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[14] viXra:2609.0092 [pdf] replaced on 2026-09-30 04:17:17
Authors: Serge Trudel
Comments: 15 Pages.
This paper presents an alternative recursive piecewise formula that generates all prime numbers only once and in increasing order using only elementary arithmetic operations, square roots and the floor function. The formula does not require previously known prime numbers as input or a precomputed list of primes. Its operation can be interpreted as an algebraic representation of the sieving mechanism underlying the sieve of Eratosthenes through the use of sets of nested equations.The formula is not claimed to improve upon the computational complexity of the sieve of Eratosthenes. Rather, its purpose is theoretical: it provides an explicit and elementary recursive formulation of the sieve mechanism without factorials, trigonometric functions (like the famous 1964 formula of Willans [1], [2] that tautologically defines the prime numbers), or constants whose construction depends on prior knowledge of the primes. The formula was discovered empirically, and no proof of its correctness is provided in this paper.At the end of the article, a comparison is made between prime numbers and lucky numbers, with a similar set of nested equations generating naturally all lucky numbers.
Category: Number Theory
[13] viXra:2609.0088 [pdf] replaced on 2026-09-30 20:31:57
Authors: James DeCoste
Comments: 6 Pages.
This paper presents a new visual framework for understanding the reduction and elimination steps used to solve Catalan's Conjecture, famously known as Mihailescu's Theorem. Rather than relying solely on separate algebraic equations, this approach maps the exponents of the Diophantine equation onto a two-dimensional coordinate grid. By applying established historical milestones as structural filters, we show how the vast majority of the infinite solution space can be systematically scrubbed.First, we utilize Chao Ko's and Lebesgue's proofs to eliminate all even powers from the grid space. Second, we apply the prime power reduction rule to remove composite odd powers, simplifying the problem to unique odd prime exponents. Prime exponents are all that remain. Finally, we demonstrate a domino effect where the systematic removal of a baseline prime automatically filters out all future higher-power multiples.The resulting grid view does not replace the algebraic proofs, but instead organizes them into a cohesive, scannable architecture. This visual model bridges the gap between raw algebra and structural geometry, offering a valuable utility for educational demonstrations and algorithmic filtering tools in number theory.
Category: Number Theory
[12] viXra:2609.0087 [pdf] submitted on 2026-09-29 20:52:29
Authors: Bahbouhi Bouchaib
Comments: 7 Pages.
For an even integer E, Pmin(E) is defined as the smallest prime p for which E-p is also prime. This paper studies Pmin as the first Goldbach synchronization encountered from the small-prime boundary. A longitudinal BOUSSOLE series contains 100 studied scales of E = 10^N + 2 and reaches E = 10^4000 + 2. A separate frozenpredictive model was tested on 2000 new even integers E = 10^N + c, with N = 20,40,...,400 and c = 1002,1004,...,1200. The empirical scale D(E) = log(E) log log(E) was used with Q50 = 0.258D, Q90 = 0.790D and Q95 = 1.088D. Observedcoverage was 51.5%, 79.4% and 85.4%, respectively. The median Pmin/D was 0.24893, close to the frozen median coefficient, while the upper tail was much heavier than predicted. BOUSSOLE therefore uses prediction as a searchcompass, not as a stopping rule. Finally, Goldbach is reformulated as the finiteness of Pmin(E), and the future analytical target is uniform positivity of a boundary counting function R(E,H). The computational evidence is not presented as a proof of Goldbach's conjecture.
Category: Number Theory
[11] viXra:2609.0084 [pdf] submitted on 2026-09-27 21:50:14
Authors: Jose Risomar Sousa
Comments: 35 Pages.
This paper presents new formulae for the harmonic numbers of order k, Hu2096(n), and for the partial sums of two Fourier series associated with them, denoted here by Cᶻu2096(n) and Sᶻu2096(n). I believe this new formula for Hu2096(n) is an improvement over the digamma function, ψ, because it is simpler and stems from Faulhaber's formula, which provides a closed form for the sum of powers of the first n positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for ζ(2k+1), and a new generating function for ζ(2k+1), among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature.
Category: Number Theory
[10] viXra:2609.0083 [pdf] submitted on 2026-09-27 21:49:33
Authors: Jose Risomar Sousa
Comments: 10 Pages.
This paper presents formulae for the sum of the terms of a harmonic progression of order k with integer parameters, HPu2096(n), and for the partial sums of its two associated Fourier series, Cᶻu2096(a, b, n) and Sᶻu2096(a, b, n). HPu2096(n) is built from the ground up, with a power series for 1/(aj + b)ᵏ that is summed over j using Faulhaber's formula. These new formulae are a generalization of the formulae created in a previous paper and were achieved using a slightly modified version of the reasoning employed before.
Category: Number Theory
[9] viXra:2609.0071 [pdf] submitted on 2026-09-25 18:50:32
Authors: Marcin Barylski
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Prime numbers can be represented in many ways. This work focuses on representing them using other primes: p, q, and r as: p×q±r. The aim of this research is to explore interesting properties of such prime representation, including the smallest product of p, q, and r that can represent a given prime number and 4 related hypotheses that were formulated when exploring this topic.
Category: Number Theory
[8] viXra:2609.0060 [pdf] submitted on 2026-09-22 01:02:37
Authors: Maodong Ye
Comments: 16 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper proposes a new representation of integers, revealing significant information regarding their divisibility. Based on this novel integer representation, it provides necessary and sufficient conditions for numbers of the form $n^2+1$ to be prime. Meanwhile, the Goldbach's conjecture has been proven.
Category: Number Theory
[7] viXra:2609.0058 [pdf] submitted on 2026-09-20 13:00:18
Authors: Marko V. Jankovic
Comments: 8 Pages.
In this paper, entropy of natural numbers is going to be introduced. It will be shown that every natural number canbe associated with a family of discrete probability distributions and their entropies. After an introduction of a new Tesla'sentropy measure, a usefulness of the concept is going to be demonstrated by a short proof of Fermat's Last Theorem, basedon a simplified ABC theorem for prime numbers.
Category: Number Theory
[6] viXra:2609.0043 [pdf] submitted on 2026-09-14 20:41:36
Authors: David Adam, Guillaume Estienne
Comments: 10 Pages. In French
Chang introduced Carlitz multiple polylogarithms (CMPs), generalizing in positive characteristic the Carlitz polylogarithms of Anderson and Thakur, and established a linear independence result for their ∞-adic algebraic values, from which follows the function field analogue of Goncharov's direct sum conjecture for multiple zeta values u200bu200b(MZV). Chen showed the analogue of this result in the v-adic setting, under the assumption that the place v is of degree one. The aim of this article is to show that Chen's result is valid for any finite place.
Category: Number Theory
[5] viXra:2609.0030 [pdf] submitted on 2026-09-12 18:08:50
Authors: Piren Mo
Comments: 6 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
A constructive refinement of prime-type classification via nested primorial sievematrices. We introduce a primorial-based additive representation of integers (Mo numbers)and prove a residue separation property: for any prime q, primes in the arithmetic progressionq mod ps# cannot lie outside the nested Mo-type MTq. The Mo-types form a strictly descendinghierarchy MT2 ⊃ MT3 ⊃ MT5 ⊃ MT11 ⊃ ···, and we prove each deepest-type class containsinfinitely many primes (a constructive refinement of Dirichlet’s theorem without analytic densityestimates).We further classify twin prime candidates into coupled types such as MT(5,7) and MT(11,13),showing that the Twin Prime Conjecture is equivalent to the divergence of true twin pairs withinMT(5,7). As a rigorous corollary, Zhang’s bounded-gap theorem admits a Mo-type formulation:some even gap N ≤ 246 yields infinitely many prime pairs in a coupled Mo-type progression.The framework reframes classical distribution problems as explicit sieve-matrix counting,leaving open the quantitative density of twin pairs for future work
Category: Number Theory
[4] viXra:2609.0027 [pdf] submitted on 2026-09-11 20:47:48
Authors: Agum Abdillah
Comments: 40 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)
This work develops a theta-geometric framework for studying the Hankel positivity problem associated with the Riemann Hypothesis.Starting from the centered Riemann xi-function, the framework exploits the even spectral symmetry to introduce a folded spectral coordinate and a meromorphic resolvent whose poles encode the squared centered zeros. Passing to reciprocal spectral variables leads to a canonical moment sequence whose Stieltjes moment property is characterized by the positivity of two associated families of Hankel matrices.The framework then connects this spectral-moment structure with the theta-function representation of the xi-function. A tilted theta measure is introduced, yielding an exact expectation representation for the folded resolvent and a differential hierarchy governing its derivatives. The resulting analysis leads to an adjoint differential operator and a two-variable kernel representation, together with an exact inhomogeneous differential identity in which boundary contributions are retained explicitly.The central outcome is an exact reduction of the spectral question to an all-orders Hankel positivity problem. The work distinguishes rigorously established identities and equivalences from numerical diagnostics and does not claim a proof of the Riemann Hypothesis. Instead, it provides a structured theta-geometric and moment-theoretic framework for investigating the remaining positivity problem.
Category: Number Theory
[3] viXra:2609.0024 [pdf] submitted on 2026-09-11 20:36:06
Authors: Pawel Jan Piskorz
Comments: 4 Pages.
We propose a procedure which allows to determine the only admissible exponents in Diophantineequations of three kinds: Fermat, Markov and Catalan. First, we use a set of rewriting rules for each Diophantine equation. Second, we obtain the parametrized versions of the Diophantine equations. Next, we differentiate each of parametrized equations obtaining the identities for Fermat, Markov and Catalan equations.
Category: Number Theory
[2] viXra:2609.0022 [pdf] submitted on 2026-09-08 06:47:50
Authors: Yong Zhao
Comments: 4 Pages.
For any integer n greater than 1 such that 2n+1 is prime, we construct interval (n,2n). By synthesizing the prime number theorem, the Bertrand—Chebyshev theorem and Euler's proof to analyze the gap between 2n+1 and primes in the interval (n,2n), we prove the Polignac's Conjecture.For any integer greater than 1, we construct intervals (n,2n) and (2n,4n). By synthesizing the Bertrand—Chebyshev theorem and the prime number theorem to analyze the gap of primes between the consecutive intervals, we prove the Polignac's Conjecture.
Category: Number Theory
[1] viXra:2609.0016 [pdf] submitted on 2026-09-07 18:15:38
Authors: Min Min Oo
Comments: 19 Pages. (Note by viXra Admin: Please cite and listed scientific references!)
This paper develops a Prime Pair Framework for studying twin primes using structured sequences of odd composites. The base sequenceO_1=9+6r_1, r_1≠((1+2x)^2-13)/6 " or" ((1+2x)^2-7)/6 provides intervals containing two odd integers, which may form twin primes if both are prime. When the general sequencesO_a=(2a+1)^2+(2+4a) r_a (a={2ⓜ,3ⓜ,4ⓜ,u2026ⓜ,k})and O_1are in proper alliance, the conditions are formed through lower, upper and double twin prime formation mechanisms.
Category: Number Theory
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