[#P14] Perfect cuboid problem

Problem. A perfect cuboid exists: there are positive integers \(a,b,c\) for which the four numbers \(\sqrt{a^2+b^2}\), \(\sqrt{a^2+c^2}\), \(\sqrt{b^2+c^2}\), and \(\sqrt{a^2+b^2+c^2}\) are integers.

Cuboid with edges, face diagonals, and space diagonal.

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Definitions and notation

1Context

Euler bricks exist, but no example is known whose space diagonal is also integral.

2Problem setup

Definition 1 (A perfect cuboid). A perfect cuboid is a rectangular box whose three edge lengths, three face diagonals, and space diagonal are positive integers.

Remark 1. An Euler brick has integer edges and face diagonals, with no requirement on the space diagonal.

3What counts as a solution

Exhibit positive integers satisfying all four square conditions, or prove that no positive integer triple can satisfy them.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Euler bricks provide positive integer edges and all three integral face diagonals. The integral space diagonal remains the missing condition. Exact unresolved remainder: Find positive integers a,b,c for which all three face diagonals and the space diagonal are integers, or prove no such triple exists.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Euler bricks provide positive integer edges and all three integral face diagonals. The integral space diagonal remains the missing condition. Exact unresolved remainder: Find positive integers a,b,c for which all three face diagonals and the space diagonal are integers, or prove no such triple exists.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Euler bricks provide positive integer edges and all three integral face diagonals. The integral space diagonal remains the missing condition. Exact unresolved remainder: Find positive integers a,b,c for which all three face diagonals and the space diagonal are integers, or prove no such triple exists.^[1] Relevance to this problem For Perfect cuboid problem, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Euler bricks provide positive integer edges and all three integral face diagonals. The integral space diagonal remains the missing condition. Exact unresolved remainder: Find positive integers a,b,c for which all three face diagonals and the space diagonal are integers, or prove no such triple exists. Open full record →
Exhibit positive integers satisfying all four square conditions, or prove that no positive integer triple can satisfy them.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Exhibit positive integers satisfying all four square conditions, or prove that no positive integer triple can satisfy them.^[1] Relevance to this problem For Perfect cuboid problem, record perfect-cuboid-problem-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Exhibit positive integers satisfying all four square conditions, or prove that no positive integer triple can satisfy them. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Exhibit positive integers satisfying all four square conditions, or prove that no positive integer triple can satisfy them. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited scholarly paper describes the existence or nonexistence of a perfect cuboid as a famous unsolved problem. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Many parametrizations, congruence restrictions, and large searches are known. Claimed nonexistence proofs require especially careful review.

Recorded example 1. The edges 44, 117, and 240 form an Euler brick with integer face diagonals 125, 244, and 267; its space diagonal is not an integer.

Computational notes

A bounded search can exclude candidates only within its parameter range.

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Perfect cuboid problem.” TheoremDB. P14. Problem statement; statement text SHA-256 79098856464dbfee119ab16bddf7129c3638a1b5d437b06512e35cb19073ef70. https://theoremdb.org/statement/?ref=P14

BibTeX

@misc{theoremdb-problem-79098856464dbfee119ab16bddf7129c3638a1b5d437b06512e35cb19073ef70,
  title = {{Perfect cuboid problem}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 79098856464dbfee119ab16bddf7129c3638a1b5d437b06512e35cb19073ef70},
  url = {https://theoremdb.org/statement/?ref=P14}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Ruslan Sharipov, “On Walter Wyss's no perfect cuboid paper”. arXiv:1704.00165 (2017). Ruslan Sharipov, arXiv:1704.00165, abstract and analysis of a claimed resolution. ↗preprint · primary source · arXiv:1704.00165, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited scholarly paper describes the existence or nonexistence of a perfect cuboid as a famous unsolved problem. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Abstract and error analysis.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Packet-linked scholarly status source and critique of a claimed proof.Source named by the research packet.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P14] Perfect cuboid problem

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Computational notes

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