An Elementary Obstruction to the Existence of a Perfect Cuboid

TL;DR

This paper presents an elementary number-theoretic obstruction to the existence of a perfect cuboid, using divisibility and congruence arguments to rule out possible solutions without advanced algebraic tools.

Contribution

It introduces a novel elementary approach based on prime divisibility and congruences to demonstrate the non-existence of perfect cuboids, avoiding complex algebraic methods.

Findings

Derives structural restrictions on cuboid configurations

Establishes an infinite descent preventing perfect cuboid existence

Provides an elementary proof avoiding Gaussian integers

Abstract

We study arithmetic constraints arising from the three faces meeting along the space diagonal of a rectangular cuboid. Using a propagation mechanism along this diagonal, based on the appearance of a minimal odd prime in certain triangular remainders, we derive strong structural restrictions on possible configurations. These constraints induce an infinite descent along the space diagonal, preventing the existence of a compatible integral structure. This approach provides an elementary obstruction to the existence of a perfect cuboid, relying only on divisibility and congruence arguments, and avoiding the use of Gaussian integers or classical quadratic factorizations.