Asymptotic approach to the perfect cuboid problem

Abstract

The perfect cuboid problem is an old famous unsolved problem in mathematics concerning the existence or non-existence of a rectangular parallelepiped whose edges, face diagonals, and space diagonal are of integer lengths. Recently Walter Wyss has published a paper claiming a solution of this problem. The purpose of this paper is to check out Walter Wyss's result.

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

“…For any two positive coprime integer numbers p = q the tenth-degree polynomial Q pq (t) = t 10 + (2 q 2 + p 2 ) (3 q 2 − 2 p 2 ) t 8 + (q 8 + 10 p 2 q 6 + + 4 p 4 q 4 − 14 p 6 q 2 + p 8 ) t 6 − p 2 q 2 (q 8 − 14 p 2 q 6 + 4 p 4 q 4 + + 10 p 6 q 2 + p 8 ) t 4 − p 6 q 6 (q 2 + 2 p 2 ) (3 p 2 − 2 q 2 ) t 2 − q 10 p 10 (1. 1) is irreducible over the ring of integers Z.…”

“…The scope of perfect cuboids in the case of the second cuboid conjecture is restricted by the following theorem derived from [1].…”

“…In [1] the region given by the inequalities (1.2) was presented as a union of two regions which were called the linear and the nonlinear regions respectively. In this paper we present an algorithm for searching cuboids in the region (1.2).…”