| ai.viXra.org > Number Theory > 2504 |
|
 |
| ai.viXra.org > Number Theory > 2504 |
|
|
[12] ai.viXra.org:2504.0099 [pdf] replaced on 2025-05-23 16:54:43
Authors: Christopher David Rice
Comments: 64 Pages. Previous assumptions lifted to continue the formal proofs
We introduce two diagnostic tools for probing the arithmetic structure of elliptic curves over the rational numbers: a canonical summation function based on the N´eron—Tate height, and a height-based entropy index that captures the distributional complexity of rational points. Empirical evidence suggests that the asymptotic behavior of the summation function reflects the rank of the Mordell—Weil group: it remains bounded for rank 0, grows logarithmically for rank 1, and exhibits polynomial growth for higher ranks. We prove that the regularized summation function admits a meromorphic continuation near the critical point s = 1, with a pole of order equal to the rank and a leading Laurent coefficient—denoted Λ(E)—matching the expected arithmetic invariants under the Birch and Swinnerton-Dyer conjecture. The entropy index also increases with rank and may serve as a complexity-based proxy in cases where explicit point enumeration is difficult. Together, these tools form a new analytic framework for investigating the Birch and Swinnerton-Dyer conjecture.
Category: Number Theory
[11] ai.viXra.org:2504.0095 [pdf] submitted on 2025-04-24 20:02:34
Authors: Deskuma [Doe]
Comments: 13 Pages. (Note by ai.viXra.org Admin: For the last time, please use real author name - both first and last name) Supplement to: https://ai.viXra.org/abs/2504.0081)
This document provides structural reinforcement for the symmetry-based formulation of the Riemann Hypothesis (RH), as introduced in the companion paper "Only One Line Knows No Drift." We present theoretical proofs, numerical visualizations, and analytic tools that support the claim that the critical line Re(s) = 1/2 is the unique axis of zero drift in the complex phase of ζ(s). This supplement includes proofs of key lemmas, phase drift models, π-jump structure, and appendices detailing auxiliary methods. All source code and reproducible scripts are available at: https://github.com/Deskuma/riemann-hypothesis-ai* This is a supplementary paper to the main theory paper (https://ai.viXra.org/abs/2504.0081).
Category: Number Theory
[10] ai.viXra.org:2504.0084 [pdf] submitted on 2025-04-22 19:55:53
Authors: Khazri Bouzidi Fethi
Comments: 5 Pages. Assisted by Claude
We present a rigorous proof of the Riemann hypothesis based on the analysis of a new energy function E(σ,t). This approach relies on studying the strict convexity of E(σ,t) and establishes a contradiction in the hypothesis of the existence of non- trivial zeros of the zeta function outside the critical line ℜ(s) = 1/2. Our method combines classical results from analytic number theory with new precise quantitative estimates, leading to a complete proof that all non-trivial zeros of the Riemann zeta function lie on the critical line and are simple.
Category: Number Theory
[9] ai.viXra.org:2504.0081 [pdf] submitted on 2025-04-21 19:13:38
Authors: Deskuma [Doe]
Comments: 23 Pages. (Note by ai.viXra.org: No pseudonym is permited - Please replace article with real author name) 12 figures + 6 appendix visuals. This is a collaborative AI-assisted research exploring a phase-based structural proof of the Riemann Hypothesis.
We propose a symmetry-based formulation of the Riemann Hypothesis (RH) by analyzingthe angular behavior of the Riemann zeta function ζ(s) in the critical strip. Using the phase function θ(t; σ) = 2 arctan(Im ζ/Re ζ), we uncover a geometric structure that isolates thecritical line Re(s) = 1/2 as the unique axis where angular drift vanishes. This phase-basedapproach connects zero-point alignment with rotational symmetry, supported by derivativeanalysis and numerical visualization. We further demonstrate that phase jumps of π coincideprecisely with nontrivial zeros on the critical line, while deviations from σ = 0.5 inducemeasurable drift and symmetry breaking. Our results provide a structural reformulation ofRH: only Re(s) = 1/2 supports drift-free dynamics in ζ(s), implying that nontrivial zerosmust lie on this line. All figures, code, and visualizations are available for replication at the linked repository [https://github.com/Deskuma/riemann-hypothesis-ai].
Category: Number Theory
[8] ai.viXra.org:2504.0075 [pdf] submitted on 2025-04-21 00:42:51
Authors: An Frost
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
This paper introduces a conceptual model based on the traditional Nine-Ring Puzzle toreinterpret the persistent difficulty in proving the Goldbach Conjecture. We hypothesizethat the numerical universe may possess a structural parity—either fundamentally odd oreven—which governs how numerical decompositions can occur. Just as a Nine-Ring Puzzle can only be solved when approached with the correct parity sequence, the conjecture may resist proof because it is being approached from a structurally incompatible direction. If the universe is built upon an odd-parity structure (as with prime numbers), then constructing even numbers from primes is valid. However, attempting to reverse-engineer primes from even numbers may ultimately reach a deadlock—not through an isolated error, but because the entire path is invalid from the outset, a fact that only becomes evident in the final stages. This paper offers not a proof, but a structural-philosophical explanation of the conjecture’s elusiveness.
Category: Number Theory
[7] ai.viXra.org:2504.0066 [pdf] submitted on 2025-04-19 11:15:42
Authors: Daoudi Rédoane
Comments: 14 Pages.
Below I found certain formulas about number theory. The proofs are very complex and I’ll submit them soon. For example there are formulas that require advanced mathematical tools like L’hôpital’s rule, the Residue theorem, Jordan’s lemma, the Cauchy Principal Value, the Dirichlet series expansions, the Wallis product.
Category: Number Theory
[6] ai.viXra.org:2504.0065 [pdf] submitted on 2025-04-19 22:43:50
Authors: Daoudi Rédoane
Comments: 14 Pages.
In this paper I try to proof certain formulas found in my previous paper. I use several mathematical tools like L’hôpital’s rule, the Residue theorem, Jordan’s lemma, the Cauchy Principal Value, the Dirichlet series expansions, the Wallis product.
Category: Number Theory
[5] ai.viXra.org:2504.0061 [pdf] submitted on 2025-04-19 22:35:45
Authors: Dobri Bozhilov
Comments: 11 Pages. Assisted by ChatGPT
This paper presents a novel method for discovering large prime numbers based on symmetric triplets centered on known primes. By extending the idea behind the Goldbach conjecture, we assume that every prime number is surrounded by other primes at equal distances, forming "symmetric triples." The method identifies the most frequent prime gaps, scales them according to the density of the distribution using the Riemann approximation, and applies them around a large, known prime number. In an experimental run on a modest 8-core cloud server, we discovered 20 new 300-digit prime numbers in just 40 seconds. This approach significantly reduces the number of required checks compared to brute-force and offers a practical way to generate guaranteed primes for cryptographic applications. Potential future applications include testing the method on powerful supercomputers or adapting it to Mersenne numbers, which are easier to verify.
Category: Number Theory
[4] ai.viXra.org:2504.0049 [pdf] submitted on 2025-04-14 16:59:51
Authors: Khazri Bouzidi Fethi
Comments: 4 Pages.
We introduce an original geometric interpretation of the Riemann zeta function based on angular transformations of prime number distributions. By establishing a correspondence between primality and complex angular measures, we develop a framework that ofers new insights into the non-trivial zeros of ζ(s). Our numerical investigations reveal distinctive convergence patterns along the critical line σ = 0.5 , suggesting an angular equilibrium condition. While this approach does not constitute a proof of the Riemann Hypothesis, it provides an intuitive geometric lens through which to explore this fundamental problem in number theory. The framework remains fully compatible with established results in analytic number theory while ofering fresh computational perspectives.
Category: Number Theory
[3] ai.viXra.org:2504.0029 [pdf] submitted on 2025-04-11 18:59:24
Authors: Dobri Bozhilov
Comments: 5 Pages.
We present a new prime number generation algorithm based on the symmetry assumption derived from the Goldbach Conjecture. The algorithm significantly reduces the search space by targeting symmetric prime pairs around each number. Using this method, we discovered 10 new 200-digit primes in under an hour on a standard laptop. Theoretical background, pseudocode, and experimental results are provided.
Category: Number Theory
[2] ai.viXra.org:2504.0023 [pdf] submitted on 2025-04-08 18:59:08
Authors: Giovanni Di Savino
Comments: 2 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
We do not have time to verify and we cannot go faster, the AI It has time but, like us, it cannot exceed the speed of light and, like us, it will not be able to intervene in real time on realities taking place at considerable distances such as the landing of the Curiosity space probe on Mars in 2021, with videos and data that were known with a 7-minute delay due to the distance Mars↔Earth.
Category: Number Theory
[1] ai.viXra.org:2504.0005 [pdf] submitted on 2025-04-02 16:21:17
Authors: Dobri Bozhilov
Comments: 5 Pages. https://selfie-church.com/riemann (Note by ai.viXra.org Admin: Please cite and list sceintific references)
We propose a geometric-functional hypothesis supporting the Riemann Hypothesis, grounded in number theory and inspired by the Pythagorean theorem. By treating the complex argument of the Riemann zeta function as a vector in the complex plane, we analyze the modulus of the function in relation to the real and imaginary parts of its input. We argue that only the critical line ℜ(s) = 1/2 yields a balanced vector structure that satisfies both the Pythagorean identity and the necessary conditions for ζ(s) to vanish. Additional reasoning involving vector alignment between known nontrivial zeros and geometric constraints supports the uniqueness of the critical line. The work represents a collaborative exploration between a human researcher and an artificial intelligence (ChatGPT), highlighting a novel approach to one of mathematics’ most profound unsolved problems.
Category: Number Theory