On a pair of cubic equations associated with perfect cuboids

TL;DR

This paper investigates two cubic equations related to rational perfect cuboids, analyzing their reducibility over rational numbers and identifying six special cases where they are reducible.

Contribution

It introduces a symmetry-based approach to study the cubic equations associated with rational perfect cuboids and finds six special reducibility cases.

Findings

Six special reducibility cases identified

Cubic equations depend on two rational parameters

Analysis advances understanding of perfect cuboid related equations

Abstract

A perfect cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The existence of such cuboids is neither proved, nor disproved. A rational perfect cuboid is a natural companion of a perfect cuboid absolutely equivalent to the latter one. Its edges and face diagonals are rational numbers, while its space diagonal is equal to unity. Recently, based on a symmetry reduction, it was shown that edges of a rational perfect cuboid are roots of a certain cubic equation with rational coefficients depending on two rational parameters. Face diagonals of this cuboid are roots of another cubic equation whose coefficients are rational numbers depending on the same two rational parameters. In the present paper these two cubic equations are studied for reducibility. Six special cases of their reducibility over the field of rational numbers are found.