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[14] viXra:2410.0177 [pdf] replaced on 2025-03-08 21:34:56
Authors: Imran Ansari
Comments: 21 Pages.
In this article, detailed study on the distribution of the Goldbach prime pairs for the even integers of form $ 2(n + 1)^2 $ were carried out. The experimental proof of the formulated conjectures were given using the algorithm. The results suggest that, there will always be Goldbach prime pair for expression, $ (n)(n-1) < p_{f} < (n+1)^{2} < p'_{f} < 2(n+1)^{2} - (n)(n-1) $, were $ n = 1, 2, 3, ....$ Additionally, the gap between Goldbach first prime pair, that is, $ (p'_{f} - p_{f})$ was found to be always less than its corresponding $n$ value after $n = 2538 $, hence, $Gap(p'_{f} - p_{f}) << n$, where, $n = 2539, 2540, 2541, ....$
Category: Number Theory
[13] viXra:2410.0167 [pdf] replaced on 2024-12-03 21:40:38
Authors: Daniel Gerrad Boyle
Comments: 20 Pages.
This document proves that the Collatz Conjecture is true for the Natural numbers excluding zero. Use is made of the probability distribution of even and odd numbers in supposed diverging Collatz sequences to establish that Collatz sequences do not diverge, having a finite number of terms, and are bounded. Finally proof by contradiction, the pigeon hole principle and proof by induction are used to prove that the Collatz Conjecture is true via two theorems.
Category: Number Theory
[12] viXra:2410.0158 [pdf] submitted on 2024-10-26 17:10:18
Authors: Andrey Belov
Comments: 11 Pages.
The article contains formulas that can specify sequences of both simple and composite numbers. The existence of such formulas proves that the appearance of primes in a series of natural numbers can be predicted. This means that the problem of factoring large numbers can be solved without waiting for the creation of appropriate quantum computing devices.
Category: Number Theory
[11] viXra:2410.0103 [pdf] submitted on 2024-10-18 04:14:07
Authors: Seiji Tomita
Comments: 5 Pages.
It is known that the equation X^4+Y^-(Z^4+W^4) = N has infinitely many rational solutions for any rational number N.We present three new solutions for the equation X^4+Y^4-(Z^4+W^4) = N.
Category: Number Theory
[10] viXra:2410.0098 [pdf] replaced on 2024-10-19 05:14:58
Authors: Naiyf S. Alsaud
Comments: 6 Pages.
A simple and elegant proof of Riemann hypothesis with a deduction of a closed form of the non-trivial zeros of zeta function.
Category: Number Theory
[9] viXra:2410.0097 [pdf] submitted on 2024-10-17 20:36:33
Authors: Mostafa Senhaji
Comments: 12 Pages. In French (Note by viXra Admin: Title and abstract should be in English and no carton/graphics whould be used as part of the headings!))
The Collatz Conjecture, also known as the Syracuse Problem, presents an intriguing challenge in mathematics, stating that for any sequence of positive integers defined by a specific transformation, the sequence always eventually joins the trivial cycle (4.2 ,1). This phenomenon, which seems anecdotal at first glance, is based on fundamental principles of number theory and the dynamics of sequences. This article aims to provide a rigorous proof of the conjecture using a methodical and academic approach, structured around several key axes: 1. Recursion Analysis: We develop a robust recurrence framework to demonstrate that any sequence, under the Collatz transformation rule, converges to the cycle (4,2,1). This framework is based on the systematic observation of the terms of the sequence and their behavior under the repeated application of the rule. 2.Framing of the Terms of the Sequence: The in-depth study of the framing of the terms of the sequence allows us to quantify the tendency towards convergence. By providing precise upper and lower bounds, we illustrate how the terms gradually approach the trivial cycle. 3.Analysis of Behaviors Under Transformation: We analyze the dynamic behaviors of the sequences when they are subjected to the transformation defined by the conjecture. This analysis highlights the underlying regularities and structures that promote convergence. By reintegrating these elements into a rigorous framework, we precisely affirm that the Collatz conjecture is indeed valid for the set of positive integers. This proof not only enriches our understanding of the conjecture but also of the dynamic mechanisms underlying iterative sequences in number theory.
Category: Number Theory
[8] viXra:2410.0096 [pdf] submitted on 2024-10-17 20:36:51
Authors: Mostafa Senhaji
Comments: 17 Pages. In French (Note by viXra Admin: Title and abstract should be in English and no carton/graphics whould be used as part of the headings!))
Goldbach's conjecture, stated by Christian Goldbach in 1742, is one of the most intriguing puzzles in mathematics. Despite centuries of attempts, complete formal demonstration has always escaped the mathematicians. However, thisarticle proposes an innovative approach: aformal proof of the conjecture of Goldbach. The journey begins with a retrospective on the history of conjecture, exploring its origins, previous attempts to resolve it, as well as the advances important theoretical. Next, he explores the contemporary developments and ideasrecent ones that have shed light on this enigma. Finally, the article presents the formal demonstration as expected from the conjecture. Each step is carefully examined, revealing the foundations theoretical and the implications of the results. This mathematical odyssey aspires to inspire next generation of mathematicians and offers a significant contribution to the resolution of an age-old enigma.
Category: Number Theory
[7] viXra:2410.0091 [pdf] submitted on 2024-10-16 19:35:53
Authors: Daniel Eduardo Ruiz
Comments: 4 Pages.
We analyze intervals of numbers whose prime factors are all $> y$. In this article we stablish and prove a lower bound of prime numbers contained in the interval $[x^a,(x+1)^a)$ where $a,x in mathbb{R}^+$ and $x>2$, $a geq 2$, then if $a=2$, the interval $[x^2,(x+1)^2)$ contains at least 1 prime.
Category: Number Theory
[6] viXra:2410.0090 [pdf] submitted on 2024-10-16 19:41:18
Authors: Daniel Eduardo Ruiz
Comments: 5 Pages.
In this paper we prove the fundamental contradiction about the Riemann Hypothesis, expressing a function as a product and given the following summation, where $R$ is the set of all solutions of $R(x)=0$:begin{equation}frac{R'(x)}{R(x)}=sum_{rin R}left(frac{1}{x-r}ight)end{equation}And considering a regularization for hypertranscendental functions, then the expression applied in Riemann Zeta function of $frac{1}{2}$, or the logarithmic derivative, where $R_t$ is the set of tirivial zeros:begin{equation}frac{zeta'(frac{1}{2})}{zeta(frac{1}{2})}eq sum_{rin R_t}left(frac{1}{frac{1}{2}-r}ight)end{equation}
Category: Number Theory
[5] viXra:2410.0063 [pdf] submitted on 2024-10-11 16:38:36
Authors: Laurent Nedelec
Comments: 41 Pages. In French
Since the Syracuse problem has remained unsolved for 80 years, it is possible that it is formally unsolvable. This text offers an « approximate » proof. Using the concepts of inverse graph and an original logical trick (the axis of verified integers), the Collatz conjecture is progressively verified, although no definitive conclusion is reached. Other approaches can build upon the ideas presented in this text to achieve an even more accurate approximation. The text is in French, and an English version is expected to be completed within two months.
Category: Number Theory
[4] viXra:2410.0050 [pdf] submitted on 2024-10-09 07:41:58
Authors: Andrej Liptaj
Comments: 2 Pages.
Assuming the Dirichlet series of $qleft(xight)$ is known, we derive a recursive formula for the Dirichlet-series coefficients of $sqrt{q^{2}left(xight)+alpha}$, $alphainmathbb{C}$.
Category: Number Theory
[3] viXra:2410.0043 [pdf] submitted on 2024-10-08 04:37:19
Authors: Juan Elias Millas Vera
Comments: 1 Page.
In this paper we use the concept of density of a subset of Natural numbers to an stadistical approach to Goldbach Conjecture.
Category: Number Theory
[2] viXra:2410.0016 [pdf] submitted on 2024-10-03 20:32:18
Authors: Theophilus Agama
Comments: 8 Pages.
In this short note, we study the distribution of the $ell$ function, particularly on the primes. We study various elementary properties of the $ell$ function. We relate the $ell$ function to prime gaps, offering a motivation for a further studies of this function.
Category: Number Theory
[1] viXra:2410.0013 [pdf] submitted on 2024-10-03 20:26:48
Authors: Wiroj Homsup, Nathawut Homsup
Comments: 2 Pages. (Note by viXra Admin: Both authors should be listed on the submission form and please don't use all caps for author names!)
We prove that the second smallest Laplacian eigenvalue of a Collatz graph is greater than zero. Thus the Collatz graph is connected.
Category: Number Theory
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