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[30] viXra:2504.0189 [pdf] submitted on 2025-04-29 10:21:58
Authors: Julian Beauchamp
Comments: 3 Pages.
In this paper, we observe some nice identities for the products of Fibonacci-like sequences. While these identities can hardly be original discoveries, I have been unable to find them elsewhere.
Category: Number Theory
[29] viXra:2504.0174 [pdf] submitted on 2025-04-28 20:15:27
Authors: Lautaro Fesembeck
Comments: 10 Pages. Correspondence: lautaro.math@gmail.com
We model the distribution of nontrivial zeros of the Riemann zeta function through a dynamic equilibrium principle. By defining a disturbance field associated with prime distributions and constructing a corresponding global energy functional, we show that any deviation from the critical line Re(s) = 1/2 necessarily increases global energy. Through analysis of local perturbations, global independence, and symmetry properties implied by the functional equation of ζ(s), we demonstrate that only the critical line configuration minimizes total energy. This provides a new and rigorous resolution of the Riemann Hypothesis via energy minimization methods.
Category: Number Theory
[28] viXra:2504.0173 [pdf] submitted on 2025-04-27 20:05:31
Authors: Christian I. G. Winsor
Comments: 10 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper presents a new approach to understanding the Collatz Conjecture. The conjecture asks whether a simple process (repeatedly halving even numbers, and tripling odd numbers then adding one) will always eventually reach the number one, no matter which positive whole number you start with. In this work, I introduce a way to group numbers based on their properties and show that, by following a specific set of steps, every number can be reduced to a smaller group. By combining this method with results that have already been checked by computer for smaller numbers, I provide a logical framework that supports the idea that the Collatz process always ends at one.
Category: Number Theory
[27] viXra:2504.0164 [pdf] submitted on 2025-04-26 12:50:22
Authors: Immense Raj Subedi
Comments: 5 Pages.
The Collatz Conjecture states that every positive integer eventually reaches 1 through a specific iterative process. This paper presents a novel approach to proving the conjecture by categorizing natural numbers and establishing a key mapping between odd numbers and numbers of a particular form. This structural approach simplifies the problem and provides a comprehensive proof of convergence
Category: Number Theory
[26] viXra:2504.0160 [pdf] replaced on 2025-11-16 00:10:23
Authors: Abdelmajid Ben Hadj Salem
Comments: 7 Pages. The proof is corrected.
In this paper, we assume that the explicit abc conjecture of Alan Baker and the conjecture c [smaller than] R^{1.63} are true, we give a proof of the abc conjecture and we propose the constant K(epsilon). Some numerical examples are provided.
Category: Number Theory
[25] viXra:2504.0141 [pdf] submitted on 2025-04-21 20:54:23
Authors: Jochen Kiemes
Comments: 7 Pages.
We present a novel reformulation of the Collatz conjecture by leveraging the binary structure of positive integers, focusing on the sequence of odd terms. Through an analysis of leading and trailing bit-position dynamics, we derive a substantial lower bound of at least 17,026,679,261 steps for any hypothetical non-trivial cycle, offering new insights into its structural constraints.
Category: Number Theory
[24] viXra:2504.0136 [pdf] replaced on 2025-05-01 17:22:42
Authors: V. Barbera
Comments: 6 Pages.
This paper presents some considerations on the stopping time of the 3n+1 problem. In particular, it presents an algorithm for finding residue classes that have a given stopping time.
Category: Number Theory
[23] viXra:2504.0134 [pdf] submitted on 2025-04-20 06:54:44
Authors: Chenglian Liu, Sonia Chien-I Chen, Ruopengyu Xu
Comments: 2 Pages.
This paper reveals a profound mathematical cascade linking three classical number theory phenomena: 1) Bernoulli numbers $B_{n}$ with denominator $6$ ($n equiv 2 pmod{6}$), 2) Special values of Riemann $zeta$-function at even integers, and 3) Enhanced Goldbach partition counts for $x equiv 0 pmod{6}$. We demonstrate their intrinsic connections through von Staudt-Clausen theorem, modular form theory, and statistical verification ($n leq 10^4$, $x leq 10^4$). A $3.2$times$ enhancement ratio in Goldbach partitions emerges as direct consequence of prime number symmetry modulo $6$.
Category: Number Theory
[22] viXra:2504.0133 [pdf] submitted on 2025-04-20 06:57:07
Authors: Chenglian Liu, Sonia Chien-I Chen, Ruopengyu Xu
Comments: 2 Pages.
This paper establishes a rigorous connection between modular arithmetic constraints and enhanced Goldbach partition counts through dual-channel prime pair combinations. We demonstrate that even numbers $ x equiv 0 pmod{6} $ exhibit $3.2times$ higher partition counts than $ x equiv 2 pmod{6} $ due to symmetric prime distribution modulo 5. The mechanism is visualized through a novel combinatorial diagram (Fig. 1) and supported by statistical analysis ($ x leq 10^4 $).
Category: Number Theory
[21] viXra:2504.0128 [pdf] submitted on 2025-04-19 08:36:11
Authors: Chenglian Liu, Sonia Chien-I Chen, Ruopengyu Xu
Comments: 2 Pages.
This paper establishes a novel connection between two classical number theory phenomena: 1) Bernoulli numbers $B_n$ with denominator $6$ ($n equiv 2 pmod{6}$) governed by the von Staudt-Clausen theorem, and 2) the enhanced Goldbach partitions for even numbers $x equiv 0 pmod{6}$. We demonstrate their complementary modular symmetry through analytic number theory tools and computational verification. A unified framework is proposed using Rankin-Selberg convolution of modular forms, revealing shared sieve-theoretic mechanisms in prime number distribution.
Category: Number Theory
[20] viXra:2504.0127 [pdf] submitted on 2025-04-19 08:38:59
Authors: Chenglian Liu, Sonia Chien-I Chen, Ruopengyu Xu
Comments: 2 Pages.
This paper establishes a deep connection between two classical number theory phenomena through modular form-L-function unification: 1) Bernoulli numbers $ B_n $ with denominator 6 ($ n equiv 2 pmod{6} $) governed by von Staudt-Clausen theorem, and 2) enhanced Goldbach partition counts $ G(x) $ for even numbers $ x equiv 0 pmod{6} $. We demonstrate their complementary modular symmetry via:begin{itemize} item Rankin-Selberg convolution of weight-1/weight-2 modular forms item Analytic continuation of associated L-functions item Computational verification ($ n leq 10^4 $, $ x leq 10^4 $)end{itemize}The unified framework reveals that $68.2$% of Bernoulli denominators and $79.4$% of Goldbach enhancements obey modular arithmetic constraints.
Category: Number Theory
[19] viXra:2504.0125 [pdf] replaced on 2026-05-04 13:57:40
Authors: Piren Mo
Comments: 35 Pages.
We found that all prime numbers can be expressed in a common new form. Which serves as a necessary but not sufficient condition for a number to be prime.And based on this expression, we have studied the distribution of prime numbers and twin primes, and we are able to predict primes within a certain interval following known primes.
Category: Number Theory
[18] viXra:2504.0118 [pdf] submitted on 2025-04-17 20:15:36
Authors: Aditya Bagchi
Comments: 21 Pages.
This paper introduces a deterministic framework for validating the conjecture by classifying integers into distinct types based on modulo 16 residues. Positive odd integers are expressed as 16k+m, where m∈{1,3,5,7,9,11,13,15}, representing Types 1 through 8. Positive even integers are expressed as 16k+mu2032, where mu2032∈{0,2,4,6,8,10,12,14} representing EV1 through EV8.The paper considers even numbers as intermediates between two successive odd integers in the Collatz sequence. Under the 3x+1 operation, odd types exhibit distinct divisibility factors (d) that govern their transformations. For instance:AType 1 transforms into Types 1, 3, 5, or 7.B) Type 2 transforms into Types 3 or 7.C) Types 3 and 7 can transform into any odd type.D) Type 4 transforms into Type 2 or 6.E) Type 5 transforms into Type 2, 4, 6 or 8.F) Type 6 transforms into Type 1or 5.G) Type 8 transforms into Type 4 or Type 8 further.Depth First Search (DFS) algorithms identify 911 looping sequences, of which 49 are increasing, and the rest are decreasing. All looping sequences are shown to terminate within finite cycles, and transformations converge universally to 1. The conjecture’s universality is established by proving non-existence of infinite looping and unbound growth. The pigeonhole principle comes into play.
Category: Number Theory
[17] viXra:2504.0112 [pdf] submitted on 2025-04-18 17:06:40
Authors: Victor Sorokine
Comments: 2 Pages.
The number A + B - C contains unnecessary factor.
Category: Number Theory
[16] viXra:2504.0110 [pdf] submitted on 2025-04-16 01:52:04
Authors: Jose Acevedo Jimenez
Comments: 3 Pages.
In this article, it is proven that for every n≥3, there exists at least one artificial prime number q such that F_n<q<F_2n , where F_k denotes the k-th number in the Fibonacci sequence. This result is obtained using the Bertrand—Chebyshev theorem and relies on a fundamental property of divisibility within the Fibonacci sequence. Although it does not imply the infinitude of classical primes in the sequence, it does guarantee the existence of infinitely many artificial primes distributed within it.
Category: Number Theory
[15] viXra:2504.0097 [pdf] replaced on 2025-10-27 23:58:58
Authors: Wing K. Yu
Comments: 13 Pages.
This paper improves my previous proof of the Legendre conjecture by reducing some redundant statements, improving some corollaries, and simplifying two data tables.
Category: Number Theory
[14] viXra:2504.0096 [pdf] replaced on 2025-04-18 01:19:02
Authors: Zhenghao Wu
Comments: 21 Pages.
We provide a general analytic formula to construct all existing starting odd numbers that obey our desired finite arbitrarily long Collatz trajectory, meaningthat these starting odd numbers obey our pre-designated maximum factors of 2 at each iteration of the reduced Collatz map. We also provide another generalanalytic formula for finding the resulting odd numbers after N iterations of the reduced Collatz map. These formulas shed light on the structure of Collatztrajectories and other properties. We can also use this information to find in finite steps all existing Collatz trajectories that become 1 after any finite N iterations.We also will see that the "location" of all of the 1’s in Collatz Conjecture can be found by solving a special case of the discrete log problem.
Category: Number Theory
[13] viXra:2504.0090 [pdf] replaced on 2025-04-22 12:11:40
Authors: Jayme Mendes
Comments: 10 Pages.
This article presents an algorithm for efficiently generating all prime numbers within the interval $[m,n]$, where $mgeq 3$. The algorithm is developed from the demonstration that non-prime numbers in this range can be obtained from certain points in the region between two rectangular hyperbolas and two straight lines. The method does not perform factorization tests and does not require prior knowledge of any prime number, which makes it easier to obtain large primes for $n-m=q=constant$, when the time and memory complexities become equal to ${cal O}(log n)$ and ${cal O}(1)$, respectively.
Category: Number Theory
[12] viXra:2504.0079 [pdf] submitted on 2025-04-12 22:15:36
Authors: Natalia Tanyatia
Comments: 16 Pages. https://github.com/NataliaTanyatia/Optimal-Prime.git (Note by viXra Admin: The article should start with article title, author name and abstract; Please submit article written with AI assistance to ai.viXra.org)
We construct a unified symbolic and geometric framework that links the recursive generation of prime numbers to the problem of closest hypersphere packing in Euclidean space. Beginning with a purely logical definition of primes and building an iterative formula that filters primes based on modular constraints, we establish a symbolic system for exact prime counting and approximation. We then transition from arithmetic to geometry by introducing sphere-packing principles in various dimensions, particularly focusing on both furthest-touching and closest-touching configurations. By analyzing simplex-based Delaunay lattices and maximizing local sphere contact, we show how prime indices emerge naturally as layers in the radial expansion of optimally packed lattices. This construction culminates in a symbolic proof of the Riemann Hypothesis by bounding the prime counting function with a geometric analogy. The result is a cohesive theory in which logical prime filtration, packing density, and analytic continuation of Dirichlet series converge in a single constructively grounded model.
Category: Number Theory
[11] viXra:2504.0076 [pdf] submitted on 2025-04-11 18:30:49
Authors: Jose Acevedo Jimenez
Comments: 6 Pages.
This paper introduces the concept of artificial primes, defined within a specific subset SSS of positive integers greater than one. A number q∈S is considered an artificial prime if no other element d∈S, with d≠q, divides q. Focusing on the subset of Fibonacci numbers greater than 1, we analyze the behavior of artificial primes in this sequence. Remarkably, the counting function of artificial primes among the first n Fibonacci numbers (with n≥3) matches the classical prime counting function π(n), which enumerates the number of primes less than or equal to n. This correspondence highlights a surprising structural parallel between classical prime distribution and internal divisibility properties within recursive numerical sequences.
Category: Number Theory
[10] viXra:2504.0072 [pdf] replaced on 2025-08-20 20:27:57
Authors: Rosario D'Amico
Comments: 12 Pages. https://dx.doi.org/10.23755/rm.v54i0.1662 keywords: Goldbach' Conjecture, Prime numbers, Probability
This paper aims to provide a set of considerations that allow us to see a possible solution to the problematic issue of Goldbach's "strong" conjecture, which amounts to asserting that any even natural number greater than 2 can be written as the sum of two prime numbers that are not necessarily distinct. Specifically, we will show mathematically that a hypothetical scenario in which no even composite number exists as a sum of two primes is impossible. This will be done by adopting a probabilistic method much simpler than the arithmetical attempts already present in literature.
Category: Number Theory
[9] viXra:2504.0065 [pdf] replaced on 2025-04-14 00:54:42
Authors: Samuel Bonaya Buya
Comments: 20 Pages.
This paper investigates the relationship between prime gaps and the Riemann zeta function, focusing on the stringent conditions under which the Riemann Hypothesis (RH) holds and the circumstances under which it is falsified. Through the analytic continuation of primes, we derive an exact prime gap theorem and an alternative formulation of the zeta function. A key result reveals that the zeta function ��(log �������� + ����) generates infinite number of zeroes outside the critical strip. Other result reveals that a zero is generated independently of ��, providing a potential counterexample to RH. This challenges the assumption that all non-trivial zeta zeros lie on the critical line ℜ(��)=1 2 . Numerical analysis supports thetheoretical framework, demonstrating that prime gaps and zeta zeros are deeply interconnected. These findings suggest that while RH is useful in number theory, it cannot be an absolute truth, requiring a revised understanding of prime number distribution.
Category: Number Theory
[8] viXra:2504.0045 [pdf] submitted on 2025-04-06 22:20:23
Authors: Abdelmajid Ben Hadj Salem
Comments: 5 Pages. Submitted to the journal Bulletin of the London Mathematical Society. Comments welcome.
In this paper, we consider the abc conjecture. Assuming that the conjecture c<rad^{1.63}(abc) is true, we give the proof that the abc conjecture is true.
Category: Number Theory
[7] viXra:2504.0027 [pdf] submitted on 2025-04-04 22:07:40
Authors: Gabriele Bonamini
Comments: 9 Pages. (Note by viXra Admin: Please cite and list scientific references; Please submit article written with AI assistance to ai.viXra.org)
In this paper, we introduce and develop the concept of "the terminator of a sequence", defined as the term that would be reached by extending the construction of a given infinite succession indefinitely. Unlike the traditional limit, whose value may be abstract or not defined in the target domain, the terminator is an actual element of the sequence.This approach leads to the introduction of new operators which formalize the ideas of "infinity by construction" and "infinitesimal by construction", respectively, which allows to re-design infinite and infinitesimal quantities as concrete mathematical entities that arise directly from the sequence’s construction, rather than as mere abstract mathematical concepts.
Category: Number Theory
[6] viXra:2504.0021 [pdf] submitted on 2025-04-03 05:37:52
Authors: Song Fei
Comments: 24 Pages.
Abstract This paper introduces the Langlands Watch-Tate (LW-Tate) framework, an extension of the Langlands Watch (LW) framework first proposed in [1] , to prove the Tate Conjecture for all K3 surfaces over mathbb{Q} . We establish that ensuremath{text{rank}text{Pic}(X)=text{ord}_{s=1}L(H^{2}(X),s)} holds universally, covering both finite and infinite automorphism groups, by decomposing H^{2}(X_{overline{mathbb{Q}}},mathbb{Q}_{ell}(1)) into irreducible representations under text{Aut}(X) and associating each with weight 2 automorphic forms on Shimura varieties. Building on LW’s hierarchical structure, LW-Tate’s novel integration of symmetry and modularity resolves a major conjecture in arithmetic geometry. Furthermore, we extend LW-Tate to Calabi-Yau threefolds , explaining text{ord}_{s=2}L(H^{3}(Y),s)=text{rank}text{Pic}(Y) , showcasing its potential to address higher-dimensional Tate Conjectures and cementing its role as a transformative tool in the Langlands Program.
Category: Number Theory
[5] viXra:2504.0014 [pdf] submitted on 2025-04-02 21:00:51
Authors: Bahbouhi Bouchaib
Comments: 14 Pages.
This article presents two methods A and B for breaking an even number into two primes. Both methods are inspired by the equations 6x ± 1 because all prime numbers and their multiples except 3 are 6x ± 1. Both methods can be very useful for even conversion in sums of two primes or to study the Goldbach's strong conjecture. Both methods can have applications in computer science.
Category: Number Theory
[4] viXra:2504.0011 [pdf] submitted on 2025-04-01 02:16:52
Authors: Muhammad Ahmad Abdullah
Comments: 41 Pages.
This paper seeks to explore the nature of primality by investigating relationships between prime numbers. For centuries, mathematicians have been intrigued by the unpredictable distribution of primes. The Seven-Set Prime Number Theorem (SSPNT) offers a novel approach to address this challenge. By building on Ulam's Spiral, SSPNT uses simple sets and visual patterns to gain insights into primality. This methodology not only provides a fresh perspective on prime number theory but also has potential applications in coding theory and cryptography, revealing hidden patterns and underlying structures.
Category: Number Theory
[3] viXra:2504.0010 [pdf] submitted on 2025-04-01 21:20:04
Authors: Minjia Shi, Lu Wang, Patrick Solé
Comments: 12 Pages.
A criterion for Lehmer's conjecture in terms of the spherical designs held in the shells of the lattice E8 was derived by de La Harpe, Pache and Venkov circa 2005. We check that this criterion is satisfied by combining spherical designs, harmonic polynomials, weighted theta series, and Deligne's bound on the modulus of the τ function.
Category: Number Theory
[2] viXra:2504.0007 [pdf] submitted on 2025-04-01 20:55:54
Authors: Samuel Bonaya Buya
Comments: 5 Pages. Original article.
In this research prime gaps will be investigated through their zeta behaviour. A formulation will be presented that links prime gaps to singularities in ��(��). This is achieved byidentifying a zeta function for Goldbach partition and extending it to the Euler product. A zeta function is formulated that encodes information about Goldbach partitions. We beginthe paper by examining the logarithmic form of a complex variable and it’s decompostion to real and imaginary parts.
Category: Number Theory
[1] viXra:2504.0003 [pdf] submitted on 2025-04-01 21:35:51
Authors: Madhukar Jadhav
Comments: 2 Pages.
A perfect number is a positive integer N that equals the sum of its proper divisors. While even perfect numbers have been classified, theexistence of an odd perfect number remains an unsolved problem. In this paper, we establish a fundamental contradiction in the divisor structure of any hypothetical odd perfect number. Specifically, we demonstrate that the largest proper divisor must be half of N, but for an odd N, this results in a non-integer, violating the necessary conditions for perfection. Consequently, we conclude that no odd perfect number can exist.
Category: Number Theory
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