ProofAtlas

Discrete geometry · rational distances · Diophantine geometry

Erdős–Ulam Problem

Collaboration beta

Fresh prime and squareclass patterns alone can occur in collinear models, so the missing ingredient must use genuinely planar direction geometry.

|P-Q|∈ℚ(P,Q∈S, P≠Q)
Known results and sources
Problem-first open-research thumbnail for Erdős–Ulam Problem, showing an unresolved mathematical structure without a completion mark or navigation arrow.
Can a dense planar set have every distance rational? The page presents source-reported partial structure without claiming a proof.

Research problem

Exact mathematical statement

Does there exist a Euclidean-dense set S⊂R² such that the distance between every two distinct points of S is rational?

|P-Q|∈ℚ(P,Q∈S, P≠Q)|P-Q|\in\mathbb Q\quad(P,Q\in S,\ P\ne Q)

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 Erdős–Ulam Problem: 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 Erdős–Ulam Problem stands

Open problem

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

Useful failureSquareclass-rank-only contradiction

The dense collinear model preserves the arithmetic patterns while defeating the intended planar contradiction. The norm-one direction torus and overlapping quadrilateral equations retain geometric information absent from the collinear model.

Route status · Narrowed route
Main reductionCurrent reduction

Encode each oriented edge by a norm-one element and combine fresh modular row collapses with overlapping quadrilateral identities.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeEliminate coordinates across overlapping quadrilateral identities.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Erdős–Ulam Problem in numbers

819retained lines of mathematical investigation819 in the current working snapshot
Argument development
697 · 85%
Explored or eliminated routes
20 · 2%
Computational analysis
12 · 1%
Open obligations
31 · 4%
Definitions and setup
59 · 7%
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 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

13 selected steps

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

13 selected steps

Scroll horizontally to explore the route

Working route overview for Erdős–Ulam ProblemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can a dense planar set have every distance rational? — Depends on missing premiseCan a dense planar set haveevery distance rational?Current reduction — Depends on missing premiseCurrent reductionDirection-torus reduction — Depends on missing premiseDirection-torus reductionClosing target — Depends on missing premiseClosing targetCollinear arithmetic model — Depends on missing premiseCollinear arithmetic modelFree squareclass subsystem — Depends on missing premiseFree squareclass subsystemRetired arithmetic routes — Depends on missing premiseRetired arithmetic routesSquareclass-rank-only contradiction — stoppedSquareclass-rank-onlycontradictionEliminate coordinates across overlapping quadrilateral identities. — OpenEliminate coordinates acrossoverlapping quadrilateralidentities.Derive a height-free normalized-leading-form relation. — OpenDerive a height-freenormalized-leading-formrelation.Convert modular row collisions into exact cyclic or collinear structure. — OpenConvert modular rowcollisions into exact cyclicor…Torus row rigidity — OpenTorus row rigidityNo direct closing theorem — OpenNo direct closing theorem
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 routeSquareclass-rank-only contradiction

The dense collinear model preserves the arithmetic patterns while defeating the intended planar contradiction. The norm-one direction torus and overlapping quadrilateral equations retain geometric information absent from the collinear model.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Eliminate coordinates across overlapping quadrilateral identities.Suggested move: Start with the retained collision expansion.
Ready to work on
02
Derive a height-free normalized-leading-form relation.Suggested move: Test finite-field and moduli-space formulations.
Ready to work on
03
Torus row rigidity

The goal is a height-free torus row-rigidity theorem.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
No direct closing theorem

No located theorem directly supplies the required rigidity.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Convert modular row collisions into exact cyclic or collinear structure.Suggested move: Avoid fixed-twist, Kummer-rank, reciprocity-only, and height-only routes.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 21, 2026
Current statusOpen problem

The problem remains open unconditionally; important results resolve it under major Diophantine conjectures or constrain configurations under additional hypotheses.

[1][2]
External progress

What the literature has established

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

  1. Peer reviewedRational distances from given rational points in the plane 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. Peer reviewedA solution of the Erdős–Ulam problem assuming the Bombieri–Lang conjecture supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet…[2]
2 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusErdős–Ulam Problem
Related problemErdős–Ulam Problem

The problem remains open unconditionally; important results resolve it under major Diophantine conjectures or constrain configurations under additional hypotheses.

[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. 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 statements5 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma2 of 72
  • counterexample1 of 71
  • 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 statementCan a dense planar set have every distance rational?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementFree squareclass subsystemintermediate
  • retained route statementCollinear arithmetic modelintermediate
  • retained route statementDirection-torus reductionintermediate
  • retained route statementRetired arithmetic routesintermediate
  • 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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 failureSquareclass-rank-only contradictionreported failure
  • Research targetEliminate coordinates across overlapping quadrilateral identities.open
  • Research targetDerive a height-free normalized-leading-form relation.open
  • Research targetConvert modular row collisions into exact cyclic or collinear structure.open
  • Research targetTorus row rigidityopen
  • Research targetNo direct closing theoremopen
  • Narrowed routeSquareclass-rank-only contradictionThe dense collinear model preserves the arithmetic patterns while defeating the intended planar contradiction. The norm-one direction torus and overlapping quadrilateral equations retain geometric information absent from the collinear model.
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 bridgeEliminate coordinates across overlapping quadrilateral identities.

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 pointEliminate coordinates across overlapping quadrilateral identities.

Erdős–Ulam 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

Fresh prime and squareclass patterns alone can occur in collinear models, so the missing ingredient must use genuinely planar direction geometry.

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

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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
    Rational distances from given rational points in the planepeer reviewed result · Pietro Corvaja, Amos Turchet, Umberto Zannier · Geometriae Dedicata · 2025 · accessed Aug 21, 2026
  2. 2
    A solution of the Erdős–Ulam problem assuming the Bombieri–Lang conjecturepeer reviewed result · Jafar Shaffaf · Discrete & Computational Geometry · 2018 · 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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