Existence of a four-dimensional Euler brick

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 four-dimensional rectangular box has positive edge lengths $a_1,a_2,a_3,a_4$. The diagonal in the coordinate face determined by edges $i,j$ has length $\sqrt{a_i^2+a_j^2}$.

Statement recorded in the supplied source.

Do there exist $a_1,a_2,a_3,a_4\in\mathbb N$ and $d_{ij}\in\mathbb N$ for $1\le i<j\le4$ such that \[ a_i^2+a_j^2=d_{ij}^2\qquad(1\le i<j\le4)? \] Construct such a box or prove its nonexistence. [S2]

Short English statement

Find four positive integer edge lengths such that each of the six face diagonals also has integer length, or prove that no such choice exists.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9300001 (AMR-092-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Tim S. Roberts, Unsolved problems in number theory, logic, and cryptography 4DEulerBrick problem statement. https://unsolvedproblems.org/index_files/4DEulerBrick.htm.