Number Theory

2608 Submissions

[3] viXra:2608.0008 [pdf] submitted on 2026-08-02 22:26:26

Collatz Conjecture and a Discrete Log

Authors: Zhenghao Wu
Comments: 5 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We study finite trajectories of the reduced Collatz map with prescribed $2$-adic exponents. By deriving explicit congruence formulas for the terminal and initial values of such trajectories, we prove that the problem of finding all length-$N$ trajectories that terminate at $1$ reduces to solving a special case of the discrete logarithm problem: $3x+1equiv 2^{m}pmod{3^{N}}$. Consequently, the Collatz conjecture is equivalent to the statement that the arithmetic progressions of starting numbers obtained from the solutions of these discrete logarithm equations tile the odd natural numbers.
Category: Number Theory

[2] viXra:2608.0007 [pdf] submitted on 2026-08-02 00:36:14

Euler Perfect Box (Best Way)

Authors: Taha Muhammad
Comments: 3 Pages.

For centuries, the existence of a Perfect Cuboid—an Euler Perfect Box where the three edges, three face diagonals, and the internal space diagonal are all positive integers—has remained one of the most resilient open challenges in number theory. This manuscript delivers a definitive proof establishing the absolute non-existence of such a system. By decomposing the structural framework into three exhaustive geometric classes (scaloid, isosceles, and equilateral), we implement an original polynomial transformation to isolate unbreakable irrationality barriers. We demonstrate that the arithmetic parameters governing the system are mutually exclusive across all integer domains, proving that an Euler Perfect Box is structurally impossible.
Category: Number Theory

[1] viXra:2608.0002 [pdf] submitted on 2026-08-02 00:16:55

Proof of the Riemann Hypothesis

Authors: Johannes Debouto
Comments: 13 Pages. In French

The field of real numbers is essential to many scientific disciplines. The first formal constructions—which appeared long after the field's initial discovery—involve properties crucial to its modern application, most notably the property of completeness. One of the most famous of these constructions implicitly relies on the Cantor-Dedekind axiom, a concept that has often sparked controversy within the mathematical community. This work aims to demonstrate the validity of the Riemann hypothesis within the axiomatic framework of ZF plus the Cantor-Dedekind axiom, utilizing a result concerning Euler's totient function derived from the study of cyclotomic polynomials.
Category: Number Theory