A note on the first cuboid conjecture

TL;DR

This paper advances the understanding of the first cuboid conjecture by showing the associated polynomial has no integer roots, contributing to the broader effort of constructing perfect Euler cuboids.

Contribution

It provides a partial result towards proving the first cuboid conjecture by demonstrating the polynomial's lack of integer roots.

Findings

The polynomial related to the conjecture has no integer roots.

Partial progress towards the irreducibility aspect of the conjecture.

Supports the conjecture by ruling out certain types of solutions.

Abstract

Recently the problem of constructing a perfect Euler cuboid was related with three conjectures asserting the irreducibility of some certain three polynomials depending on integer parameters. In this paper a partial result toward proving the first cuboid conjecture is obtained. The polynomial which, according to this conjecture, should be irreducible over integers is proved to have no integer roots.