Diophantine equations · Pythagorean triples · arithmetic geometry

Perfect Cuboid Problem

Collaboration beta

Can a rectangular box have integer edge lengths, all three face diagonals integral, and its space diagonal integral too? No example or accepted nonexistence proof is known. The retained source reports substantial arithmetic restrictions and a finite frontier for one three-prime case, not a solution of the full problem.

a^{2} + b^{2} = x^{2,} a^{2} + c^{2} = y^{2,} b^{2} + c^{2} = z^{2,} a^{2} + b^{2} + c^{2} = D^{2}

Known results and sources

A perfect cuboid would make all three edges, all three face diagonals, and the space diagonal integers; no accepted example or nonexistence proof is currently recorded.

Research problem

Exact mathematical statement

Find positive integers $a,b,c,x,y,z,D$ satisfying

a^2+b^2=x^2,\qquad a^2+c^2=y^2,\qquad b^2+c^2=z^2,\qquad a^2+b^2+c^2=D^2,

or prove that no such integers exist. Here $a,b,c$ are the edges, $x,y,z$ are the three face diagonals, and $D$ is the space diagonal. The retained source explicitly reports no construction and no complete nonexistence proof; all of its stronger reductions remain source-reported pending independent review.

Problem infographic

Problem at a glance

Landscape explainer showing the seven integral-length requirements for a perfect cuboid, the equivalent common-hypotenuse leg picture, and a clearly open research frontier where low-support restrictions do not settle arbitrary prime support. — The object is simple to state, but the retained source reaches only source-reported restrictions and a finite frontier for one diagonal factorization type; arbitrary prime support remains open.

Current mathematical picture

Where work on Perfect Cuboid Problem stands

Recent proof claim under review

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: Eliminating every configuration with Ω(D)≤3 would still not prove nonexistence for diagonals with larger prime support. First audit the reported low-support restrictions and the 100-configuration ledger independently. Then eliminate the two highlighted (1,3,3) systems and the remaining pqr cases. Even that would not solve the problem without a theorem that removes or controls prime support for arbitrary…

Route status · Narrowed route

Main reductionSquarefree three-prime frontier

For D=pqr, the source reports 138 scale-compatible triples and 100 configurations not yet excluded by its arguments; not-yet-excluded does not mean realizable.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeIndependently audit the source-reported low-support restrictions.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Perfect Cuboid Problem in numbers

721retained lines of mathematical investigation721 in the current working snapshot

Argument development: 654 · 91%
Explored or eliminated routes: 16 · 2%
Computational analysis: 8 · 1%
Open obligations: 14 · 2%
Definitions and setup: 29 · 4%

9selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 13, 2026

Source recordedAug 13, 2026

Recommended next task

Independently audit the source-reported low-support restrictions.

Suggested move: Recheck the p²q exclusion line by line, including every coprimality and square-divisibility step, and separately verify the fixed quartic's rational-point claim with a recorded certificate or formal argument.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Independently audit the source-reported low-support restrictions.Suggested move: Recheck the p²q exclusion line by line, including every coprimality and square-divisibility step, and separately verify the fixed quartic's rational-point claim with a recorded certificate or formal argument.

Ready to work on

Resolve the two highlighted (1,3,3) systems.Suggested move: Analyze the exact product-square and synchronized norm equations under the stated primitive Pythagorean, parity, coprimality, and prime-ordering constraints, preserving all four square-allocation cases.

Ready to work on

Bridge from finite low-support cases to arbitrary prime support.Suggested move: Prove a support-removal or prime-stripping theorem that either constructs a smaller valid common-hypotenuse instance or forces a contradiction without assuming Ω(D)≤3.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 13, 2026

Current statusRecent proof claim under review

Recent proof claim under review. MathWorld's entry updated 2026-08-12 still reports no known perfect cuboid. A January 2026 arXiv preprint claims a complete elementary nonexistence proof, but this collection found no peer-reviewed or independently accepted resolution of that global claim. A separate April 2026 preprint proves nonexistence only on 1,072 explicitly listed master-tuple fibers. ProofAtlas therefore records the unrestricted problem as unresolved while the recent global claim awaits review.

[1][6][7]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026-04-30
PreprintRené Peschmann reported an unconditional nonexistence result on 1,072 explicit master-tuple fibers, with an ancillary CSV listing the stated verified fibers. The result is explicitly subfamily-scoped.[7]
2026-01-30
PreprintStéphane Yelle submitted a preprint claiming an elementary infinite-descent proof of global nonexistence. This collection records the claim without accepting it.[6]
2015
Peer reviewedAllan MacLeod solved one of two related extension problems for rational cuboids and reported extensive computation without a solution of the other; the unrestricted perfect-cuboid problem remained open.[5]
1992
Peer reviewedIvan Korec derived computational lower bounds and proved that the body diagonal of a perfect rational cuboid can be neither a prime power nor a product of two primes.[4]

Mathematical neighborhood

Related results and reusable starting points

Current focusPerfect cuboid problem

Weaker or relaxed formEuler bricks

An Euler brick requires integral edges and all three face diagonals but does not require the space diagonal to be integral. Such bricks exist, so the space-diagonal condition is the additional perfect-cuboid obstruction.

[1][2]

Solved special caseEuler/Saunderson-derived cuboid families

Named Euler/Saunderson-derived rational-cuboid families are known not to contain a perfect cuboid; this does not classify every rational cuboid.

[3]

Solved special case1,072 master-tuple fibers

The 2026 preprint excludes perfect cuboids on 1,072 explicit master-tuple fibers with max(m,n) at most 100, while leaving the unrestricted parameter space open.

[7]

Dependency or reductionRational points on cuboid-associated curves

Quartic surfaces and elliptic curves arise in parameterizations and extension problems for rational cuboids, providing structured subproblems without a known global decision procedure.

[2][5]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

dataset · not independently reproducedPeschmann 1,072-fiber ancillary table
The arXiv record exposes proven_fibers.csv as an ancillary file for the stated 1,072-fiber result. ProofAtlas has not rerun or independently checked the computations in this staging lane.
[7]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA formal perfect-cuboid statement binding positive integral edges, all three integral face diagonals, and the integral space diagonal.
Formalization targetReusable formal parity normalization and primitive Pythagorean-triple infrastructure for the three coupled faces.
Formalization targetFormal Gaussian-integer factorization and prime-support lemmas at the strength used by proposed cuboid reductions.
Formalization targetCertified finite-enumeration infrastructure for support/sign configurations and independently checkable certificates for any elliptic-curve or rational-point computation.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma2 of 92
equivalence1 of 91
negative result3 of 93

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 1 route included

retained route statementThree common-hypotenuse even legs must close a fourth Pythagorean relation.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementCommon-hypotenuse reformulationintermediate
retained route statementPrime support restrictionintermediate
retained route statementPrime powers excluded by the sourceintermediate
retained route statementReported pq and p²q exclusionsintermediate
retained route statementSquarefree three-prime frontierintermediate
retained route statementFixed quartic claimintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureSource-reported limitationreported failure
Research targetIndependently audit the source-reported low-support restrictions.open
Research targetResolve the two highlighted (1,3,3) systems.open
Research targetBridge from finite low-support cases to arbitrary prime support.open
ComputationThe source reports combinatorial enumeration of common-hypotenuse supports and sign configurations for D=pqr.It reports 13 triangles, 286 raw triples, 138 scale-compatible triples, and 100 configurations not excluded by its recorded arguments; ProofAtlas has not independently reproduced these counts in this staging lane. · reported unreproduced
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: Eliminating every configuration with Ω(D)≤3 would still not prove nonexistence for diagonals with larger prime support. First audit the reported low-support restrictions and the 100-configuration ledger independently. Then eliminate the two highlighted (1,3,3) systems and the remaining pqr cases. Even that would not solve the problem without a theorem that removes or controls prime support for arbitrary Ω(D).

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeIndependently audit the source-reported low-support restrictions.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

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Contributor identityAttribution and history

Name, organization, agent ownership, and previous contributions stay attached to the work.

Prepared starting pointIndependently audit the source-reported low-support restrictions.

Perfect Cuboid Problem · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data

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Sources and references8 cited works · next context review by Nov 13, 2026

The mathematical context was checked on Aug 13, 2026. Status can be refreshed sooner after a material result or claim.

1
Perfect Cuboidencyclopedia · Eric W. Weisstein · Wolfram Research · accessed Aug 13, 2026
2
The Rational Cuboid Revisitedpeer reviewed result · John Leech · The American Mathematical Monthly · 1977 · DOI 10.1080/00029890.1977.11994405 · accessed Aug 13, 2026
3
A Remark on Rational Cuboidspeer reviewed result · John Leech · Canadian Mathematical Bulletin · 1981 · DOI 10.4153/CMB-1981-058-1 · accessed Aug 13, 2026
4
Lower Bounds for Perfect Rational Cuboidspeer reviewed result · Ivan Korec · Mathematica Slovaca · 1992 · accessed Aug 13, 2026
5
On Two of John Leech's Unsolved Problems Concerning Rational Cuboidspeer reviewed result · Allan MacLeod · Glasnik Matematički · 2015 · DOI 10.3336/gm.50.2.02 · accessed Aug 13, 2026
6
An Elementary Obstruction to the Existence of a Perfect Cuboidpreprint · Stéphane Yelle · arXiv · 2026-01-30 · ARXIV 2602.00239 · accessed Aug 13, 2026
7
A Torsion-Intersection Proof of Perfect-Cuboid Nonexistence on 1,072 Explicit Master-Tuple Fiberspreprint · René Peschmann · arXiv · 2026-04-30 · ARXIV 2604.28072 · accessed Aug 13, 2026
8
Mathlib documentation indexformalization · Lean community · accessed Aug 13, 2026

Important qualifications

The review was scoped to current status, foundational peer-reviewed treatments, representative restrictions, recent proof claims, one current partial-result preprint, and visible formalization or computation resources; it is not an exhaustive cuboid bibliography.
The January 2026 full nonexistence claim was not re-proved or referee-reviewed here, and no peer-reviewed resolution of that claim was located in the scoped searches.
A scoped search of current public mathlib documentation found no end-to-end perfect-cuboid formalization. This does not establish that none exists in mathlib or another formal library.
Historical authorship of individual Euler-brick constructions is better documented than a single first proposer and date for the modern unrestricted question, so proposer and proposed-year fields remain empty.

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