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 10000 instances of $a$, $b$, and $u$.

AmSTeX, 8 pages, amsppt style

Cited by in corpus (19)

• A note on the first cuboid conjecture
• A note on the second cuboid conjecture. Part I
• A note on the third cuboid conjecture. Part I
• On an ideal of multisymmetric polynomials associated with perfect cuboids
• Perfect cuboids and multisymmetric polynomials
• A biquadratic Diophantine equation associated with perfect cuboids
• On the equivalence of cuboid equations and their factor equations
• Inverse problems associated with perfect cuboids
• On singularities of the inverse problems associated with perfect cuboids
• On two algebraic parametrizations for rational solutions of the cuboid equations
• A note on rational and elliptic curves associated with the cuboid factor equations
• A note on solutions of the cuboid factor equations
• On two elliptic curves associated with perfect cuboids
• Asymptotic estimates for roots of the cuboid characteristic equation in the linear region
• A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture
• Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture
• A note on invertible quadratic transformations of the real plane
• On a pair of cubic equations associated with perfect cuboids
• A fast modulo primes algorithm for searching perfect cuboids and its implementation