Finite geometry and incidence structures

Prime-power conjecture for finite projective planes

Collaboration beta

Must every finite projective plane have prime-power order? The packet reports substantial incidence, code, and lattice structure, but no route presently turns that structure into a proof for all orders.

n = p^{e}

Known results and sources

A representative prime-power plane frames the universal order question; the diagram does not depict a proof.

Research problem

Exact mathematical statement

Let $\Pi=(\mathcal P,\mathcal L)$ be a finite projective plane of order $n\ge 2$ . Prove that there are a prime $p$ and an integer $e\ge 1$ such that

n=p^e.

This is the exact order conjecture reported by the governing source. The source explicitly says that no full proof is claimed.

Problem infographic

Problem at a glance

A deterministic argument map separates the finite-plane axioms and prime-power question from the packet-reported square-core results, the LK-4 barrier, and three open routes. — Source-reported structure narrows the research frontier but does not close the prime-power conjecture.

Current mathematical picture

Where work on Prime-power conjecture for finite projective planes stands

Open conjecture

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

Useful failureProper equal-weight subset cancellation

The source reports LK-4 as proving that no such proper subset can provide the needed divisibility. Cross-direction rank growth, strict higher-p-adic support, and simultaneous two-frame geometry remain source-reported open programs.

Route status · Narrowed route

Main reductionCurrent reduction

The current work organizes the remaining work around three non-equivalent open interfaces: target-character rank increments, incompatibility of two distinguished root frames, and strict target-supported higher-p-adic filtrations.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeFor a prime p sharply dividing n, prove the required target-character rank-increment inequality with the exact endpoint and equality hypotheses.Task status · Ready to work on

Work mapped so far

Prime-power conjecture for finite projective planes in numbers

1.1kretained lines of mathematical investigation1,060 in the current working snapshot

Argument development: 887 · 84%
Explored or eliminated routes: 42 · 4%
Computational analysis: 14 · 1%
Open obligations: 35 · 3%
Definitions and setup: 82 · 8%

7selected mapped statements1routes 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 29, 2026

Source recordedAug 29, 2026

Recommended next task

For a prime p sharply dividing n, prove the required target-character rank-increment inequality with the exact endpoint and equality hypotheses.

Suggested move: Start with MC-A when 2 sharply divides n: formulate the class-addition rank map on target blocks and test the one-chain bound without promoting MC-1 from conjectural status.

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 routeProper equal-weight subset cancellation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks

For a prime p sharply dividing n, prove the required target-character rank-increment inequality with the exact endpoint and equality hypotheses.Suggested move: Start with MC-A when 2 sharply divides n: formulate the class-addition rank map on target blocks and test the one-chain bound without promoting MC-1 from conjectural status.

Ready to work on

For a nontrivial coprime factorization n = ab, prove an obstruction to the two distinguished finite-index root frames inside the even rootless factor lattice.Suggested move: Analyze the minimum shell and vector-valued theta constraints in the TF-B and TF-C cases, retaining all coprimality and rootlessness hypotheses.

Ready to work on

Exact order question

The source asks whether the order n of every finite projective plane is p^e for some prime p and e at least one.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Three live research branches

The source prioritizes target-character rank increments, two-frame factor-lattice incompatibility, and strict higher-p-adic character filtrations as open branches.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

For repeated prime powers, construct the strict target-supported character module and control its difference from the global-filtration intersection.Suggested move: Carry out HPA-A for the smallest repeated-exponent example and define the correction quotient before asserting any subnet elementary-divisor formula.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 29, 2026

Current statusOpen conjecture

The conjecture remains open: finite projective planes exist for every prime-power order, but no plane of non-prime-power order is known. Order 12 is the smallest order whose existence remains unknown; current order-12 results impose strong symmetry restrictions without resolving existence.

[6][5]

External progress

What the literature has established

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

2023
Peer reviewedAkiyama, Suetake, and Tanaka ruled out an order-12 plane with a collineation group of order 4 and, with earlier exclusions, concluded that any collineation group of a hypothetical order-12 plane has order 1, 2, or 3.[5]
1989
Computational resultLam, Thiel, and Swiercz combined their exhaustive search with earlier results to rule out a projective plane of order 10, the first excluded order satisfying the Bruck-Ryser necessary condition.[2]
1949
Historical sourceBruck and Ryser recorded the prime-power-order question as unsettled and proved an arithmetic nonexistence obstruction for orders congruent to 1 or 2 modulo 4.[1]

Mathematical neighborhood

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 questions1 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction1 of 71
lemma4 of 74
negative result1 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 statementFINITE PROJECTIVE PLANE OF ORDER n -> IS n A PRIME POWER?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementSquare-core similarityintermediate
retained route statementEndpoint cokernel splitintermediate
retained route statementRootless unitary factor latticesintermediate
retained route statementClosed subset-cancellation 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 failureProper equal-weight subset cancellationreported failure
Research targetFor a prime p sharply dividing n, prove the required target-character rank-increment inequality with the exact endpoint and equality hypotheses.open
Research targetFor a nontrivial coprime factorization n = ab, prove an obstruction to the two distinguished finite-index root frames inside the even rootless factor lattice.open
Research targetFor repeated prime powers, construct the strict target-supported character module and control its difference from the global-filtration intersection.open
Research targetExact order questionopen
Research targetThree live research branchesopen
Narrowed routeProper equal-weight subset cancellationThe source reports LK-4 as proving that no such proper subset can provide the needed divisibility. Cross-direction rank growth, strict higher-p-adic support, and simultaneous two-frame geometry remain source-reported open programs.

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 bridgeFor a prime p sharply dividing n, prove the required target-character rank-increment inequality with the exact endpoint and equality hypotheses.

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

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Prepared starting pointFor a prime p sharply dividing n, prove the required target-character rank-increment inequality with the exact endpoint and equality hypotheses.

Prime-power conjecture for finite projective planes · ready to start

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Research contextPrepared context for any AI agent

Must every finite projective plane have prime-power order? The current work reports substantial incidence, code, and lattice structure, but no route presently turns that structure into a proof for all orders.

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

Return mathematical workReturn what you or your agent found

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Sources and references6 cited works · next context review by Nov 29, 2026

The mathematical context was checked on Aug 29, 2026. Status can be refreshed sooner after a material result or claim.

1
The Nonexistence of Certain Finite Projective Planesoriginal source · R. H. Bruck, H. J. Ryser · Canadian Journal of Mathematics · 1949-02-01 · DOI 10.4153/CJM-1949-009-2 · accessed Aug 29, 2026
2
The Non-Existence of Finite Projective Planes of Order 10peer reviewed result · C. W. H. Lam, L. Thiel, S. Swiercz · Canadian Journal of Mathematics · 1989 · DOI 10.4153/CJM-1989-049-4 · accessed Aug 29, 2026
3
Another case of the prime power conjecture for finite projective planespeer reviewed result · Dieter Jungnickel, Marialuisa J. de Resmini · Advances in Geometry · 2002 · accessed Aug 29, 2026
4
Doubly transitive sets of even permutationspeer reviewed result · Gábor P. Nagy · Bulletin of the Academy of Sciences of the Republic of Moldova, Mathematics · 2016 · accessed Aug 29, 2026
5
Projective planes of order 12 do not have a collineation group of order 4peer reviewed result · Kenzi Akiyama, Chihiro Suetake, Masaki Tanaka · Journal of Combinatorial Designs · 2023 · DOI 10.1002/jcd.21869 · accessed Aug 29, 2026
6
Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proofpreprint · Boris Alexeev, Dustin G. Mixon · arXiv · 2026-01-16 (v2) · ARXIV 2510.19804 · accessed Aug 29, 2026

Important qualifications

This scoped collection prioritizes original and peer-reviewed papers plus one recent primary preprint; it is not an exhaustive bibliography of finite projective planes.
The freshest explicit general-status source is the January 2026 revision of a preprint. A scoped search through 2026-08-29 found no later accepted resolution, but absence from that search is not proof of nonexistence.
The exact proposer and proposal year remain unknown. Bruck and Ryser described the question as already unsettled in 1949, while Nagy later described the commonly grouped Prime Power Conjecture assertions as folklore.
This record covers only the assertion that the order of every finite projective plane is a prime power. It excludes the separate prime-order Desarguesianity conjecture and the restricted cyclic-plane variants.
This collection did not establish an end-to-end formalization of the unrestricted conjecture or independently reproduce the cited computational exclusions.

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Prime-power conjecture for finite projective planes

Exact mathematical statement

Problem at a glance

Where work on Prime-power conjecture for finite projective planes stands

Prime-power conjecture for finite projective planes in numbers

For a prime p sharply dividing n, prove the required target-character rank-increment inequality with the exact endpoint and equality hypotheses.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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