Number Theory

2610 Submissions

[6] viXra:2610.0077 [pdf] submitted on 2026-10-19 01:54:07

A Fourier View of the Riemann Hypothesis

Authors: Jianglin Feng
Comments: 9 Pages.

An elementary proof of the Riemann hypothesis is provided using Fourier analysis on the alternating zeta function η(s). The real series |η(s)|^2, whose zeros are equivalent to the non-trivial zeros, increases along Im(s) in the left half of the critical strip 0<Re(s)<1/2.
Category: Number Theory

[5] viXra:2610.0061 [pdf] submitted on 2026-10-16 01:23:39

Goldbach's Conjecture with Twin Primes Only - Potential Proof

Authors: James DeCoste
Comments: 7 Pages. Contact: jbdecoste@eastlink.ca

Following a structural breakthrough in proving the eternal persistence of twin primes, this paper presents a unified, geometric proof of Harvey Dubner's variation of Goldbach's Conjecture. This variation asserts that every even integer greater than 4,208 can be expressed as the sum of two twin primes.By mapping even integers to a symmetrical, triangular mirror grid, we show that the addition of numbers converts into a physical intersection along a linear center axis. We demonstrate that numbers divisible by 6 act as a structural shield, yielding a regular 2x combination advantage over non-divisible numbers.Using an inductive contradiction modeled after Euclid, and backed by the infinite pool of twin primes established in my prior work, we prove that the creation velocity of row combinations outpaces the logarithmic thinning of the primes. This ensures that the possible combination count never drops back down to zero past the final historical exception of 4,208, cementing the proof.
Category: Number Theory

[4] viXra:2610.0046 [pdf] submitted on 2026-10-13 01:51:56

Optimal Mersenne Primes Search Using Modular Groups

Authors: Jabari Zakiya
Comments: 8 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)

Mersenne Primes have the form Mp = 2^p — 1. They are very rare. As of October 2024 only 52 have been confirmed, the last being the largest known prime. Traditionally, every sequential p value has beenused to compute|test whether 2^p — 1 is prime, requiring first generating a complete sequential list of candidate primes. Presented here is anoptimally efficient and fast process to generate candidate Mp values to test without having to use any primes to directly compute 2^p — 1, usingMersenne Prime Generators (MPG). With them, we can sequentially generate all possible Mp values starting with a p of some known Mp, without any further input or intervention in the search process.
Category: Number Theory

[3] viXra:2610.0037 [pdf] submitted on 2026-10-10 00:48:07

Note to Analytical Derivation of Viswanath's Constant

Authors: Miroslav Súkeník, Magdaléna Súkeníková
Comments: 3 Pages.

In the article, we examine the control functions that enable the analytical derivation of Viswanath's constant.
Category: Number Theory

[2] viXra:2610.0020 [pdf] submitted on 2026-10-06 00:35:46

A Rigorous Proof of the Riemann Hypothesis via Geometric Characterization of the Dirichlet Eta Function: Half-Angle Residual Dynamics and Invariant Level-Crossing Barriers ```[span_0](start_span)[span_0](end_span)

Authors: Ali Rıza Şahin
Comments: 13 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This paper establishes a rigorous real-analytic and geometric framework governing the distribution of the non-trivial zeros of the Dirichlet eta function $eta(s)$ across the critical strip $mathcal{S}={sinmathbb{C}:00$ simultaneously across dual orthogonal projections with an exact $pi/4$ phase shift. We establish that this simultaneous locking condition enforces an overdetermined phase alignment and a structural polarization disparity between the dual channels, providing an absolute real-variable obstruction that strictly confines all non-trivial zeros to the critical line $mathbb{R}(s)=1/2$. By demonstrating that this geometric barrier is mathematically insurmountable, this work establishes a rigorous and definitive proof of the Riemann Hypothesis.
Category: Number Theory

[1] viXra:2610.0007 [pdf] submitted on 2026-10-02 12:49:36

A Sum of Three Squares in Which One is the Product of the Other Two

Authors: Walter A. Kehowski
Comments: 10 Pages.

The identity x^2 + (x+1)^2 + (x(x+1))^2 = (x^2+x+1)^2 is the motivation for study of the equation d^2 + e^2 + n^2 = x^2, where d and e are conjugate divisors of n, that is, n=de and $d1 fall empirically into two families: a triangular family mathscr{T}, and a Fibonacci family mathscr{F}, with virtually no overlap. It is conjectured that the two families comprise all n with |N_{n}|>1.
Category: Number Theory