Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture

TL;DR

This paper develops reverse asymptotic estimates for roots of a key Diophantine equation related to the second cuboid conjecture, aiding in understanding the non-existence of perfect cuboids within a specific subclass.

Contribution

It introduces reverse asymptotic expansions for the roots of the defining equation when the second parameter is fixed and the first tends to infinity, complementing previous results.

Findings

Derived reverse asymptotic estimates for roots with fixed second parameter

Discussed implications for the perfect cuboid problem

Extended understanding of root behavior in the second cuboid conjecture

Abstract

A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The second cuboid conjecture specifies a subclass of perfect cuboids described by one Diophantine equation of tenth degree and claims their non-existence within this subclass. This Diophantine equation has two parameters. Previously asymptotic expansions and estimates for roots of this equation were obtained in the case where the first parameter is fixed and the other tends to infinity. In the present paper reverse asymptotic expansions and estimates are derived in the case where the second parameter is fixed and the first one tends to infinity. Their application to the perfect cuboid problem is discussed.