ProofAtlas

Diophantine equations · elliptic curves · computational number theory

Magic Square of Squares

Collaboration beta

Large exact searches and arithmetic reductions rule out broad families, yet no square or impossibility proof is known.

aij=xij2, xij∈ℤ>0distinct;∑j=13aij=S (i=1,2,3),∑i=13aij=S (j=1,2,3),a11+a22+a33=a13+a22+a31=S
Known results and sources
Problem-first open-research thumbnail for Magic Square of Squares, showing an unresolved mathematical structure without a completion mark or navigation arrow.
Can all nine entries be distinct squares? The page presents source-reported partial structure without claiming a proof.

Research problem

Exact mathematical statement

Determine whether a nontrivial 3×3 magic square can have nine distinct positive perfect-square entries.

aij=xij2, xij∈ℤ>0distinct;∑j=13aij=S (i=1,2,3),∑i=13aij=S (j=1,2,3),a11+a22+a33=a13+a22+a31=Sa_{ij}=x_{ij}^2,\ x_{ij}\in\mathbb Z_{>0}\text{ distinct};\quad \sum_{j=1}^3a_{ij}=S\ (i=1,2,3),\quad \sum_{i=1}^3a_{ij}=S\ (j=1,2,3),\quad a_{11}+a_{22}+a_{33}=a_{13}+a_{22}+a_{31}=S

This remains open; the source reports partial results and route constraints, not a complete proof.

Problem infographic

Problem at a glance

Three-panel source-bound explainer for Magic Square of Squares: exact target, strongest retained result, and the unresolved closing bridge.
The source separates the exact target, retained partial results, and the still-open bridge.

Current mathematical picture

Where work on Magic Square of Squares stands

Open problem

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

Useful failureExtending two-prime cancellation one exponent at a time

No known norm-product cycle cancels all three prime powers to an absolute constant. Cyclic Gaussian lifts, type-13 cover arithmetic, and a global sum-freeness argument remain viable.

Route status · Narrowed route
Main reductionCurrent reduction

Express solutions through an additive configuration in a rational circle set or through offset spectra, then control simultaneous high-precision Gaussian purity equations across active split primes.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeClose type-13 three-prime compatibility.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. 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

Magic Square of Squares in numbers

1.5kretained lines of mathematical investigation1,471 in the current working snapshot
Argument development
1,231 · 84%
Explored or eliminated routes
35 · 2%
Computational analysis
88 · 6%
Open obligations
49 · 3%
Definitions and setup
68 · 5%
7selected 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 21, 2026
Source recordedAug 21, 2026

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

11 selected steps

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

11 selected steps

Scroll horizontally to explore the route

Working route overview for Magic Square of SquaresA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can all nine entries be distinct squares? — Depends on missing premiseCan all nine entries bedistinct squares?Current reduction — Depends on missing premiseCurrent reductionGlobal sum-free route — Depends on missing premiseGlobal sum-free routeRational normal form — Depends on missing premiseRational normal formCenter-root search — Depends on missing premiseCenter-root searchClosing target — Depends on missing premiseClosing targetTwo-active-prime closure — Depends on missing premiseTwo-active-prime closureExtending two-prime cancellation one exponent at a time — stoppedExtending two-primecancellation one exponent ata…Close type-13 three-prime compatibility. — OpenClose type-13 three-primecompatibility.Treat type 6 and other extremal faces. — OpenTreat type 6 and otherextremal faces.Pass from squarefree support to arbitrary exponents. — OpenPass from squarefree supportto arbitrary exponents.
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 routeExtending two-prime cancellation one exponent at a time

No known norm-product cycle cancels all three prime powers to an absolute constant. Cyclic Gaussian lifts, type-13 cover arithmetic, and a global sum-freeness argument remain viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Close type-13 three-prime compatibility.Suggested move: Derive a cyclic product of three quotient equations.
Ready to work on
02
Treat type 6 and other extremal faces.Suggested move: Develop an independent type-6 quotient or cover analysis for its larger primitive relation.
Ready to work on
03
Pass from squarefree support to arbitrary exponents.Suggested move: Lift any squarefree support theorem to arbitrary exponent depth.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 21, 2026
Current statusOpen problem

The distinct-positive 3×3 square-entry problem remains open; computational and algebraic papers provide reductions and exclusions rather than a complete construction or impossibility proof.

[1][2]
External progress

What the literature has established

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

  1. PreprintA New Transformation of the Magic Square of Squares supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[2]
  2. PreprintA Partial Residue Categorization of the Magic Square of Squares supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[1]
2 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusMagic Square of Squares
Related problemMagic Square of Squares

The distinct-positive 3×3 square-entry problem remains open; computational and algebraic papers provide reductions and exclusions rather than a complete construction or impossibility proof.

[1][2]

Formalization opportunities

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

  • Formalization targetA formal statement matching the exact public target and all quantifiers.
  • Formalization targetFormal libraries for the principal mathematical structures used by the strongest route.
  • Formalization targetA checked closing argument for the source-identified open bridge.

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 updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
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
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
Research-record correctionWe removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Claim connections clarified
Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

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.

4 standing statements3 proposed statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma2 of 72
  • equivalence1 of 71
  • computational claim1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementCan all nine entries be distinct squares?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementRational normal formintermediate
  • retained route statementCenter-root searchintermediate
  • retained route statementTwo-active-prime closureintermediate
  • retained route statementGlobal sum-free routeintermediate
  • 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
  • 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 failureExtending two-prime cancellation one exponent at a timereported failure
  • Research targetClose type-13 three-prime compatibility.open
  • Research targetTreat type 6 and other extremal faces.open
  • Research targetPass from squarefree support to arbitrary exponents.open
  • Research targetThree-active-prime consequencecompleted reported
  • Research targetType-13 cyclesuperseded
  • Narrowed routeExtending two-prime cancellation one exponent at a timeNo known norm-product cycle cancels all three prime powers to an absolute constant. Cyclic Gaussian lifts, type-13 cover arithmetic, and a global sum-freeness argument remain viable.
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 bridgeClose type-13 three-prime compatibility.

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

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointClose type-13 three-prime compatibility.

Magic Square of Squares · 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

Large exact searches and arithmetic reductions rule out broad families, yet no square or impossibility proof is known.

  • 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
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references2 cited works · next context review by Nov 21, 2026

The mathematical context was checked on Aug 21, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    A Partial Residue Categorization of the Magic Square of Squarespreprint · Christian Woll · arXiv · 2018 · accessed Aug 21, 2026
  2. 2
    A New Transformation of the Magic Square of Squarespreprint · Christian Wolird · arXiv · 2023 · accessed Aug 21, 2026

Important qualifications

  • This was a bounded status and identity check, not an exhaustive bibliography, priority review, or legal review.
  • Private packet claims were not treated as external authority; submitted links and attachments were not executed or actively rendered.
  • No absence claim is inferred from the bounded search, and recent preprints remain subject to ordinary scholarly review.

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