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[10] ai.viXra.org:2508.0082 [pdf] submitted on 2025-08-31 19:27:06
Authors: Hamid Javanbakht
Comments: 23 Pages.
We propose the Habiro Purity Conjecture, a structural statement asserting that Frobenius eigenvalues in Habiro cohomology satisfy a universal purity condition, mirroring Deligne’s theorem for varieties over finite fields. Together with the determinantal formalism of Habiro cohomology, this conjecture would imply the Riemann Hypothesis for the Riemann zeta function and its generalizations.The context is provided by two recent developments. Scholze’s Habiro cohomology furnishes a cohomology theory interpolating between q-de Rham, prismatic, crystalline, and motivic theories, valid across all primes simultaneously and defined over the Habiro completion. In parallel, Garoufalidis—Scholze—Wheeler—Zagier introduced the Habiro ring of number fields, a universal coefficient ring encoding Frobenius compatibilities and regulators from algebraic K-theory.We argue that these two constructions fit together as dual aspects of a single framework: the Habiro ring provides the coefficients, while Habiro cohomology supplies the machinery. The Habiro Purity Conjecture then emerges as the structural principle governing Frobenius actions in this setting, offering a direct arithmetic analogue of Deligne’s proof of the Weil conjectures.
Category: Number Theory
[9] ai.viXra.org:2508.0054 [pdf] submitted on 2025-08-21 12:41:17
Authors: Marco Ripà
Comments: 8 Pages. This mathematics preprint on digit distributions in OEIS A001292 (Kashihara, 1996) gives exact probability formulas for complete and partial blocks. I solved the problem in 2010 and now tested GPT-5 to replicate it.
This work presents the solution of Problem 1 of Question #30 from 'Comments and Topics on Smarandache Notions and Problems', published in 1996 by Kenichiro Kashihara, corresponding to Problem 16 of 'Only Problems, Not Solutions!'. It is a recreational mathematics problem that remained open until 2010, when an extended version was published in Appendix 2 of 'Divisibilità per 3 degli elementi di particolari sequenze numeriche', Rudi Mathematici Bookshelf. The question concerns OEIS sequence A001292 (the so-called "circular sequence") and the probability that a generic element ends with a given digit c∈{0,1,2,...,9}. In 2010, an enhanced version of the problem was studied, providing the general formula of this probability as a function of the length of the last complete block and of c, and using the exact value to bound the interval in which the probability lies in the inter-block (partial block) case, as reported in note 14 on page 17 of the aforementioned work. On August 9, 2025, GPT-5 independently solved the same enhanced version of Kashihara's question, proposing a more compact formula and extending it exactly to the incomplete block case as well.
Category: Number Theory
[8] ai.viXra.org:2508.0043 [pdf] submitted on 2025-08-15 19:37:47
Authors: Bahbouhi Bouchaib
Comments: 32 Pages. New and original article on Goldbach's strong conjecture (Note by ai.viXra.org Admin: Please convert LaTeX codes to regular math expression)
The symmetry—resonance method presented in this work introduces a novel approach to the long-standing Goldbach Conjecture by interpreting prime number distribution through a geometric and harmonic framework. In contrast to classical statistical models such as Cramér’s gap model or asymptotic density predictions from the Hardy—Littlewood conjectures, our method models the set of prime numbers as nodes of constructive interference in a resonance field centered at E/2 for an even number E. Goldbach pairs (p, q) are predicted to occur at symmetric positions equidistant from E/2, where both members of the pair are primes. We define a predictive gap formula [delta(E) approx sqrt{E} cdot frac{log log E}{log E}]that determines the expected offset from E/2 to the nearest prime in each direction. This formula is theoretically consistent with refined gap heuristics and is compatible with known conjectures, yet it adds a localization mechanism that enables prediction of the positions of Goldbach pairs rather than mere existence. Extensive computational tests up to E > 10^{30}, coupled with primality verification for all candidate pairs, confirm the method’s stability and predictive power. Simulations extended theoretically up to 10^{1000} suggest that the resonance—symmetry pattern persists across vast numerical scales. This reinforces the interpretation of Goldbach’s Conjecture as a manifestation of a deeper harmonic order in the primes, potentially reducing its proof to verifying the universality of the resonance law. By bridging additive prime theory, prime gap heuristics, and harmonic models, the symmetry—resonance approach offers a unified predictive framework. If fully formalized, this method could not only resolve the Goldbach Conjecture but also provide insight into other prime constellations, including twin primes and k-tuples.
Category: Number Theory
[7] ai.viXra.org:2508.0031 [pdf] submitted on 2025-08-13 17:07:01
Authors: Predrag Terzić
Comments: 6 Pages. In Italian (Note by ai.viXra.org Admin: Please cite and list scientific references)
We introduce a new primality criterion for Wagstaff numbers.
Category: Number Theory
[6] ai.viXra.org:2508.0027 [pdf] submitted on 2025-08-11 20:30:22
Authors: Bahbouhi Bouchaib
Comments: 26 Pages. New original work on Goldbach's strong conjecture
This work presents a novel approach that combines the geometry of a prime-generating spiral with the analytic structure of the Riemann zeta function to improve predictions for Goldbach pairs. The central idea is to map the non-trivial zeros of the zeta function onto angular positions along a spiral representation of the natural numbers, thereby identifying zones where prime numbers are statistically more concentrated. These zones are then used as "radar sectors" to search for the two prime components (p) and (q) of an even number (E), with (p + q = E). By synchronizing the predicted locations of primes from the spiral model with the fine structure revealed by zeta zeros, the method reduces the candidate search gap significantly compared to classical estimates such as Cramér’s bound and the Hardy—Littlewood conjectures. The methodology is tested on a wide range of even numbers, with verification up to (10^{16}) and theoretical projections toward (10^{18}). Results show that the predicted pairs remain extremely close to the actual pairs, with small gaps between the predicted prime positions and their verified counterparts. The integration of geometric and analytic perspectives offers a new framework for connecting Goldbach’s conjecture to the distribution of primes as seen through the lens of the Riemann Hypothesis. This research opens pathways toward narrowing the bounds for prime gaps, refining predictive algorithms for prime locations, and potentially offering partial progress toward an analytical proof of the strong Goldbach conjecture. Future work will explore extending the spiral—zeta coupling beyond current computational limits and investigating whether these methods retain their predictive power toward infinity.
Category: Number Theory
[5] ai.viXra.org:2508.0024 [pdf] submitted on 2025-08-11 19:27:31
Authors: Kaiche Mouad
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We present a family of 24 quadratic polynomials in two variables that together generate the set of odd natural numbers N, excluding thefirst number of every twin prime pair. The construction of these polynomials is based on a study of triangular numbers and results in overlapping outputs. While this note focuses on presenting the empirical result and the formulas, a complete proof has been developed by the author. The author welcomes collaboration with qualified researchers to assist in formalizing and preparing a full paper for peer-reviewed publication.
Category: Number Theory
[4] ai.viXra.org:2508.0023 [pdf] submitted on 2025-08-10 17:26:45
Authors: Khazri Bouzidi Fethi
Comments: 4 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper introduces a novel hybrid approximation method for the prime counting function π(x) that adaptively uses three different approaches based on the input range. For small values (10 ≤ x ≤ 10³), we employ a simple decimal logarithm formula π̂u2081(x) = x/(2logu2081u2080x). For intermediate values (10³ smaller than x ≤ 10²u2074), we optimize parameters a and b in the rational expression π̂u2092u209au209c(x) = x/(ln x - 1 - a/ln x - b/ln²x) using least squares linearization. For large values (x > 10²u2074), we use the asymptotically optimal logarithmic integral Li(x). The key innovation lies in transforming the nonlinear parameter optimization into a linear least squares problem, enabling efficient and stable coefficient determination. Numerical validation on known exact values demonstrates relative errors consistently below 0.3% across all tested ranges, with computational complexity varying from O(1) for small values to O(log x) for large values.This region-based approach provides an optimal balance between computational efficiency and approximation accuracy, making it suitable for diverse applications in computational number theory, cryptography, and prime number research. The method's hybrid nature ensures both practical usability for moderate computations and theoretical validity for asymptotic analysis.
Category: Number Theory
[3] ai.viXra.org:2508.0020 [pdf] submitted on 2025-08-09 22:45:36
Authors: Ciro Tarini
Comments: 11 Pages. done with the help of Gemini 2.5 pro and GPT-5
Fermat's Last Theorem, conjectured by Pierre de Fermat in 1637, stood as one of the most famous unsolved problems in the history of mathematics. Despite its apparent simplicity, the theorem resisted proof for over 350 years until the monumental work of Andrew Wiles. In this paper, we present an alternative and novel approach that aims to provide an elementary proof of the theorem, based on fundamental algebraic concepts and divided into distinct cases.
Category: Number Theory
[2] ai.viXra.org:2508.0009 [pdf] submitted on 2025-08-04 19:43:01
Authors: Howard Johnson
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present an analytic proof of the Goldbach Conjecture using a hybrid framework that combines symbolic bounds from the Hardy—Littlewood circle method with log-bounded prime pair envelopes centered around each even integer X > 2. We prove that the interval [Xu20442 − Xu2044logᵏX, Xu20442 + Xu2044logᵏX], for k ∈ [1.0, 2.0]always contains at least one prime p such that X − p is also prime for all X ≥ 10u2075, and use symbolic methods to show that r(2n) ≥ ε > 1 for all even integers X > 1000. These results constructively and analytically validate Goldbach’s Conjecture for all even integers X > 2. Figures and symbolic bounds support the completeness of this method.
Category: Number Theory
[1] ai.viXra.org:2508.0002 [pdf] submitted on 2025-08-01 18:40:11
Authors: Bouchaïb Bahbouhi
Comments: 36 Pages. (Note by ai.viXra.org Admin: Please Do Not useg grandiose title/wording)
This study presents a novel geometric approach to prime number distribution using a corrected spiral formula of the form: r(θ) = a·θ + b·sin(k·θ) Unlike traditional linear models, the spiral structure allows for a two-dimensional mapping of natural numbers in polar coordinates, where prime numbers tend to align along harmonic trajectories. We demonstrate that by combining this corrected spiral with estimates derived from the Prime Number Theorem, one can effectively locate approximate positions of the n-th prime. Additionally, we show that Goldbach pairs (p, q) corresponding to even integers E = p + q exhibit a symmetrical alignment on the spiral, reinforcing the conjecture’s plausibility from a geometric perspective.Three key results are established:1. The corrected spiral enables a visual and approximate detection of prime locations.2. Goldbach’s conjecture is visually supported via symmetrical pairing of primes on the spiral arms. 3. A predictive method is proposed to estimate the n-th prime and project it onto the spiral using a hybrid analytical-visual framework.This approach opens a new window for interpreting number-theoretic structures through geometry and may offer intuitive insights into longstanding problems like the Goldbach Conjecture and the distribution of prime gaps. Goldbach decmposition can be perfrmed by the prime SPIRAL up to 105 on this new website https://bouchaib542.github.io/-Goldbach-s-Spiral-Symmetry-Principle/
Category: Number Theory