been developed and proven, thus solving the Goldbach Conjecture. Therefore, we call this a "back door" proof of the Goldbach Conjecture.
Category: Number Theory

[2979] viXra:2306.0065 [pdf] submitted on 2023-06-13 01:29:47

Collatz Problemi İspatı (Proof of the Collatz Problem)

Authors: Mesut Kavak
Comments: 4 Pages. In Turkish

Pozitif bir tam sayı seçildiğinde, sayı çift ise 2'ye bölünür; aksi takdirde 3 ile çarpılır ve bundan sonra sonuca 1 eklenir. Sonucun tek veya çift olması şartından dolayı problemin gerekli seçeneği ile aynı işlem tekrarlandığında, 0 ve 1'den farklı olan her pozitif tamsayı 1'e indirgenebilir mi?

Choosing a positive integer is divisible by 2 if the number is even; otherwise it is multiplied by 3 and then 1 is added to the result. Can every positive integer other than 0 and 1 be reduced to 1 when the same operation is repeated with the required option of the problem, provided that the result is odd or even?
Category: Number Theory

[2978] viXra:2306.0061 [pdf] submitted on 2023-06-13 01:37:05

İkiz Asallar Kestirimi İspatı (Proof for Twin Prime Conjecture)

Authors: Mesut Kavak
Comments: 2 Pages. In Turkish

İkiz asallar, aralarındaki fark 2 olan asal sayılardır. Sonsuz sayıda ikiz asal sayı var mıdır?

Twin primes are prime numbers that differ by 2. Are there an infinite number of twin prime numbers?
Category: Number Theory

[2977] viXra:2306.0060 [pdf] submitted on 2023-06-12 15:48:46

Proof for Twin Prime Conjecture

Authors: Mesut Kavak
Comments: 2 Pages.

"Twin primes are prime numbers that differ by 2. Are there infinitely many twin primes?"
Category: Number Theory

[2976] viXra:2306.0059 [pdf] submitted on 2023-06-12 18:42:34

Proof for Collatz Conjecture

Authors: Mesut Kavak
Comments: 4 Pages.

When a positive integer is chosen, if the number is even, it is divided by 2; otherwise, it is multiplied by 3 and after that 1 is added to the result. Due to the condition that the result is odd or even, the same operation is repeated with the required option of the problem, every positive integer other than 0 and 1 can the integer be reduced to 1?
Category: Number Theory

[2975] viXra:2306.0049 [pdf] submitted on 2023-06-11 01:25:39

Considerations on the 3n+1 Problem

Authors: V. Barbera
Comments: 6 Pages.

This paper presents some considerations on the 3n+1 problem. In particular on the next odd elements in the sequence lower than the starting number.
Category: Number Theory

[2974] viXra:2305.0179 [pdf] submitted on 2023-05-31 01:22:07

[proof Of] the Goldbach’s Conjecture

Authors: Aniket Bhattacharjee
Comments: 2 Pages.

In this paper, I want to present the proof to one of the most famous conjecture - The Goldbach’s Conjecture.
Category: Number Theory

[2973] viXra:2305.0178 [pdf] submitted on 2023-05-31 01:20:23

A Simple Markov Chain for the Collatz Problem

Authors: Wiroj Homsup, Nathawut Homsup
Comments: 4 Pages.

We show that the iteration of the Collatz function is represented by a simple three states Markov chain. This simple model is implemented to show the probabilistic convergence of the algorithm to the equilibrium point set {1,2}.
Category: Number Theory

[2972] viXra:2305.0165 [pdf] submitted on 2023-05-28 20:08:28

Sequences of Prime Numbers

Authors: Emmanuil Manousos
Comments: 4 Pages.

In this article we present sequences - networks of prime numbers.
Category: Number Theory

[2971] viXra:2305.0159 [pdf] submitted on 2023-05-26 17:30:16

Voyage on Two Syracuse

Authors: Berkouk Mohamed
Comments: 27 Pages. In French (Correction made by viXra Admin - Please conform!)

Dans une matrice carrée, vu qu’il y a deux suites, une horizontale de Syracuse (S0 ,S1,u2026Sm)Constituant les lignes de la matrice, puis des suites des valeurs prises de 0 à n pour un S0 donné, ces valeurs s’organisant à leurs tours verticalement dans les colonnes.le fait qu’elles soit déterminées respectivement par deux formules récurrentes issues des mêmes instructions de Collatz , méritent le nom de « deux Syracuse »

In a square matrix, since there are two sequences, a Syracuse horizontal (S0 ,S1,u2026Sm)Constituting the rows of the matrix, then the series of values u200bu200btaken from 0 to n for a given S0, these values u200bu200bbeing organized in turn vertically in the columns. the fact that they are respectively determined by two recurring formulas from the same instructions of Collatz, deserve the name of "two Syracuse"
Category: Number Theory

[2970] viXra:2305.0153 [pdf] submitted on 2023-05-24 08:43:40

Proof of the Collatz Conjecture

Authors: Henok Tadesse
Comments: 13 Pages.

This paper presents new insights towards proving the Collatz conjecture.
Category: Number Theory

[2969] viXra:2305.0151 [pdf] submitted on 2023-05-24 20:31:12

On Prime Number Generating Formula

Authors: Yuji Masuda
Comments: 1 Page.

Although there have been many attempts to obtain prime number generating formulas, the purpose of this study was to deepen our basic knowledge of prime numbers in order to create this prime number generating formula. I hope that this research will help to deepen our understanding of prime numbers.
Category: Number Theory

[2968] viXra:2305.0149 [pdf] submitted on 2023-05-23 05:36:14

Solution to the Collatz Conjecture

Authors: Samuel Ferrer Colas
Comments: 5 Pages. The chart is only a hints. Do not consider it as part of the core proof

The Collatz or 3x + 1 conjecture is perhaps the simplest stated yet unsolved problem in mathematics in the last 70 years. It was circulated orally by Lothar Collatz at the International Congress 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 proof of this conjecture, by finding an upper bound to the Collatz sequence and, as a consequence, a contradiction.
Category: Number Theory

[2967] viXra:2305.0136 [pdf] submitted on 2023-05-20 00:42:27

Proofs of ABC Conjecture

Authors: Dmitri Martila
Comments: 4 Pages. (Corrections made by viXra Admin to conform with scholarly norm)

Several crucial properties of ABC conjecture are presented and proven. Therefore, the ABC conjecture is proven.
Category: Number Theory

[2966] viXra:2305.0133 [pdf] submitted on 2023-05-18 08:35:19

The Graphical Method of the Kakutani’s Problem

Authors: Xingyuan Zhang
Comments: 10 pages. This is my second proof of the Collatz conjecture called graphical method. It can be only 6 pages. It has also other solutions or ideas, such as only using a table and in binary. My first proof is at viXra: 2301.0154.

In this paper we had given another elementary proof of the Kakutani’s problem by using the Kakutani’s Angle, it holds. By detailed analysis of the properties of both forward and inverse operations of the proposition, we had some important conclusions: 1, there hasn’t any triple in the forward path numbers; 2, there have an infinity number of inverse path numbers which had been defined as similar numbers in one time of inverse operation; 3, on the figure of Kakutani’s Angle, the operation path of any odd is unique; 4, the inverse operations can start with any odd, and all of the path numbers on the countless paths is getting larger and larger, on the contrary, to do forward operations for any inverse path number, it must go back to the starting point or to 1. It’s not difficult to prove if understanding its operational mechanism and grasping the methods.
Category: Number Theory

[2965] viXra:2305.0122 [pdf] submitted on 2023-05-17 08:06:06

Pythagorean Triples and the Binomial Formula

Authors: Kurmet Sultan
Comments: 3 Pages.

The article shows the possibility of compiling Pythagorean triples using the binomial formula and provides a Theorem that is an alternative proof of the infinity of Pythagorean triples and confirmation of the close connection of the Pythagorean Theorem with the binomial formula.
Category: Number Theory

[2964] viXra:2305.0112 [pdf] submitted on 2023-05-15 11:24:22

Disproof of the Riemann Hypothesis and the Non-Trivial Zeros of the Zeta Function

Authors: Henok Tadesse
Comments: 7 Pages.

This paper disproves the Riemann hypothesis by disproving the non-trivial zeros of the Riemann zeta function.
Category: Number Theory

[2963] viXra:2305.0107 [pdf] submitted on 2023-05-15 02:08:00

A proof of Riemann Hypothesis.

Authors: Liu Yajun
Comments: 2 Pages.

we give a proof of Riemann Hypothesis.
Category: Number Theory

[2962] viXra:2305.0039 [pdf] submitted on 2023-05-06 01:51:12

An Elementary Proof of Collatz's Conjecture

Authors: Chongxi Yu
Comments: 3 Pages.

Any even or odd number can be written as one of 10x + 1, 10x + 3, 10x + 5, 10x + 7, 10x + 9, 10x + 0, 10x + 2, 10x + 4, 10x + 6, or 10x + 8, (x= 1, 2, 3,u2026.n); all 10x + 1, 10x + 3, 10x + 5, 10x + 7, 10x + 9, 10x + 0, 10x + 2, 10x + 4, 10x + 6, or 10x + 8, can be transferred in to 5 x 2y, y = 1, 2, 3, 4, 5, 6, 8,u2026m by repeating two arithmetic operation (3x + 1 and dividing 2). When y is an odd number, 3 times y plus 1 will always yield one even number, if the even number is not one of 2n, then the even number divide 2 once or more, a new odd number y’ will be yielded, but the new odd number must be different from the original y, 3 times y’ + 1 will yield another new even number, if the new even number is not one of 2n, then the new even number divide 2 once or more, a new odd number y’’ will be yielded, so on, every dividing operation will yield one new odd number which is different from previous odd number, every time 3 + 1 will yield a new even number which is different from previous even number, these operations can be going unlimited and infinite different even numbers will be yielded until reach one of 2n which is less than total even number, but is also infinite, that is: by an infinite number of repeating two arithmetic operation (3x + 1 and dividing 2), one of 2n must be reach, then 5 x 1 will be reach, final 1 will be reach, this statement must be true, then the Collatz’s conjecture will be the Collatz’s theorem
Category: Number Theory

[2961] viXra:2305.0031 [pdf] submitted on 2023-05-03 01:59:31

Several Short Proofs of the Riemann Hypothesis

Authors: Dmitri Martila
Comments: 4 Pages. Rejected by all journals

I am showing that zeta(x+iy)=0 for 0<x<1 implies zeta(1/2+iy)=0. Several short proofs of the Riemann Hypothesis.
Category: Number Theory

[2960] viXra:2305.0029 [pdf] submitted on 2023-05-03 07:19:46

Syracuse Conjecture Quadrature

Authors: Rolando Zucchini
Comments: 37 Pages.

After circa 2300 years (Circle Quadrature; Archimèdès, Syracuse 287 — 212 BC) the history of mathematics repeats itself in a different problem.The conjecture of Syracuse, or Collatz conjecture, is approached from a completely dissimilar point of view than many previous attempts. One of its features suggests a process that leads to Theorem 2n+1, whose demonstration subdivided the set of odd numbers in seven subsets which have different behaviors applying algorithm of Collatz. It allows us to replace the Collatz cycles with the cycles of links, transforming their oscillating sequences in monotone decreasing sequences. By Theorem of Independence we can manage cycles of links as we like, also to reach very high horizons and when we decide go back to lower horizons. In this article it’s proved that Collatz conjecture is not fully demonstrable. In fact, if we consider the banal link n < 2n, there are eight cycles which connect each other in an endless of possible links. It is a type of Circle Quadrature, but its statement is confirmed. In other words: BIG CRUNCH (go back to 1) is always possible, but BIG BANG (to move on) has no End.
Category: Number Theory

[2959] viXra:2305.0020 [pdf] submitted on 2023-05-03 01:10:00

The Symmetry of N-domain and Prime Number Conjectures

Authors: Yajun Liu
Comments: 2 Pages. (Author name added to article by viXra Admin - Please conform!)

In this paper, we discuss the symmetry of N-domain and we find that using the symmetry characters of Natural Numbers we can give proofs of the Prime Conjectures: Goldbach Conjecture、Polignac’s conjecture and Twins Prime Conjecture.
Category: Number Theory

[2958] viXra:2305.0018 [pdf] submitted on 2023-05-03 01:20:47

L1/2(0 1/2 1) Space and Quantum Time-Space with Energy

Authors: Yajun Liu
Comments: 5 Pages. (Note by viXra Admin: Author's name format should be first name followed by last name)

In this paper, We constructed a Time-Space with energy model just considering the velocity of the light C and the Plank constant h and "1/" a_g (a_g is the strength of gravition (m/s2)) This model has a geometry space (complex) and just provide a probability to combine the Gravitation and Electric-Magnetics field under a basic structure of quantum Time-Space with energy. We hope to throw a little bit light on the big picture of uniting the quantum mechanics and General relative theory.KeywordsQuantum Time-Space with energy Unified Field Theory
Category: Number Theory

[2957] viXra:2305.0016 [pdf] submitted on 2023-05-03 01:11:37

Solution Conditions

Authors: Hajime Mashima
Comments: 14 Pages.

For Fermat’s Last Theorem, the condition that holds when there is inverse element.
Category: Number Theory

[2956] viXra:2304.0226 [pdf] submitted on 2023-04-29 07:41:41

Application of Ohm's Law to Numbers ~ac and DC Numbers and Impedance Numbers~

Authors: Yuji Masuda
Comments: 1 Page.

Each number has its own meaning. In this chapter, I was able to make explicit the relationship between the meaning of numbers and Ohm's law.
Category: Number Theory

[2955] viXra:2304.0222 [pdf] submitted on 2023-04-28 23:47:35

The Asymptotic Squeeze Principle and the Binary Goldbach Conjecture

Authors: Theophilus Agama
Comments: 6 Pages.

In this paper, we prove the special squeeze principle for all sufficiently large $nin 2mathbb{N}$. This provides an alternative proof for the asymptotic version of the binary Goldbach conjecture in cite{agama2022asymptotic}.
Category: Number Theory

[2954] viXra:2304.0218 [pdf] submitted on 2023-04-27 22:08:49

New Prime Number Theory

Authors: Budee U. Zaman
Comments: 12 Pages.

This paper introduces a novel approach to estimating the distribu- tion of prime numbers by leveraging insights from partition theory, prime number gaps, and the angles of triangles. Application of this methodology to infinite sums and nth terms, and propose several ways of defining the nth term of a prime number. By using the Ramanujan infinite series of natural numbers, I am able to derive an infinite series of prime numbers value . Overall, this work represents a significant contribution to the field of prime number theory and sheds new light on the relationship between prime numbers and other mathematical concepts.
Category: Number Theory

[2953] viXra:2304.0209 [pdf] submitted on 2023-04-26 07:48:59

Complex Circles of Partition and the Squeeze Principle

Authors: Berndt Gensel, Theophilus Agama
Comments: 12 Pages.

In this paper we continue the development of the circles of partition by introducing the notion of complex circles of partition. This is an enhancement of such structures from subsets of the natural numbers as base sets to the complex area as base and bearing set. The squeeze principle as a basic tool for studying the possibilities of partitioning of numbers is demonstrated.
Category: Number Theory

[2952] viXra:2304.0192 [pdf] submitted on 2023-04-24 07:12:55

Irrationality of Pi Using Just Derivatives

Authors: Timothy W. Jones
Comments: 6 Pages.

The quest for an irrationality of pi proof that can be incorporated into an analysis (or a calculus) course is still extant. Ideally a proof would be well motivated and use in an interesting way the topics of such a course. In particular $e^{pi i}$ should be used and the more easily algebraic of derivatives and integrals -- i.e. derivatives. A further worthy goal is to use techniques that anticipate those needed for other irrationality and, maybe even, transcendence proofs. We claim to have found a candidate proof.
Category: Number Theory

[2951] viXra:2304.0183 [pdf] submitted on 2023-04-23 02:02:21

How to Make a Conjecture About the More Compact Sequence with no 3 Terms in Arithmetic Progression Using Mathematica© and Www.oeis.org

Authors: Edoardo Gueglio
Comments: 5 Pages.

This is an example of how with a mathematical software you can make a mathematicalconjecture and help to prove it.
Category: Number Theory

[2950] viXra:2304.0181 [pdf] submitted on 2023-04-22 09:21:33

The Randomness in the Prime Numbers

Authors: Ihsan Raja Muda Nasution
Comments: 4 Pages.

The prime numbers has very irregular pattern. The problem of finding pattern in the prime numbers is the long-standing open problem in mathematics. In this paper, we try to solve the problem axiomatically. And we propose some natural properties of prime numbers.
Category: Number Theory

[2949] viXra:2304.0166 [pdf] submitted on 2023-04-20 23:55:17

Proof of the Triple and Twin Prime Conjectures Using the Sindaram Sieve Method

Authors: Kuiying Yan
Comments: 15 Pages.

Yitang Zhang proved in 2013 that there are infinitely many pairs of prime numbers differing by 70 million, it has been proved now that there are infinitely many pairs of prime numbers differing by 246. In this paper, we use the sievemethod invented by Snndaram in 1934 to find out the solution of triple prime numbers and twin prime numbers, and find the general solution formula of the subset, i.e, an1 + b which is result of each subset, such as 3n + 1, 5n + 2, 7n + 3, 9n + 4, 11n+ 5, 13n+ 6, 15n+ 7, 17n+ 8, · · · in 2mn+n+m, modulo x respectively (x ≤ 3 takes prime). This general solution formula is used to prove the triple prime conjecture and the twin prime conjecture.
Category: Number Theory

[2948] viXra:2304.0123 [pdf] submitted on 2023-04-18 00:38:35

Transcendental Equations: Solving Transcendental Equations Using the βw-Convergence Formula

Authors: John Evans Bwire
Comments: 14 Pages.

The main purpose of this paper is based on the general idea that an equation of this form can’t be a^x+b^x=c algebraically. In this question, the derived formula ((βw-convergence) with mathematical proof can be used to solve such an equation with ease. Since the formula is purely invented with my own approach, the article lacks references.
Category: Number Theory

[2947] viXra:2304.0122 [pdf] submitted on 2023-04-17 16:46:33

Solution of the Diophantine Brocard—Ramanujan Equation

Authors: Kurmet Sultan
Comments: 9 Pages.

The solution of the Diophantine Brocard-Ramanujan equation is obtained by proving the impossibility of representing other factorials, except for the known three, as a product of two natural numbers differing by 2. This is justified by the fact that no factorial is greater than 7! cannot be represented as a product of an increasing sequence of natural numbers, the first of which is equal to the argument of the factorial.
Category: Number Theory

[2946] viXra:2304.0115 [pdf] submitted on 2023-04-16 13:03:07

The Erdös-Borwein Constant

Authors: Edgar Valdebenito
Comments: 5 Pages.

The Erdös-Borwein constant is the sum of the reciprocals of the Mersenne numbers. It is named after Paul Erdös and Peter Borwein.
Category: Number Theory

[2945] viXra:2304.0113 [pdf] submitted on 2023-04-17 01:30:46

Proof Of The BGC & [the Use of the] Logic For Primes Prediction

Authors: Oussama Basta
Comments: 2 Pages.

The Goldbach’s Conjecture is an astonishing proposition that appears to be one of the most long-standing, renowned, and unsolved problems in number theory and in mathematics. This work herein is dedicated for proving it. The approach to be followed for the proof uses a system of equations predefined, and with the relatively simple analysis, the conjecture's proof is simple compared to the size of the problem.In the second part of this research, and with the purpose of predicting prime numbers in the known sequence of primes, the same system of equations is used, laying down a general mathematical framework that is computationally concise and can just achieve the objective. With proper selection of the coefficients of the equations in the algorithm, it’s guaranteed that prime number are among the outputs. The algorithm consists of basic arithmetic operations which is by itself a feat. The proof of the algorithm is also astoundingly straightforward and pintsized.
Category: Number Theory

[2944] viXra:2304.0071 [pdf] submitted on 2023-04-11 03:11:11

An Explicit Decomposition Formula of a Matrix in GL2(Z)

Authors: Dominique Fosse
Comments: 8 Pages.

Given three generators A, B, C of GL2(Z), we propose an explicit formula of decomposition of any element in $langle A,B,Cangle$.
Category: Number Theory

[2943] viXra:2304.0070 [pdf] submitted on 2023-04-09 04:11:35

Non-Existence of Looping in Collatz Conjecture Integers Series

Authors: Tsuneaki Takahashi
Comments: 2 Pages.

If there is looping in the series of Collatz conjecture integers, the conjecture cannot be realized. Therefore, non-existence of looping should be required for the correctness of the conjecture. Here it is tried to prove that there is no looping in it.
Category: Number Theory

[2942] viXra:2304.0056 [pdf] submitted on 2023-04-08 02:20:32

The Infinite Number of Primes Generate the Perfect Even Numbers and the Perfect Odd Numbers

Authors: Giovanni Di Savino
Comments: 4 Pages.

Perfect numbers were defined by Euclid with a proposition: "If we want as many numbers as we want starting from a unit, they are continuously arranged in double proportion, until the sum of all becomes a prime, and if the sum multiplied in the last one forms a , the product will be perfect"; Euler proved that even perfect numbers can be generated as defined by Euclid and are the result of (2^n -1) * 2^(n-1). Odd perfect numbers can be defined and generated with the proposition and algorithm with which even perfect numbers are defined and generated with the following modifications: a) the prime number 2 reported in Euler's algorithm is replaced by one of the infinite numbers first courses ≥ 3; b) the distance that the prime number must have from the result of a power of prime numbers ≥ 3^n is 2; c) with prime numbers ≥ 3, the "double proportion", reported in Euclid's proposition and generated by the number 2, becomes the triple proportion or the quintuple or.......the proportion of the nth prime number. With the modifications to situations defined as similar, "the generation of perfect odds is similar to the generation of perfect evens" and, also the algorithms with which the perfect numbers are generated are similar: the even perfect numbers are the result of ((2^n -1) * 2^(n-1))/(2-1), the odd perfect numbers are the result of ((prime ≥ 3^n -2) * prime ≥ 3^(n-1))/( first≥3-1).
Category: Number Theory

Replacements of recent Submissions

[1683] viXra:2403.0077 [pdf] replaced on 2024-03-19 03:01:07

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

[1682] viXra:2402.0032 [pdf] replaced on 2024-03-11 23:05:58

Derivation|Correction of Hardy-Littlewood Twin Prime Constant using Prime Generator Theory (PGT)

Authors: Jabari Zakiya
Comments: 7 Pages. Corrected data value in Figure 2 for m = 11.

The Hardy—Littlewood twin prime constant is a metric to compute the distribution of twin primes. Using Prime Generator Theory (PGT) it is shown it is more easily mathematically and conceptually derived, and the correct value is a factor of 2 larger.
Category: Number Theory

[1681] viXra:2401.0153 [pdf] replaced on 2024-03-06 08:45:15

Pythagorean Triples and Fermat's Theorem N = 4

Authors: Rolando Zucchini
Comments: 7 Pages.

This article contains a theorem to build the Primitive Pythagorean triples and the proof of the last Fermat’s Theorem for n = 4.
Category: Number Theory

[1680] viXra:2401.0084 [pdf] replaced on 2024-01-22 17:58:56

Riemann Hypothesis: Direct Demonstration Proposal

Authors: Vincent KOCH
Comments: 3 Pages.

In his 1859 article "On the number of prime numbers less than a given quantity", Bernhard Riemann formulated the hypothesis that all non-trivial zeros of the Zeta function have the real part 1/2.This assertion, known as the "Riemann Hypothesis", remains unproven to this day. The present paper is an attempt at a direct demonstration.
Category: Number Theory

[1679] viXra:2401.0068 [pdf] replaced on 2024-01-19 20:47:42

A Convergent Subsequence of $theta_n(x+iy)$ in a Half Strip

Authors: Young Deuk Kim
Comments: 8 Pages. Typos are fixed.

For $frac{1}{2}0$ and $ninmathbb{N}$, let $displaystyletheta_n(x+iy)=sum_{i=1}^nfrac{{mbox{sgn}}, q_i}{q_i^{x+iy}}$,where $Q={q_1,q_2,q_3,cdots}$ is the set of finite product of distinct odd primes and${mbox{sgn}}, q=(-1)^k$ if $q$ is the product of $k$ distinct primes.In this paper we prove that there exists an ordering on $Q$ such that $theta_n(x+iy)$ has a convergent subsequence.
Category: Number Theory

[1678] viXra:2401.0008 [pdf] replaced on 2024-01-23 01:28:41

Goldbach's Number Construction

Authors: Radomir Majkic
Comments: 7 Pages.

The internal structure of the natural numbers reveals the relation between the weak and the strong Goldbach's conjectures. The three prime integers structure of the odd integers alreadycontains the two prime integers base of the even integers. An explicit one-to-one correspondence between these two structures, defined asGoldbach's numbers exist. Thus, if the weak Goldbach's conjecture is true, the strong Goldbach'sconjecture should be. Hopefully, this will bring a happy end to Goldbach'sconjecture problem.
Category: Number Theory

[1677] viXra:2312.0108 [pdf] replaced on 2024-02-11 17:24:38

Complete Operations

Authors: Pith Peishu Xie
Comments: 27 Pages.

The Operator axioms have produced complete operations with real operators. Numerical computations have been constructed for complete operations. The classic calculator could only execute 7 operator operations: + operator operation(addition), - operator operation(subtraction), $times$ operator operation(multiplication), $div$ operator operation(division), ^{} operator operation(exponentiation), $surd$ operator operation(root extraction), log operator operation(logarithm). In this paper, we invent a complete calculator as a software calculator to execute complete operations. The experiments on the complete calculator could directly prove such a corollary: Operator axioms are consistent.
Category: Number Theory

[1676] viXra:2312.0016 [pdf] replaced on 2024-01-02 21:20:55

Proof for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 9 Pages.

I am going to use telescoping series and then a proof by induction. I am using Lambert's continued fraction for the base case.
Category: Number Theory

[1675] viXra:2312.0016 [pdf] replaced on 2023-12-08 09:26:48

Proof for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 9 Pages.

I am going to use telescoping series and then a proof by induction. I am using Lambert's continued fraction for the base case.
Category: Number Theory

[1674] viXra:2312.0005 [pdf] replaced on 2023-12-04 09:12:10

Proof of Fermat’s Last Theorem for Odd Primes

Authors: Minho Baek
Comments: 27 Pages.

It was already proved right that xn+yn=zn, (n>2) has no solutions in positive integers which we called Fermat’s Last Theorem (FLT) by Andrew Wiles. But his proof would be impossible in the 17th century. Since Fermat showed he proved n=even by leaving proof for n=4, many people have tried to prove the odd primes. I took the idea from Euler proof and proved in case of n=odd primes by simple method.
Category: Number Theory

[1673] viXra:2311.0137 [pdf] replaced on 2023-12-03 18:59:14

New Bounds on Mertens Function

Authors: Juan Moreno Borrallo
Comments: 6 Pages.

In this brief paper we study and bound Mertens function. The main breakthrough is the obtention of a Möbius-invertible formulation of Mertens function, which with some transformations and the application of a generalization of Möbius inversion formula, allows us to reach an asymptotic rate of growth of Mertens function that proves the Riemann Hypothesis.
Category: Number Theory

[1672] viXra:2311.0126 [pdf] replaced on 2023-11-29 06:38:42

Generalization for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 13 Pages.

I came across "The Ramanujan Machine" on the Internet and, using my intuition on those kind of stuff, I found some interesting results.
Category: Number Theory

[1671] viXra:2311.0126 [pdf] replaced on 2023-11-26 09:38:52

Generalization for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 7 Pages.

I came across "The Ramanujan Machine" on the Internet and, using my intuition on those kind of stuff, I found some interesting results.
Category: Number Theory

[1670] viXra:2311.0125 [pdf] replaced on 2024-03-07 06:20:33

No Collatz Conjecture Integer Series Have Looping

Authors: Tsuneaki Takahashi
Comments: 2 Pages.

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

[1669] viXra:2311.0125 [pdf] replaced on 2023-12-25 06:23:28

No Collatz Conjecture Integer Series Have Looping

Authors: Tsuneaki Takahashi
Comments: 3 Pages.

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

[1668] viXra:2311.0059 [pdf] replaced on 2023-11-24 16:16:41

Divisible Cyclic Numbers

Authors: Julian Beauchamp
Comments: 4 Pages.

There are known to exist a number of (multiplicative) cyclic numbers, but in this paper I introduce what appears to be a new kind of number, which we call divisible cyclic numbers (DCNs), examine some of their properties and give a proof of their cyclic property. It seems remarkable that I can find no reference to them anywhere. Given their simplicity, it would be extraordinary if they were hitherto unknown.
Category: Number Theory

[1667] viXra:2311.0052 [pdf] replaced on 2024-01-20 01:00:27

On the Incompletely Predictable Problems of Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 82 Pages. Now incorporating Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture

We validly ignore even prime number 2. Based on all arbitrarily large number of even prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with Prime number theorem for Arithmetic Progressions. With this condition being satisfied by all Odd Primes, we argue that Modified Polignac's and Twin prime conjectures are proven to be true when these conjectures are treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (which acts as proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when this hypothesis is treated as Incompletely Predictable Problem.
Category: Number Theory

[1666] viXra:2311.0052 [pdf] replaced on 2023-12-08 09:16:15

On the Incompletely Predictable Problems of Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 74 Pages. Now incorporating Hodge conjecture and Grothendieck period conjecture

We validly ignore even prime number 2. Based on all arbitrarily large number of even prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with Prime number theorem for Arithmetic Progressions. With this condition being satisfied by all Odd Primes, we argue that Modified Polignac's and Twin prime conjectures are proven to be true with these conjectures treated as Incompletely Predictable Problems. In so doing [and with the famous Riemann hypothesis being a special case], the generalized Riemann hypothesis formulated for Dirichlet L-function is also supported. By broadly applying Hodge conjecture and Grothendieck period conjecture to Dirichlet eta function (as proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true with this hypothesis treated as Incompletely Predictable Problem.
Category: Number Theory

[1665] viXra:2311.0049 [pdf] replaced on 2023-12-21 09:27:28

Collatz Conjecture Proved Ingeniously & Very Simply

Authors: A. A. Frempong
Comments: 12 Pages. Copyright © by A. A. Frempong

To prove Collatz conjecture, one would apply a systematic observation of the sequences produced by the (3n + 1)/2 process, Two main cases are covered. In Case 1, the integer can be written as a power of 2 as 2^(k) (k = 1, 2, 3,u2026), and the sequence would reach the integer 1 by repeated division by 2, In Case 2, the integer cannot be written as a power of 2, but the sequence terms of the integers reach integers equivalent to 2^(2k) (k = 2, 3,...),and by repeated division by 2, the sequences would reach the integer 1. In Case 2, when the sequence terms reach some particular odd integers such as 5, 21 and 85, the application of 3n +1 operation to these integers will result in the integers equivalent to the powers, 2^(2k) (k = 2, 3, 4,u2026). One would call these integers, the 2k-power converters. A term of the sequence must be converted to an integer equivalent to 2^(2k) (k = 2, 3, 4,u2026). There are infinitely many paths for converting integers to 2^(2k) powers on the 2^(2k)-route, a route on which a 2^(2k)-power can be divided repeatedly by 2 until the sequence reaches the integer 1. Of these conversion paths, the integer 5-path is the nearest 2^(2k) (k=2) converter path to the integer 1 on the 2^(2k)-route. Other paths to the 2^(2k)-route include the 21-path, (k = 3), the 85-path (k = 4), and infinitely, many paths. For the 5-path, when a sequence terms reach the integer, 5, the next term would be 16. Similarly, for the integers 21, and 85, the next terms, respectively, would be 64 and 256. Some non-2k power converters can follow the integer 5-path to the to 2^(4) power. The integers 2^(p)(5) (p = 1, 2, 3,u2026) will take the integer 5-path to convert to a 2k-power. The integers, 2^(p)(21), 2^(p)(85) with (p = 1, 2, 3,u2026), will take, respectively, the integer 21-path, and the 85-path, to reach 2k-powers. There are infinitely many 2^(2k)-power converters, 2k-powers, 2k-power converter paths, and descendants, 2^(p)(C) (p = 1, 2, 3,u2026) of C, By repeated division by 2, the sequence of every positive integer that reaches the equivalent integer, 2^(2k), would reach the integer 1, Therefore, using the approaches in Cases 1 and 2, the sequence of every positive integer would eventually reach the integer 1.
Category: Number Theory

[1664] viXra:2311.0049 [pdf] replaced on 2023-11-14 21:49:35

Collatz Conjecture Proved Ingeniously & Very Simply

Authors: A. A. Frempong
Comments: 12 Pages. Copyright © by A. A. Frempong

Collatz conjecture states that beginning with a positive integer, if one repeatedly performs the following operations to form a sequence of integers, the sequence will eventually reach the integer one; the operations being that if the integer is even, divide it by 2, but if the integer is odd, multiply it by 3 and add one; and also, use the result of each step as the input for the next step.One would note the patterns of the sequence terms as the Collatz process reaches the equivalent powers, 2^(2k) (k = 2, 3, ...), and the sequence reaches the integer 1 by repeated division by 2. Two main cases are covered. In Case 1, the integer can be written as a power of 2 as 2^(k) (k=1,2,3,...), and in this case, the sequence would reach the integer one by repeated division by 2, i.e., 2^(k-1), 2^(k-2), 2^(k-3,),...,2^(k-k). In Case 2, the integer cannot be written as a power of 2, but the sequence terms reach the equivalent power, 2^(2k) (k = 2, 3,...) and by repeated division by 2, the sequence will reach the integer 1. In Case 2, when the sequence terms reach some particular integers such as 5, 21 and 85, the application of 3n + 1 to these integers will result in the powers, 2^(2k). One would call these integers, the 2k-power converters. There are infinitely many 2k-power converters as there are 2^(2k) powers. There are infinitely many paths for converting integers to 2^(2k) powers. Of these paths, the integer 5-path, is the nearest 2^(2k) converter path to the integer 1 on the 2^(2k)-route. Other integers can follow the integer 5-path to 16 as follows: Let n be an integer whose sequence terms would reach 16, and let n ± r = 5, where r is the net change in the sequence terms before the integer 5; and one uses the positive sign if n < 5, but the negative sign if n > 5. One will call the following, the 5-path 2k-converter formula: 3(n ± r) + 1 = 16. By the substitution axiom, using this formula, the sequence of every positive integer that cannot be written as a power of 2, would reach the integer, 16, and continue to reach the integer 1. Therefore, the sequence of every positive integer would reach the integer 1.
Category: Number Theory

[1663] viXra:2310.0145 [pdf] replaced on 2024-01-15 08:50:48

A Truly Easy Proof: Pi is Irrational

Authors: Timothy Jones
Comments: 5 Pages. Expanded content per comments received.

Using the sum of the derivatives of an integer polynomial with Euler's formula we prove that pi is irrational. We show how the technique can be used to show e and pi's transcendence.
Category: Number Theory

[1662] viXra:2310.0145 [pdf] replaced on 2023-11-15 13:49:39

An Truly Easy Proof: Pi is Irrational

Authors: Timothy W. Jones
Comments: 2 Pages. There was an error in the previous: conflated composition with evaluation.

Using the derivative of an integer polynomial composed with Euler's formula we prove that pi is irrational.
Category: Number Theory

[1661] viXra:2310.0110 [pdf] replaced on 2023-10-30 21:07:58

Nuanced Truth of Collatz Conjecture

Authors: Pierre Lamothe
Comments: 15 Pages. In French

The algebra of transition functions between elements of generalized Collatz sequences has revealed the true nature of cycles. The structure of the cyclic invariant enables us to demonstrate both:
a) The Syracuse conjecture cannot be substantiated as absolute veracity because no matter the length, a random cycle is theoretically possible.
b) The Syracuse conjecture can only hold true in practice with exceedingly probable status due to the exponential decay of a cycle’s probability as a function of length.
Category: Number Theory

[1660] viXra:2310.0083 [pdf] replaced on 2023-10-25 10:02:01

An Algorithm for Finding the Factors of Fermat Numbers

Authors: Emmanuil Manousos
Comments: 3 Pages. Dear Editor, I am replacing the article due to errors in Example 2. Thank you for hosting my work on vixra.org. Best regards Emmanuil Manousos

In this article we present an algorithm for finding the factors Q of composite Fermat numbers. The algorithm finds the Q factors with less tests than required through the equation 2n×K+1.
Category: Number Theory

[1659] viXra:2310.0002 [pdf] replaced on 2023-10-10 02:13:49

A New Closed Formula for the Riemann Zeta Function at Prime Numbers

Authors: Oussama Basta
Comments: 2 Pages. A better version

The Riemann zeta function is one of the most important functions in mathematics, but it is also one of the most difficult to compute. In this paper, we present a new closed formula for the Riemann zeta function at prime numbers. Our formula is based on a new function.
Category: Number Theory

[1658] viXra:2309.0109 [pdf] replaced on 2023-10-05 02:33:18

The Geometric Collatz Correspondence

Authors: Darcy Thomas
Comments: 28 Pages. This is a final, and more complete, version of the paper. It is better categorized number theory.

The Collatz Conjecture, one of the most renowned unsolved problems in mathematics, presents adeceptive simplicity that has perplexed both experts and novices. Distinctive in nature, it leaves manyunsure of how to approach its analysis. My exploration into this enigma has unveiled two compellingconnections: firstly, a link between Collatz orbits and Pythagorean Triples; secondly, a tie to theproblem of tiling a 2D plane. This latter association suggests a potential relationship with PenroseTilings, which are notable for their non-repetitive plane tiling. This quality, reminiscent of theunpredictable yet non-repeating trajectories of Collatz sequences, provides a novel avenue to probethe conjecture’s complexities. To clarify these connections, I introduce a framework that interpretsthe Collatz Function as a process that maps each integer to a unique point on the complex plane.In a curious twist, my exploration into the 3D geometric interpretation of the Collatz Function has nudged open a small, yet intriguing door to a potential parallel in the world of physics. A subtle link appears to manifest between the properties of certain objects in this space and the atomic energy spectral series of hydrogen, a fundamental aspect in quantum mechanics. While this connection is in its early stages and the depth of its significance is yet to be fully unveiled, it subtly implies a simple merging where pure mathematics and applied physics might come together.The findings in this paper have led me to pursue development of a new type of number I call a Cam number, which stands for "complex and massive", indicating that it is a number with properties that on one hand act like a scalar, but on the other hand act as a complex number. Cam numbers can be thought of as having somewhat dual identities which reveal their properties and behavior under iterations of the Collatz Function. This paper serves as a motivator for a pursuit of a theory of Cam numbers.
Category: Number Theory

[1657] viXra:2309.0082 [pdf] replaced on 2023-11-24 04:09:09

Theory of Electrons System

Authors: Sheng-Ping Wu
Comments: 12 Pages.

Self-consistent Lorentz equation is proposed, and is solved to electrons and the structures of particles and atomic nucleus. The static properties and decay are reasoned, all meet experimental data. The equation of general relativity sheerly with electromagnetic field is discussed as the base of this theory.
Category: Number Theory

[1656] viXra:2309.0020 [pdf] replaced on 2023-10-24 11:26:37

Hilbert and Pólya Conjecture, Dynamical System, Prime Numbers, Black Holes, Quantum Mechanics, and the Riemann Hypothesis

Authors: Mohamed sghiar
Comments: 13 Pages.

In mathematics, the search for exact formulas giving all the prime numbers, certain families of prime numbers or the n-th prime number has generally proved to be vain, which has led to contenting oneself with approximate formulas [8]. The purpose of this article is to give a simple function to produce the list of all prime numbers.And then I give a generalization of this result and we show a link with the quantum mechanics and the attraction of black Holes. And I give a new proof of lemma 1 which gave a proof of the Riemann hypothesis [4]. Finally another excellent new proof o f the Riemann hypothesisis given and I deduce the proof of Hilbert Polya's conjecture
Category: Number Theory

[1655] viXra:2308.0210 [pdf] replaced on 2023-11-07 16:00:54

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

[1654] viXra:2308.0037 [pdf] replaced on 2023-09-04 14:37:12

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

[1653] viXra:2307.0092 [pdf] replaced on 2023-08-08 22:33:02

Proving the Erdos-Straus Conjecture

Authors: Oussama Basta
Comments: 2 Pages. A better version, Email included, submitted to a journal

This paper presents a rigorous proof of the Erdős-Straus conjecture, demonstrating that the equation (w^2−F1F2) / F^2 = 1 holds true for all n ≥ 2. The Erdős-Straus conjecture, originally formulated by mathematicians Paul Erdős and Ernst G. Straus, relates to the representation of positive integers as the sum of three reciprocal fractions. Through a series of mathematical derivations and substitution, we prove the conjecture and provide insight into its implications.
Category: Number Theory

[1652] viXra:2307.0092 [pdf] replaced on 2023-08-02 10:59:15

Proving the Erdos-Straus Conjecture

Authors: Oussama Basta
Comments: 2 Pages. A better version

This paper presents a rigorous proof of the Erdős-Straus conjecture, demonstrating that the equation (w^2−F1F2) / F^2 = 1 holds true for all n ≥ 2. The Erdős-Straus conjecture, originally formulated by mathematicians Paul Erdős and Ernst G. Straus, relates to the representation of positive integers as the sum of three reciprocal fractions. Through a series of mathematical derivations and substitution, we prove the conjecture and provide insight into its implications.
Category: Number Theory

[1651] viXra:2306.0135 [pdf] replaced on 2023-08-20 23:44:53

The Elemental Property of Primes and Small Gaps Between Primes

Authors: Yung Zhao
Comments: 2 Pages.

The solution to the Twin Prime Conjecture lies in the elemental property of primes. We construct a sequence of consecutive primes, analyzing and handling them by the combination of the elemental property of primes and the Statistics theory reveal that Twin Prime Conjecture is true.
Category: Number Theory

[1650] viXra:2306.0110 [pdf] replaced on 2023-06-24 03:14:43

On the Set of Prime Numbers

Authors: Emmanuil Manousos
Comments: 5 Pages.

"The Octets of the Odd Numbers" theory categorizes the odd numbers into four categories D1, Q1, D2, Q2. We relate the distribution of octets of odd numbers in the set of integers to the distribution of prime numbers and obtain an algorithm for finding the set of prime numbers of the form D2 and Q2. The algorithm sequentially finds all prime numbers of the form D2 and Q2 in ascending order.
Category: Number Theory

[1649] viXra:2306.0061 [pdf] replaced on 2023-06-16 15:44:12

İkiz Asallar Kestirimi İspatı (Proof for Twin Prime Conjecture)

Authors: Mesut Kavak
Comments: 3 Pages. in Turkish

İkiz asallar, aralarındaki fark 2 olan asal sayılardır. Sonsuz sayıda ikiz asal sayı var mıdır?Twin primes are prime numbers that differ by 2. Are there an infinite number of twin prime numbers?
Category: Number Theory

[1648] viXra:2306.0060 [pdf] replaced on 2023-06-16 15:45:21

Proof for Twin Prime Conjecture

Authors: Mesut Kavak
Comments: 3 Pages.

"Twin primes are prime numbers that differ by 2. Are there infinitely many twin primes?"
Category: Number Theory

[1647] viXra:2306.0049 [pdf] replaced on 2024-02-05 18:41:36

Considerations on the 3n+1 Problem

Authors: V. Barbera
Comments: 9 Pages.

This paper presents some considerations on the 3n+1 problem. In particular on the next odd elements in the sequence lower than the starting number.
Category: Number Theory

[1646] viXra:2306.0049 [pdf] replaced on 2023-08-13 10:47:49

Considerations on the 3n+1 Problem

Authors: V. Barbera
Comments: 7 Pages.

This paper presents some considerations on the 3n+1 problem. In particular on the next odd elements in the sequence lower than the starting number.
Category: Number Theory

[1645] viXra:2305.0179 [pdf] replaced on 2023-07-11 20:37:53

The Proof [of] Goldbach’s Conjecture

Authors: Aniket Bhattacharjee
Comments: 2 Pages.

In this paper, I want to present the proof to one of the most famous conjecture - The Goldbach’s Conjecture.
Category: Number Theory

[1644] viXra:2305.0165 [pdf] replaced on 2023-06-15 17:12:51

Sequences of Prime Numbers

Authors: Emmanuil Manousos
Comments: 6 Pages.

The categorization of odd numbers by "The octets of the odd numbers" theory gives an algorithm for finding prime numbers.
Category: Number Theory

[1643] viXra:2305.0165 [pdf] replaced on 2023-06-06 09:51:39

Sequences of Prime Numbers

Authors: Emmanuil Manousos
Comments: 6 Pages.

By combining two theorems of the theory "the octets of odd numbers" we obtain an algorithm for finding prime numbers.
Category: Number Theory

[1642] viXra:2305.0153 [pdf] replaced on 2023-06-12 11:02:57

A Proof of the Collatz Conjecture

Authors: Henok Tadesse
Comments: A mistake in version 2, pages 18, 19,20, led to the wrong conclusion that the paper completely proved non-existence of closed loops other than the 1-4-2-1 loop. This has been corrected in this version and new insights also introduced.

Take any positive integer N. If it is odd, multiply it by three and add one. If it is even, divide it by two. Repeatedly do the same operations to the results, forming a sequence. It is found that, whatever the initial starting number we choose, the sequence will eventually descend and reach number 1, where it enters an eternal closed loop of 1- 4 - 2 - 1. This has been numerically confirmed for starting numbers up to 260. This is known as the Collatz conjecture which states that the sequence always reaches number 1. So far no proof has ever been found that this holds for every positive integer. This problem has been stated by some as perhaps the simplest math problem to state, yet perhaps the most difficult to solve. This paper makes significant advances in solving the problem by using new insights. Proving the conjecture requires proving that: 1. The sequence will not diverge to infinity 2. There is no closed loop other than the 1-4-2-1 loop. This paper completely proves the first and makes significant advance in proving the second. The new insight is that the whole of the Collatz sequence up to the point where it enters a closed loop is encoded in the starting number. A Collatz sequence diverging to infinity would mean infinite information being encoded in a finite starting number, which is impossible! Therefore, a Collatz sequence generated from a finite starting number can never diverge to infinity. Further reasoning leads to the conclusion that the Collatz sequence necessarily /always ends up in a closed loop.
Category: Number Theory

[1641] viXra:2305.0153 [pdf] replaced on 2023-05-29 10:01:30

Proof of the Collatz Conjecture

Authors: Henok Tadesse
Comments: 28 Pages.

Take any positive integer N. If it is odd, multiply it by three and add one. If it is even, divide it by two. Repeatedly do the same operations to the results, forming a sequence. It is found that, whatever the initial starting number we choose, the sequence will eventually descend and reach number 1, where it enters an eternal closed loop of 1- 4 - 2 - 1. This has been numerically confirmed to numbers up to 260. This is known as the Collatz conjecture. So far no proof has ever been found that this holds for every positive integer. This problem has been stated by some as perhaps the simplest math problem to state, yet perhaps the most difficult to solve. This paper completely solves this problem by using new insights.
Category: Number Theory

[1640] viXra:2305.0029 [pdf] replaced on 2023-08-15 07:54:51

Syracuse Conjecture Quadrature

Authors: Rolando Zucchini
Comments: 41 Pages.

After circa 2300 years (Circle Quadrature; Archimèdès, Syracuse 287 — 212 BC) the history of mathematics repeats itself in a different problem.The conjecture of Syracuse, or Collatz conjecture, is approached from a completely dissimilar point of view than many previous attempts. One of its features suggests a process that leads to Theorem 2n+1, whose demonstration subdivided the set of odd numbers in seven subsets which have different behaviors applying algorithm of Collatz. It allows us to replace the Collatz cycles with the cycles of links, transforming their oscillating sequences in monotone decreasing sequences. By Theorem of Independence we can manage cycles of links as we like, also to reach very high horizons and when we decide go back to lower horizons. In this article it’s proved that Collatz conjecture is not fully demonstrable. In fact, if we consider the banal link n < 2n, there are eight cycles which connect each other in an endless of possible links. It is a particular type of Circle Quadrature, but its statement is confirmed. In other words: BIG CRUNCH (go back to 1) is always possible, but BIG BANG (to move on) has no End.
Category: Number Theory

[1639] viXra:2305.0029 [pdf] replaced on 2023-07-05 08:04:15

Syracuse Conjecture Quadrature

Authors: Rolando Zucchini
Comments: 40 Pages.

After circa 2300 years (Circle Quadrature; Archimèdès, Syracuse 287 — 212 BC) the history of mathematics repeats itself in a different problem.The conjecture of Syracuse, or Collatz conjecture, is approached from a completely dissimilar point of view than many previous attempts. One of its features suggests a process that leads to Theorem 2n+1, whose demonstration subdivided the set of odd numbers in seven subsets which have different behaviors applying algorithm of Collatz. It allows us to replace the Collatz cycles with the cycles of links, transforming their oscillating sequences in monotone decreasing sequences. By Theorem of Independence we can manage cycles of links as we like, also to reach very high horizons and when we decide go back to lower horizons. In this article it’s proved that Collatz conjecture is not fully demonstrable. In fact, if we consider the banal link n < 2n, there are eight cycles which connect each other in an endless of possible links. It is a particular type of Circle Quadrature, but its statement is confirmed. In other words: BIG CRUNCH (go back to 1) is always possible, but BIG BANG (to move on) has no End.
Category: Number Theory

[1638] viXra:2305.0029 [pdf] replaced on 2023-06-04 09:29:06

Syracuse Conjecture Quadrature

Authors: Rolando Zucchini
Comments: 40 Pages.

After circa 2300 years (Circle Quadrature; Archimèdès, Syracuse 287 — 212 BC) the history of mathematics repeats itself in a different problem.The conjecture of Syracuse, or Collatz conjecture, is approached from a completely dissimilar point of view than many previous attempts. One of its features suggests a process that leads to Theorem 2n+1, whose demonstration subdivided the set of odd numbers in seven subsets which have different behaviors applying algorithm of Collatz. It allows us to replace the Collatz cycles with the cycles of links, transforming their oscillating sequences in monotone decreasing sequences. By Theorem of Independence we can manage cycles of links as we like, also to reach very high horizons and when we decide go back to lower horizons. In this article it’s proved that Collatz conjecture is not fully demonstrable. In fact, if we consider the banal link n < 2n, there are eight cycles which connect each other in an endless of possible links. It is a type of Circle Quadrature, but its statement is confirmed. In other words: BIG CRUNCH (go back to 1) is always possible, but BIG BANG (to move on) has no End.
Category: Number Theory

[1637] viXra:2305.0016 [pdf] replaced on 2024-03-03 19:28:43

Solution Conditions

Authors: Hajime Mashima
Comments: 16 Pages.

For Fermat’s Last Theorem, the condition that holds when there is inverse element.
Category: Number Theory

[1636] viXra:2305.0016 [pdf] replaced on 2024-02-18 19:08:01

Solution Conditions

Authors: Hajime Mashima
Comments: 12 Pages.

For Fermat’s Last Theorem, the condition that holds when there is inverse element.
Category: Number Theory

[1635] viXra:2304.0209 [pdf] replaced on 2023-04-30 08:16:39

Complex Circles of Partition and the Squeeze Principle

Authors: Berndt Gensel, Theophilus Agama
Comments: 12 Pages.

In this paper we continue the development of the circles of partition by introducing the notion of complex circles of partition. This is an enhancement of such structures from subsets of the natural numbers as base sets to the complex area as base and bearing set. The squeeze principle as a basic tool for studying the possibilities of partitioning of numbers is demonstrated.
Category: Number Theory

[1634] viXra:2304.0182 [pdf] replaced on 2023-10-13 23:33:37

Consideration of Collatz Conjecture and Its Integer Space

Authors: Tsuneaki Takahashi
Comments: 4 Pages.

Investigation is tried about approach to Collatz conjecture and its integer space.
Category: Number Theory

[1633] viXra:2304.0181 [pdf] replaced on 2023-08-26 14:27:36

The Randomness in the Prime Numbers

Authors: Ihsan Raja Muda Nasution
Comments: 4 Pages.

The prime numbers have very irregular pattern. The problem of finding pattern in the prime numbers is the long-standing open problem in mathematics. In this paper, we try to solve the problem axiomatically. And we propose some natural properties of prime numbers.
Category: Number Theory

[1632] viXra:2304.0122 [pdf] replaced on 2023-12-29 01:31:57

Solution of the Brocard Ramanujan Equation

Authors: Kurmet Sultan
Comments: 7 Pages. In Russian

The solution of the Diophantine Brocard equation is obtained by proving the impossibility of representing other factorials, except for the known three, as a product of two natural numbers differing by 2. This is justified by the fact that no factorial is greater than 7! cannot be represented as a product of an increasing sequence of natural numbers, the first of which is equal to the argument of the factorial.
Category: Number Theory

[1631] viXra:2304.0122 [pdf] replaced on 2023-12-11 20:47:23

Solution of the Brocard Ramanujan Equation

Authors: Kurmet Sultan
Comments: 16 Pages. In Russian

The solution of the Diophantine Brocard-Ramanujan equation is obtained by proving the impossibility of representing other factorials, except for the known three, as a product of two natural numbers differing by 2.
Category: Number Theory

[1630] viXra:2304.0113 [pdf] replaced on 2023-07-25 20:54:31

Proving the Goldbach Conjecture: Algebraic Proofs and Predicting Prime Numbers

Authors: Oussama Basta
Comments: 4 Pages. A better version

The Goldbach’s Conjecture is an astonishing proposition that stands as one of the most renowned and enduring unsolved problems in number theory and mathematics. This research aims to provide a proof for this remarkable conjecture. The approach to be followed for the proof is yielded by using a predefined system of equations, and with a relatively simple analysis. The proof is quite simple compared to the size of the problem. In the second part of this study, we leverage the same system of equations to develop a general mathematical framework for predicting prime numbers within the known sequence, laying down a general mathematical framework that is computationally concise and can just achieve the objective. With proper selection of the coefficients of the equations in the algorithm, it’s guaranteed that prime numbers are among the outputs. The algorithm consists of basic arithmetic operations which is by itself ground breaking. The proof of the algorithm is also astoundingly straightforward and compact.
Category: Number Theory