Perfect cuboids and irreducible polynomials

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 10 000 instances of a, b, and u.

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

“…Theorem 1.1 can be found in [1]. It stems from the results of [2] and [3]. As for the perfect cuboid problem itself, it has a long history reflected in .…”

Asymptotic approach to the perfect cuboid problem

“…The second cuboid conjecture 1.1 and the polynomial Q pq (t) in it were introduced in [1]. They are associated with the problem of constructing a perfect Euler cuboid (see [2] and for more details).…”

A note on the first cuboid conjecture

“…Note that P abu (t) is a polynomial of four variables a, b, u and t. However, in the formula (1.1) it is presented as a univariate polynomial depending on three integer parameters a, b, and u. Relying on this presentation, in [37] the theorem 1.1 was reformulated as follows.…”

“…In [37] the polynomial (1.1) was studied for reducibility and the following special cases were discovered where P abu (t) is reducible:…”

“…(1.4) depending on the integer parameters a, u and p, q. In [37] the reducibility of the polynomials (1.3), (1.4) and the reducibility of the initial polynomial (1.1) were studied numerically and three conjectures were formulated. The main goal of this paper is to prove the following partial result associated with the first cuboid conjecture 1.1.…”