ProofAtlas

Number theory · factorial Diophantine equations

Brocard's Problem

Collaboration beta

Are 4!+1, 5!+1, and 7!+1 the only factorials one below a square?

n!+1=m2⇒n∈{4,5,7}
Known results and sources
A dark-green cover shows a rising staircase of factorial blocks beneath a cyan diagonal path. Exactly three filled ivory-and-gold diamonds mark the familiar square hits; a fourth diamond is only outlined on a later step, and a separate gold-outlined circle beyond the staircase keeps the continuation visibly unresolved.
Are 4!+1, 5!+1, and 7!+1 the only factorials one below a square?

Research problem

Exact mathematical statement

Brocard's Problem asks for all positive integer solutions of

n!+1=m2.n!+1=m^2.

The known solutions are (n,m)=(4,5),(5,11),(7,71)(n,m)=(4,5),(5,11),(7,71). The open task is to prove that no solution exists for n≥8n\ge8, or to find another one.

Problem infographic

Problem at a glance

A dark-green mathematical diagram begins with a rising factorial staircase and exactly three filled square-hit diamonds followed by one outlined unresolved diamond. The staircase splits into two nonintersecting arithmetic rails, one cyan and one gold, each carrying five distinct complete-block tiles. The cyan rail is selected and enters five two-socket gate cells: four hold a tile in one socket, while the fifth is dashed and empty to show the still-open task of forcing a selected block to fail.
The packet recasts Brocard's equation as a canonical complete-block divisibility gate selected by exponent parity. It reports exact arithmetic reductions and finite certificates, but the universal step—forcing at least one selected block to fail for every n ≥ 8—remains open.

Current mathematical picture

Where work on Brocard's Problem stands

Open problem

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

Useful failureTreating the parity exactifier as a solution

PE-1 only reformulates a hypothetical solution; one must still prove a selected complete block fails its local root condition. Use PE-1 as the exact start of a block-failure or descent proof.

Route status · Narrowed route
Main reductionCurrent reduction

A hypothetical solution is equivalent to every block in one canonical exponent-parity class placing X at 0 or -1 modulo its full prime power.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve that some selected complete block fails for every n at least eight.Task status · Ready to work on

Work mapped so far

Brocard's Problem in numbers

1.1kretained lines of mathematical investigation1,150 in the current working snapshot
Argument development
973 · 85%
Explored or eliminated routes
39 · 3%
Computational analysis
28 · 2%
Open obligations
58 · 5%
Definitions and setup
52 · 5%
7selected mapped statements2routes investigated5open questions5contribution-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 28, 2026
Source recordedAug 28, 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

14 selected steps

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

14 selected steps

Scroll horizontally to explore the route

Working route overview for Brocard's ProblemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.A factorial plus one should be square only at n = 4, 5, and 7. — Depends on missing premiseA factorial plus one shouldbe square only at n = 4, 5,and…Canonical parity exactifier — Depends on missing premiseCanonical parity exactifierCurrent reduction — Depends on missing premiseCurrent reductionNeighboring-square normalization — Depends on missing premiseNeighboring-squarenormalizationClosing target — Depends on missing premiseClosing targetFinite parity scan — Depends on missing premiseFinite parity scanKnown solutions — Depends on missing premiseKnown solutionsTreating the parity exactifier as a solution — stoppedTreating the parityexactifier as a solutionJoining finite and ineffective asymptotic ranges — stoppedJoining finite andineffective asymptoticrangesProve that some selected complete block fails for every n at least eight. — OpenProve that some selectedcomplete block fails forevery…Turn the dual determinant-one normal forms into a block-aware descent contradiction. — OpenTurn the dualdeterminant-one normal formsinto…Prove the exact mixed-modulus system has no nontrivial state. — OpenProve the exactmixed-modulus system has nonontrivial…Small selected-block failure — OpenSmall selected-block failureMixed-modulus feasibility — OpenMixed-modulus feasibility
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

2 recorded
Narrowed routeTreating the parity exactifier as a solution

PE-1 only reformulates a hypothetical solution; one must still prove a selected complete block fails its local root condition. Use PE-1 as the exact start of a block-failure or descent proof.

Route status · Narrowed route
Narrowed routeJoining finite and ineffective asymptotic ranges

Without an effective threshold, the two ranges cannot be joined. Retain asymptotic inputs only with their exact quantifiers and seek an effective bridge separately.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Prove that some selected complete block fails for every n at least eight.Suggested move: Build a finite-state proof for the small selected blocks rather than extending the numerical scan.
Ready to work on
02
Turn the dual determinant-one normal forms into a block-aware descent contradiction.Suggested move: Use determinant-one normal forms and complete half-block atoms to force a descent contradiction.
Ready to work on
03
Small selected-block failure

A uniform proof must force one selected small or moving block to fail.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Mixed-modulus feasibility

The exact mixed-modulus binary system must be shown to have no nontrivial state.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Prove the exact mixed-modulus system has no nontrivial state.Suggested move: Eliminate the nontrivial states of the exact mixed-modulus binary system.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 28, 2026
Current statusOpen problem

The cited current preprint explicitly treats the assertion that n=4, 5, and 7 are the only positive solutions as open. Computations and conditional finiteness results do not settle it.

[2]
External progress

What the literature has established

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

  1. PreprintNovaković continued to state the three-solution assertion as open while distinguishing conditional finiteness results.[2]
  2. Peer reviewedBerndt and Galway reported a search through n=10^9 with no positive solutions beyond n=4, 5, and 7.[1]
2 cited sources0 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Formalization opportunities

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

  • Formalization targetA formalization needs factorial arithmetic, the exact positive-integer domain, and a proof excluding every n outside 4, 5, and 7.
  • Formalization targetFinite searches and conditional finiteness results cannot close the universal Diophantine statement without separately checked certificates and hypotheses.

Detailed research inventory

Claims, milestones, and routes in the current map

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

5 standing statements2 proposed statements5 open questions2 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction1 of 71
  • lemma1 of 71
  • computational claim2 of 72
  • equivalence2 of 72
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 2 routes included
  • retained route statementA factorial plus one should be square only at n = 4, 5, and 7.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementKnown solutionsintermediate
  • retained route statementNeighboring-square normalizationintermediate
  • retained route statementCanonical parity exactifierintermediate
  • retained route statementFinite parity scanintermediate
  • 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 failureTreating the parity exactifier as a solutionreported failure
  • Useful failureJoining finite and ineffective asymptotic rangesreported failure
  • Research targetProve that some selected complete block fails for every n at least eight.open
  • Research targetTurn the dual determinant-one normal forms into a block-aware descent contradiction.open
  • Research targetProve the exact mixed-modulus system has no nontrivial state.open
  • Research targetSmall selected-block failureopen
  • Research targetMixed-modulus feasibilityopen
  • ComputationAn independent GMP scan compares the canonical parity gate with direct square testing through n = 10,000.The finite scan reports first failed selected primes and no further solutions through its bound. Intake did not execute the program, and the scan is not proof for all n. · reported unreproduced
  • Narrowed routeTreating the parity exactifier as a solutionPE-1 only reformulates a hypothetical solution; one must still prove a selected complete block fails its local root condition. Use PE-1 as the exact start of a block-failure or descent proof.
  • Narrowed routeJoining finite and ineffective asymptotic rangesWithout an effective threshold, the two ranges cannot be joined. Retain asymptotic inputs only with their exact quantifiers and seek an effective bridge separately.
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 bridgeProve that some selected complete block fails for every n at least eight.

2 approaches have 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.

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Prepared starting pointProve that some selected complete block fails for every n at least eight.

Brocard's 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

Are 4!+1, 5!+1, and 7!+1 the only factorials one below a square?

  • 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 references2 cited works · next context review by Nov 28, 2026

The mathematical context was checked on Aug 28, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    On the Brocard-Ramanujan diophantine equation n! + 1 = m^2peer reviewed result · Bruce C. Berndt, William F. Galway · Ramanujan Journal · 2000-03 · DOI 10.1023/A:1009873805276 · accessed Aug 28, 2026
  2. 2
    A note on some polynomial-factorial diophantine equationspreprint · Saša Novaković · arXiv · 2023-08-21 · ARXIV 2308.11002 · accessed Aug 28, 2026

Important qualifications

  • This was a bounded primary-source and publisher-record search, not an exhaustive literature, priority, citation, rights, or authorship review.
  • Open status means that the cited source states or studies the problem as a conjecture or open problem and the bounded search found no statement-aligned primary resolution; it does not prove that no later claim exists.
  • Recent preprints are recorded only with their stated preprint posture and are not treated as peer-reviewed or independently verified.
  • No submitted attachment, submitted URL, packet-reported computation, or model output was treated as independent external authority.
  • No statement-aligned formalization, certificate, or independently reproduced computation was established by this search.

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