Perfect Cuboid

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A rectangular cuboid has positive edge lengths $a,b,c$, face-diagonal lengths $d,e,f$, and space-diagonal length $g$. A perfect cuboid requires all seven lengths to be positive integers.

Statement recorded in the supplied source.

Do there exist $a,b,c,d,e,f,g\in\mathbb N$ satisfying all four equations \[ a^2+b^2=d^2,\qquad b^2+c^2=e^2,\qquad c^2+a^2=f^2,\qquad a^2+b^2+c^2=g^2? \] No edge length may be zero. [S2]

Short English statement

Is there a rectangular box whose three edges, three face diagonals and space diagonal all have positive integer lengths?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1302 (ALG-006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Aubrey de Grey, Philip Gibbs and Louie Helm, Novel required properties of, and efficient algorithms to seek, perfect cuboids, arXiv:2401.06784v1 (2024), Section 1. https://arxiv.org/pdf/2401.06784.