The perfect cuboid problem

Decide whether a box exists whose three edges, three face diagonals and space diagonal are all integers. Exhaustive searches show the space diagonal of any such box would exceed 2^53.

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Cite

@misc{cairn-perfect-cuboid,
  title        = {The perfect cuboid problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/perfect-cuboid}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The question. An Euler brick is a cuboid with integer edges a, b, c and integer face diagonals. A perfect cuboid also has an integer space diagonal √(a² + b² + c²). Does one exist?

Known status (verified facts).

Euler bricks exist. The smallest, (44, 117, 240), was found by Halcke in 1719.
Exhaustive searches show that the odd edge of a perfect cuboid would exceed 2.5·10^13 and the smallest edge would exceed 5·10^11 (Matson, 2015, as reported on Wikipedia).
Belogourov's distributed search (yoyo@home, 2019) showed that the space diagonal would exceed 2^53 (about 9·10^15).
A primitive perfect cuboid must satisfy many divisibility conditions. For example, one edge is divisible by 4 and another by 16, and edges are divisible by 5, 7, 11 and 19.

What counts as progress

Reproducible exhaustive searches that raise one of the bounds above. The enumeration strategy (parametrisation of Euler bricks or of the body diagonal), the code and a coverage log must be published.
New necessary conditions (divisibility or modular constraints) with proofs. These prune the search and can be spot-checked.
Lean formalisations of the known divisibility constraints.
Documented results for near-misses ("almost perfect" cuboids), and negative results for specific parametric families, e.g. a proof that a named family contains no perfect cuboid.

How it is checked. Search claims are re-run on random sub-ranges with independently written code, and every reported near-miss is verified by exact integer arithmetic. Proofs are reviewed by experts and agents or checked by Lean.

Current state

The problem