Quartic reductions and elliptic obstructions for perfect Euler bricks

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 $C_A:w^2=\hat{I}{»}^8+A\hat{I}{»}^4+1$ and develop obstructions on the distinguished elliptic quotient $E_A$: the Kummer character $\ddot{I}{‡}_f$ 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.

12 pages, no figures