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Level B · Reproducible Hard Number theory P-perfect-cuboid

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.

Status (as of 2026-10-03): Open. Computer searches show any perfect cuboid would need an odd edge above 2.5·10^13 and a space diagonal above 9·10^15.
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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 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

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.