Perfect cuboids and irreducible polynomials

TL;DR

This paper explores the connection between perfect cuboids and the irreducibility of certain polynomials with integer parameters, providing numerical evidence supporting a conjecture about their non-existence.

Contribution

It investigates the irreducibility of specific univariate polynomials related to perfect cuboids and offers numerical verification for numerous parameter instances.

Findings

Numerical evidence supports the irreducibility conjecture.

Irreducibility implies the non-existence of perfect cuboids.

Approximately 10,000 instances verified.

Abstract

The problem of constructing a perfect cuboid is related to a certain class of univariate polynomials with three integer parameters $a$, $b$, and $u$. Their irreducibility over the ring of integers under certain restrictions for $a$, $b$, and $u$ would mean the non-existence of perfect cuboids. This irreducibility is conjectured and then verified numerically for approximately 10000 instances of $a$, $b$, and $u$.