Exponent-one blockers and a Mordell-Weil construction of Euler bricks

TL;DR

This paper investigates the properties of body cuboids, revealing a prime divisor phenomenon in their parameters and employing elliptic curve techniques to generate and analyze numerous candidate solutions, none of which are perfect cuboids.

Contribution

It introduces a verified prime divisor phenomenon in body cuboids and develops an elliptic curve-based method to generate and analyze extensive candidate solutions.

Findings

Verified exponent-one prime divisor phenomenon in all tested Master-Hits.

Generated over 1.2 million candidate solutions using elliptic curves.

None of the candidates is a perfect cuboid, supporting the open problem.

Abstract

A body cuboid is a rectangular parallelepiped with integer edges and integer face diagonals; if its space diagonal is also integer, it is a perfect cuboid, whose existence is a long-standing open problem. We make two contributions to the study of body cuboids parametrised by two coprime Pythagorean pairs $(a,b)$ and $(m,n)$ in Euclid form (Master-Hits). The first is a verified exponent-one blocker phenomenon: for every Master-Hit, the space-diagonal norm $f_1 := (W_1 U_2)^2 + (U_1 V_2)^2$ admits a prime divisor $\ell$ of exponent exactly one which is coprime to a fixed list of $29$ canonical expressions in the parameters. This is strictly stronger than the existence of any odd-exponent prime divisor: a prime of exponent $3, 5, \ldots$ would obstruct $f_1$ from being a square but carry an extra square factor; the observed obstruction is always primitive. The phenomenon is verified on all $151{,}575$ Master-Hits whose $f_1$ has been fully factorised. Two natural strengthenings fail: the largest outside-parameter prime need not be a blocker, and the smallest outside-parameter blocker need not have exponent one. The second contribution uses the elliptic fibration of the Master-Hit variety over the $(m,n)$-plane. For coprime $(m,n)$ the Master-Hit equation defines a genus-one quartic $H_{m,n}$; a quartic-to-Weierstrass normalisation gives an elliptic model $E_{m,n}$ with a rational function $\tau$ returning $t^2$. Our generator enumerates bounded Mordell-Weil combinations on $E_{m,n}(\mathbb{Q})$, lifts the points satisfying $\tau(P) \in \mathbb{Q}_{>0}^{\square}$ to admissible Euclid pairs $(a,b)$, and certifies each via exact integer arithmetic. From $61{,}829$ classical Master-Hits we generate $1{,}222{,}841$ further ones \"uber $411$ fibres. None of the resulting $1{,}284{,}670$ Master-Hits is a perfect cuboid; all fully factored records satisfy the exponent-one blocker phenomenon.