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[18] viXra:2509.0158 [pdf] submitted on 2025-09-30 00:00:47
Authors: Theophilus Agama
Comments: 3 Pages.
We develop a criterion for an addition chain to have low energy and pose a related classification problem.
Category: Number Theory
[17] viXra:2509.0155 [pdf] submitted on 2025-09-30 20:57:24
Authors: Fian Qnoz
Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific references)
Diophantine equation of the form a^3+b^3 = 2(2^5 - c^3)relates to the quadratic equation 1 + 4n + 4n^2 which further relates to 1x^3 + 4y^3 + 4z^3 = 512 whose parametric solution for x is exactly is the twice of the square of odd number > 1, i.e. 2(2n+1)^2. Considering phi (sum of the odd divisors of 2(2n+1)^2) is exactly equal to phi (sum of the even divisors of 2(2n+1)^2), some interesting properties involving Piltz functions and Jordan totients were conjectured.
Category: Number Theory
[16] viXra:2509.0152 [pdf] submitted on 2025-09-30 23:19:34
Authors: Theophilus Agama
Comments: 5 Pages.
Let E(n): 1 = s_0 smaller than s_1 = 2 smaller than u2026 smaller than s_h = n, with h on the order of n to the power 1 minus epsilon (for some small positive epsilon), be an addition chain leading to n. We develop a heuristic for the least number of primes in an addition chain for all sufficiently large targets n.
Category: Number Theory
[15] viXra:2509.0146 [pdf] submitted on 2025-09-29 02:09:09
Authors: Aditya Bagchi
Comments: 50 Pages.
This paper presents a complete proof of the conjecture by demonstrating that both of the counterargument scenarios are mathematically impossible. Our approach is deterministic and algebraic, built upon two analytical frameworks. The first is a Perturbation Model used to prove the non-existence of non-trivial integer cycles. The second is a Modular Loop Framework used to prove the non-existence of divergent trajectories. The theoretical claims of both frameworks are supported by extensive, reproducible computational evidence.
Category: Number Theory
[14] viXra:2509.0133 [pdf] submitted on 2025-09-25 09:02:17
Authors: Jérôme Chauvet
Comments: 4 Pages.
We adress here the TPC (Twin Primes Conjecture). Considering the unique growing list of twin primes as elements of a dynamical system with invariant transformation law on, we infer a lower boundary for their density along the real axis incompatible with their finiteness. Although non constructive, since we do not prove a general formula for finding twin primes pairs arbitrarily great, this proof relies for closure on the tertium non datur principle with regard to their minimal density, and thus their cardinality at infinity
Category: Number Theory
[13] viXra:2509.0123 [pdf] submitted on 2025-09-21 03:05:28
Authors: Theophilus Agama
Comments: 14 Pages.
Let E(n): 1 = su2080 < su2081 = 2 < u2026 < s_h = n be an addition chain leading to n in [2^m, 2^{m+1}). We study the distribution of the logarithmic partial sum ∑ log(s_i) on maximal consecutive steps of a given type.
Category: Number Theory
[12] viXra:2509.0114 [pdf] submitted on 2025-09-19 09:25:21
Authors: Marko V. Jankovic
Comments: 4 Pages.
t has been recently explained that Riemann rearrangement theorem is wrong [1], and that it has never been correctly proved. In [1] was demonstrated, on a famous example, that it is not the rearrangement of the elements, but rather the omission of elements of the conditionally convergent series, that would lead to a different summation result.The example that was used in [1] does not strictly follow the Riemann rearrangement method that is proposed in his theorem. It was correctly detected by Google's AI module. In this paper an example that follows the Riemann rearrangement method is going to be presented and again, it is going to be explained that the reason the "rearranged"series has a different sum is the omission of the infinite number of elements of the original series. Generally speaking, the rearrangement method has no critical impact on the summation result — the summation result depends on the sum of elements that are not included in the sum, and that is very simple to understand
Category: Number Theory
[11] viXra:2509.0102 [pdf] submitted on 2025-09-17 14:24:04
Authors: Dmitri Martila
Comments: 3 Pages.
The theta(x) < x + 0.5 ln x implies new major results, i.e., proofs of many conjectures.
Category: Number Theory
[10] viXra:2509.0097 [pdf] submitted on 2025-09-16 17:05:36
Authors: Kuan Peng
Comments: 10 Pages. (impropriate materials removed by viXra Admin)
In this article we have created the table that classifies all primitive triples, shown some properties of basic triples and discussed about the use of primitive Pythagorean triples in cryptography.
Category: Number Theory
[9] viXra:2509.0094 [pdf] submitted on 2025-09-15 08:30:51
Authors: Theophilus Agama
Comments: 9 Pages.
We prove that a certain class of infinite sequences whose finite truncation is an addition chain must have arbitrarily large gaps between their consecutive terms. This result is generic and can be applied to particular known infinite sequences with this property.
Category: Number Theory
[8] viXra:2509.0091 [pdf] submitted on 2025-09-15 20:02:10
Authors: Uggala Guru Jaganath
Comments: 7 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper introduces Jagan’s Primes, a novel recursive modular equation for the deterministic generation of prime numbers in exact order. Unlike traditional sieve-based or probabilistic methods, this framework constructs primes through a self-referential arithmetic logic that isolates non-prime residues via modular shielding. The equation operates without primality testing, leveraging recursive congruence relations to build the prime sequence from first principles. The approach offers a new perspective on prime enumeration, demonstrating that primes can emerge as a consequence of structural recursion rather than exclusion. This work contributes to the foundations of computational number theory and opens pathways for algorithmic applications in cryptography, mathematical linguistics, and symbolic modeling.
Category: Number Theory
[7] viXra:2509.0087 [pdf] submitted on 2025-09-15 20:31:42
Authors: Yonatan Zilpa
Comments: 17 Pages. (Note by viXra Admin: Please cite all listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper explores the zeros of symmetric analytic functions, focusing on the Riemann zeta function. Using the Abel-Plana formula and an auxiliary function, we investigate their distribution. Our approach reformulates the problem algebraically and employs a proof by contradiction. We demonstrate this approach by applying it to the Riemann zeta function.
Category: Number Theory
[6] viXra:2509.0073 [pdf] submitted on 2025-09-12 16:35:30
Authors: Sugahara Jotaro
Comments: 8 Pages. (Note by viXra Admin: An abstract in the article is required; please cite listed scientific referenes and submit article written with AI assistance to ai.viXra.org)
[This paper explores the a rigorous equivalence between Phase Drift and the Riemann Hypothesis]
Category: Number Theory
[5] viXra:2509.0055 [pdf] submitted on 2025-09-09 12:19:39
Authors: Mar Detic
Comments: 7 Pages.
This paper provides a comprehensive analysis of the Diophantine equations 2m2+2m =yn andm(m+2) =yn forintegersm,y ≥ 0andn ≥ 2. Wedemonstrate that the first equation has infinitely many solutions for n = 2 (via a Pell equation) and only the trivial solution for n ≥ 3 (by Erd˝os—Selfridge), while the second has no nontrivial solutions for any n ≥ 2. We explore connections to Fermat’s Last Theorem, the Beal Conjecture, and the ABC Conjecture. Additionally, we show that for odd m = 2k+1, the equation m(m+2) = yn becomes 4(k+1)2−1 = yn, connecting it to arithmetic progressions and Pell-type equations. We demonstrate that attempts to express these equations in Beal form fail, and we highlight the role of discriminants and factorization in determining the existence of solutions
Category: Number Theory
[4] viXra:2509.0037 [pdf] submitted on 2025-09-06 22:46:10
Authors: Rusin Danilo Olegovich
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
We present a complete structural proof of the Riemann Hypothesis, based on the interplay between the canonical well-ordering of nontrivial zeros and the symmetry imposed by the functional equation. Working from three established analytic properties of the Riemann zeta function — discreteness of zeros, confinement to the critical strip, and functional equation symmetry — we construct a proof that reduces the Riemann Hypothesis to a purely combinatorial statement about order and symmetry. We prove the Critical Gap Theorem: if a zero off the critical line exists, the first such zero (under the canonical ordering) must lie to the left of the critical line, forcing its functional equation partner to appear later in the ordering. This leads to a contradiction unless no such zero exists. The result is a logically complete, structurally elegant proof of the Riemann Hypothesis requiring no advanced analytic estimates — only classical properties known since the 19th century.
Category: Number Theory
[3] viXra:2509.0033 [pdf] submitted on 2025-09-05 16:30:58
Authors: Rayan Bhuttoo
Comments: 7 Pages. License: CC BY-NC-ND 4.0 (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We demonstrate that prime numbers are precisely the indecomposable elements under a novel group operation defined on a circle, providing a geometric characterization of primality.By projecting integers onto a circle via the mapping θn = arccos(n/R), we show thatprimes exhibit intense clustering at the endpoints of the diameter, while composite numbers distribute uniformly. We formalize this observation by defining an angular density function F (n) that vanishes if and only if n is prime, with a rigorous proof based on the PrimeNumber Theorem. Furthermore, we analyze the Fourier spectrum of the prime distribution,revealing a distinct high-frequency signature. Finally, we conjecture connections betweenthis harmonic signature and the nontrivial zeros of the Riemann zeta function, suggesting anew approach to understanding prime distribution through geometric and harmonic analysis.
Category: Number Theory
[2] viXra:2509.0008 [pdf] submitted on 2025-09-01 22:53:31
Authors: Ahcene Ait Saadi
Comments: 11 Pages. (Note by viXra Admin: Further repetition may not be accepted; please cite and list scientific references in the article)
This document is entitled (mysteries of prime numbers and magic matrices) Explores the relationships between prime numbers and special matrices. The main objective is to use these matrices to form triples of prime numbers and to establish mathematical conjectures. This work may have common mathematical relationships with the Golbach conjecture and Collatz conjecture. The research is only at is beginnings, I hope that young researchers will be interested in it, and why not draw mathematical theory’s from it.Key words: Prime numbers; Matrices; System of equality, Some of square of prime numbers.
Category: Number Theory
[1] viXra:2509.0005 [pdf] replaced on 2025-09-10 16:08:44
Authors: Jabari Zakiya
Comments: 14 Pages. New content added before publication.
Various bounds on p, such as Bertrand’s Postulate and Legendre’s Conjecture, propose regions around n that have at at least one prime within them. Using Prime Generator Theory, I show more precise symmetric bounds on p, such that for n a prime exists symmetrically within a distance of n^(1/2) below and above it. That is to say, a prime exists for: n — n^(1/2) < p < n and n < p < n + n^(1/2).
Category: Number Theory
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