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[12] ai.viXra.org:2506.0131 [pdf] submitted on 2025-06-29 01:22:32
Authors: Bahbouhi Bouchaib
Comments: 32 Pages.
This paper introduces a new predictive algorithm, referred to as Algorithm-T, for addressing the Strong Goldbach Conjecture. The method is based on identifying an optimal distance t from the midpoint N/2 such that both N/2 − t and N/2 + t are prime numbers. The algorithm adapts to the modular class of the even number N (of the form 6x, 6x+2, or 6x+4) and selects t accordingly—using primes for certain forms and multiples of 3 for others. By narrowing the search space around N/2, Algorithm-T offers a highly efficient and structured approach to identifying Goldbach pairs (p, q) satisfying p + q = N. Extensive computational tests show that the algorithm is robust and accurate: no failures are observed up to 106, and only one exception is found up to 109. The results demonstrate a strong correlation between the optimal t-values and the function N/log(N), providing a meaningful connection with classical results such as the Prime Number Theorem. Algorithm-T thus provides a predictive, modular, and computationally light framework for exploring the structure of prime sums. A comparative analysis of the three modular classes—6x, 6x+2, and 6x+4—reveals interesting differences in the behavior and performance of Algorithm-T. Even numbers of the form 6x show the highest success rate and fastest convergence to a valid prime pair, followed by 6x+2. The 6x+4 class, while still highly successful, occasionally requires slightly larger t-values. This variation highlights the subtle role modular structure plays in guiding the algorithm’s effectiveness. This study presents the method in detail, evaluates its empirical success across modular families, and discusses its potential contributions to the understanding and verification of Goldbach-type problems.
Category: Number Theory
[11] ai.viXra.org:2506.0115 [pdf] submitted on 2025-06-26 03:19:00
Authors: Paolo Viscariello
Comments: 16 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We propose an orbital approach to the Riemann Conjecture, introducing a new function ζ_Φ(s) derived from a discrete harmonic field regulated by φ³. We construct a harmonic operator ℛ that transforms ζ_Φ(s) into the classical zeta function ζ(s). The model produces zeros compatible with those known for ζ(s), with mean error <0.01%, and presents a theoretical basis consistent with the golden section, logarithms, and orbital symmetry. We include preface, numerical simulations, technical appendices, and falsifiable predictions.
Category: Number Theory
[10] ai.viXra.org:2506.0106 [pdf] submitted on 2025-06-23 21:02:30
Authors: Shanzhong Zou
Comments: 4 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
Assuming that the hail conjecture is wrong, there is a number set H, and none of which satisfies the conjecture. In H, there must be a smallest number. Because the minimum number cannot be found by the conjecture algorithm, thus proving that H must be empty.
Category: Number Theory
[9] ai.viXra.org:2506.0081 [pdf] replaced on 2025-11-02 11:01:56
Authors: Patrick Kotal
Comments: 18 Pages.
This paper presents a proof of the Collatz conjecture. By analyzing the dynamics of the original Collatz operations within a stochastic process model, we show that they lead to contraction due to a lower bound for the ratio a/b of the counter variables. Then we derive Bit length growth constraints which emerge from the structure of the Collatz process. We finally show that the original Collatz operations applied to any positive integer n>1 can only produce sequences that contract to 1.
Category: Number Theory
[8] ai.viXra.org:2506.0079 [pdf] submitted on 2025-06-18 21:07:25
Authors: Thoriso Motlola
Comments: 2 Pages.
This manuscript introduces and formalizes the concept of *Servitude of Numbers*, a novel mathematical theory exploring the symmetry between distinct arithmetic operations that yield identical results, such as addition equaling multiplication under specific conditions. Building upon this foundation, the work presents several original identities and inverse transformations, proposing new mathematical routes to solving square numbers and modeling numerical relationships. The manuscript also introduces the *Osiroht Theorem*, a theoretical construct describing infinite angular division as a metaphor for infinite dimensional reality, wherein spatial angles converge toward but never reach a defined boundary (180°). Together, these discoveries form a philosophical and mathematical framework that challenges traditional notions of operation, dimension, and numerical behavior. This work contributes to theoretical mathematics, with potential implications for physics, cosmology, and metaphysical interpretation.
Category: Number Theory
[7] ai.viXra.org:2506.0043 [pdf] submitted on 2025-06-09 17:50:06
Authors: Daoudi Rédoane
Comments: 1 Page.
This paper aims to present several formulas concerning number theory. In particular, I establish deep connections between various constants such as Euler-Mascheroni constant, pi, square roots, logarithms, the exponential function, and the gamma function.
Category: Number Theory
[6] ai.viXra.org:2506.0032 [pdf] submitted on 2025-06-07 21:04:43
Authors: Bryan Clem
Comments: 4 Pages. (Note by ai.viXra.org Admin: For the last time, please cite and list sceintific references)
The Bailey Earl Prime Key Conjecture is a proposed mathematical framework that redefines number theory & classification, extends prime identification, and factorization through a structured, self explanatory distribution system, and with infinite Keys. Unlike traditional systems that treat primes as isolated or randomly distributed entities, the Infinite Key System organizes all natural numbers into infinite hierarchical keys. Each number is assigned to the first key where it functions as both a multiple and a prime within that key’s internal logic, establishing a geometric and structural symmetry across the entire number system. At its core, this system eliminates the need for probabilistic or trial-based methods in prime detection and factorization. It offers a method to predict and classify primes to infinity by ensuring that every number finds its proper place in a perfectly organized key structure, where all primes are inherently embedded. This unlocks the potential for deterministic prime prediction, minimal-computation sieving, and refined control over number theory functions. [Truncated by ai.viXra.org Admin]
Category: Number Theory
[5] ai.viXra.org:2506.0031 [pdf] submitted on 2025-06-07 20:57:48
Authors: Seojoon Lee
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
The Collatz conjecture remains one of the most well-known open problems in mathematics. In this paper, we propose a novel framework for analyzing the Collatz sequence using function composition and modular arithmetic. By defining the functions F(x) = 3x + 1 and G(x) = x/2 , and encoding the iterative structure of the Collatz process as a composition of F and G operations.
Category: Number Theory
[4] ai.viXra.org:2506.0023 [pdf] replaced on 2025-06-06 22:09:15
Authors: Hannah McCoy, Stephen Raphael Manning
Comments: 42 Pages.
We introduce a novel modular resonance approach to the Riemann Hypothesis by constructing a modified zeta function, ζ_mod(s), derived from a deterministic sieve of modular prime residues. This function admits full analytic continuation, is defined via a Mellin transform of a modular theta kernel, and forms an Euler product analogue to ζ(s). A real scaling transformation α ≈ 1.0083 establishes a spectral bijection betweenζ_mod(s) and ζ(s), such that: ζ(s) = ζ_mod(αs) for all s ∈ ℂ {1} Using operator analysis, bounded tail convergence, and zero alignment, we demonstrate that the non-trivial zeros of ζ(s) correspond precisely to those of ζ_mod(s), all lying on the critical line Re(s) = 1/2. This constitutes a conditional yet comprehensivesymbolic framework linking classical and modular prime structures through analytic, algebraic, and spectral equivalence. The method provides a potential new path toward a constructive proof of the Riemann Hypothesis and reveals previously unseen resonance patterns in the distribution of primes.
Category: Number Theory
[3] ai.viXra.org:2506.0011 [pdf] submitted on 2025-06-04 00:03:06
Authors: Steve Costello
Comments: 27 Pages.
This paper introduces a novel structural approach to the Collatz Conjecture using arooted binary tree framework based on odd integers. The infinite tree, rooted at 1, encompasses all positive integers and provides a systematic method for establishing parent-child relationships among Collatz predecessors. In particular, a rule-based path construction is defined to resolve ambiguity in the standard inverse Collatz map, especially for even integers of the form 2�� where �� > 2. Thisframework offers new insight into the conjecture’s universal convergence property.
Category: Number Theory
[2] ai.viXra.org:2506.0004 [pdf] submitted on 2025-06-03 01:07:02
Authors: Anatoly Galikhanov
Comments: 23 Pages.
We introduce a pro-étale geometric object D∞ arising naturally from the tower of Artin-Schreier extensions in characteristic 2, equipped with a canonical endofunctor O whose fixed points correspond to automorphic representations of GL2(AF2). The main theorem establishes that invariant predicates on D∞ parametrize cuspidal automorphic representations, preserving L-functions. We provide complete proofs using ∞-categorical techniques, explicit computations for small cases, and establish connections to discrete conformal field theory. As applications, we resolve the Carlitz-Drinfeld uniformization conjecture for function fields and compute previously unknown motivic cohomology groups. Our approach differs fundamentally from coalgebraic models by working internally in topoi and connecting to arithmetic geometry.
Category: Number Theory
[1] ai.viXra.org:2506.0003 [pdf] replaced on 2025-06-03 14:38:10
Authors: Giuseppe Fierro
Comments: 4 Pages.
This article proves the logical equivalence between a conjecture which, as far as we know, has not been previously published, which asserts the existence of at least two primes symmetric about every integer n ≥ 2, and the famous Goldbach Conjecture. The symmetric conjecture states that for every integer n ≥ 2, there exists at least an integer k ∈ [0,n−2] such that both n−k and n+k are prime. Reformulating Goldbach’s conjecture in the form "for every n ≥ 2, there exist primes p,q such that p + q = 2n", we show that the two conjectures are equivalent.
Category: Number Theory