Quartic reductions and elliptic obstructions for perfect Euler bricks

TL;DR

The paper investigates the longstanding perfect Euler brick problem by translating it into a question about specific quartic equations, developing obstructions via hyperelliptic curves, and computationally verifying no solutions up to a certain range.

Contribution

It reduces the problem to hyperelliptic curves and elliptic obstructions, providing new theoretical insights and computational evidence against solutions within tested parameters.

Findings

No solutions found for parameters up to 10^3

Obstructions on elliptic quotients exclude several families of solutions

Reduction to genus-3 hyperelliptic curves links the problem to advanced number theory

Abstract

We show that the perfect Euler brick (perfect cuboid) problem is equivalent to the following elementary question: do there exist coprime integers $a, b, m, n$ such that the two expressions $(2(a^2-b^2)mn)^2 + ((a^2+b^2)(m^2-n^2))^2$ and $(4abmn)^2 + ((a^2+b^2)(m^2-n^2))^2$ are simultaneously perfect squares? Despite their near-identical structure (differing only in the first summand), no solution has ever been found. We reduce this quartic pair to a one-parameter family of genus-3 hyperelliptic curves $C_A\colon w^2 = \lambda^8 + A\lambda^4 + 1$ and develop obstructions on the distinguished elliptic quotient $E_A$: the Kummer character $\chi_f$ is non-trivial on the 4-torsion, and 2-descent arguments exclude several families of square classes. Computationally, we verify that no solution exists for parameters up to $10^3$. These results do not yet exclude perfect Euler bricks unconditionally; the remaining gap and possible approaches (including a genus-5 covering obstruction and connections to $\mathbb{Q}(\sqrt{2})$) are discussed.