A fast modulo primes algorithm for searching perfect cuboids and its implementation

TL;DR

This paper introduces a fast modulo primes algorithm to efficiently search for solutions to a complex Diophantine equation related to the long-standing perfect cuboid problem, with implementation details on a 32-bit Windows system.

Contribution

It presents a novel, efficient algorithm leveraging modulo primes for searching solutions to a key Diophantine equation in perfect cuboid research.

Findings

Developed a fast modulo primes sieve algorithm

Implemented the algorithm on 32-bit Windows platform

Enhanced the search efficiency for perfect cuboid solutions

Abstract

A perfect cuboid is a rectangular parallelepiped whose all linear extents are given by integer numbers, i. e. its edges, its face diagonals, and its space diagonal are of integer lengths. None of perfect cuboids is known thus far. Their non-existence is also not proved. This is an old unsolved mathematical problem. Three mathematical propositions have been recently associated with the cuboid problem. They are known as three cuboid conjectures. These three conjectures specify three special subcases in the search for perfect cuboids. The case of the second conjecture is associated with solutions of a tenth degree Diophantine equation. In the present paper a fast algorithm for searching solutions of this Diophantine equation using modulo primes seive is suggested and its implementation on 32-bit Windows platform with Intel-compatible processors is presented.