Number theory · Diophantine equations · Egyptian fractions

Erdős–Straus Conjecture

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The conjecture asks whether every fraction 4/n with n at least 2 splits into three positive unit fractions. Huge finite ranges and many residue classes are known, but no argument covers every integer.

\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}

Erdős Problem #242

Known results and sources

$A single fraction 4 over n faces three positive unit-fraction tiles across an equality marked by a question, with sparse residue-class geometry and an open status badge.$

Erdős–Straus asks whether every 4/n for n at least 2 is a sum of three positive unit fractions; finite checks and residue reductions do not prove the universal claim.

Research problem

Exact mathematical statement

For every integer $n\ge 2$ , do there exist positive integers $x,y,z$ such that

\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}?

The variables are required to be positive integers. Finite computational verification, residue-class formulas, and a solution of a stronger restricted conjecture would not by themselves change this exact universal statement unless the implication to every $n$ is proved.

Problem infographic

Problem at a glance

Problem-first explainer showing the exact three-unit-fraction equation, prime and residue reductions, two source-reported sublinear arithmetic frontiers, and the unproved saturation bridge. — The source organizes the hard prime case into exact branches and sublinear frontiers, but the simultaneous saturation escape and the full universal conjecture remain unproved.

Current mathematical picture

Where work on Erdős–Straus Conjecture stands

Open conjecture

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

Useful failureFinite or fixed-form coverage

The source explicitly deprioritizes a larger finite scan without a new record, counterexample, or structural law and forbids restarting several fixed-window routes. A proof through a new saturation theorem or a return to the complete Type T/O face complex remains viable.

Route status · Narrowed route

Main reductionHard residue and complete branches

The source restricts the unresolved work to primes congruent to 1 modulo 24 and two complete parametrized branches.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve a saturation theorem on a nontrivial infinite family of hard primes.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

Erdős–Straus Conjecture in numbers

2.8kretained lines of mathematical investigation2,792 in the current working snapshot

Argument development: 2,446 · 88%
Explored or eliminated routes: 82 · 3%
Computational analysis: 104 · 4%
Open obligations: 62 · 2%
Definitions and setup: 98 · 4%

8selected mapped statements1routes investigated4open questions4contribution-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.

Proposed byPaul Erdős and Ernst StrausProposed1948Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove a saturation theorem on a nontrivial infinite family of hard primes.

Suggested move: Use the factorization triangle and exact gcd-equality criteria to couple at least two cube-root faces rather than testing each face independently.

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 routeFinite or fixed-form coverage

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Prove a saturation theorem on a nontrivial infinite family of hard primes.Suggested move: Use the factorization triangle and exact gcd-equality criteria to couple at least two cube-root faces rather than testing each face independently.

Ready to work on

Find a forcing or obstructing invariant for the minus-Euclidean unit-side formulation.Suggested move: Study length-three negative continued fractions and the determinant-frieze matrix form while preserving the source’s mod-8 and mod-3 sieve.

Ready to work on

Simultaneous saturation escape

Forcing at least one cube-root face to saturate is the first unverified implication and is not proved in the source.

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

Ready to work on

Prepare a rigorous fallback if the carry-one unit-side strengthening fails.Suggested move: Use the complete refutation protocol, then return in order to the sublinear face complex, minimal diagonal twins, primitive linear form, and crossed square-divisor form.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The universal three-unit-fraction statement remains open. Peer-reviewed 2026 work and recent divisor parametrizations sharpen special structures, and primary and maintained records report finite verification through 10^18, but none of these supplies an infinite proof.

[1][4][5]

External progress

What the literature has established

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

2026
PreprintA divisor-based parametrization was proved complete for a scoped class of decompositions, without resolving all cases of the conjecture.[6]
2026
Peer reviewedChamberland proved an if-and-only-if representation criterion for Type II prime solutions and explicitly retained the full conjecture as unsolved.[5]
2025
PreprintMihnea and Bogdan report improving computational verification to 10^18, and the maintained Erdős Problems entry records verification for all n at most 10^18. This finite checkpoint does not prove the universal conjecture.[4][1]
2014
Computational resultSalez reported the earlier computational checkpoint through 10^17; this historical finite verification was later superseded by the 10^18 checkpoint and was never a proof of the universal conjecture.[3]

Mathematical neighborhood

Related results and reusable starting points

Current focusErdős–Straus conjecture

Dependency or reductionPrime and Type I/II reductions

It suffices to treat prime denominators; Type I and Type II parametrizations further organize the prime problem without closing every prime.

[5]

Related problemDivisor-parametrized decomposition classes

The divisor parametrization recovers exactly a constrained decomposition family and provides arithmetic structure, but it is not equivalent to a full solution as currently stated.

[6]

Formal and computational footholds

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

computation · not independently reproducedVerification through 10^18
The 2025 primary arXiv record reports improving the computational bound to 10^18, corroborated by the maintained Erdős Problems entry. This intake did not execute the computation, and finite verification cannot prove the infinite statement.
[4][1]

Formalization opportunities

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

Formalization targetAn exact formal statement over positive natural-number denominators and unit fractions.
Formalization targetFormal prime reduction and residue-class identities covering all elementary cases.
Formalization targetA formally checked infinite argument closing the remaining prime classes; finite computation is insufficient.

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.

6 standing statements2 proposed statements4 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction4 of 84
lemma2 of 82
equivalence1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included

retained route statementCan every 4/n be split into three positive unit fractions?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact universal statementintermediate
retained route statementReduction to primesintermediate
retained route statementHard residue and complete branchesintermediate
retained route statementSublinear exact outer frontierintermediate
retained route statementMinus-Euclidean unit-side formintermediate
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
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 failureFinite or fixed-form coveragereported failure
Research targetProve a saturation theorem on a nontrivial infinite family of hard primes.open
Research targetFind a forcing or obstructing invariant for the minus-Euclidean unit-side formulation.open
Research targetPrepare a rigorous fallback if the carry-one unit-side strengthening fails.open
Research targetSimultaneous saturation escapeopen
Narrowed routeFinite or fixed-form coverageThe source explicitly deprioritizes a larger finite scan without a new record, counterexample, or structural law and forbids restarting several fixed-window routes. A proof through a new saturation theorem or a return to the complete Type T/O face complex remains 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 bridgeProve a saturation theorem on a nontrivial infinite family of hard primes.

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 pointProve a saturation theorem on a nontrivial infinite family of hard primes.

Erdős–Straus Conjecture · ready to start

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Research contextPrepared context for any AI agent

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Current routes and known obstacles
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Sources and references6 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Erdős Problem #242maintained problem list · Erdős Problems · accessed Aug 14, 2026
2
Counting the number of solutions to the Erdős–Straus equation on unit fractionspeer reviewed result · Christian Elsholtz, Terence Tao · Journal of the Australian Mathematical Society · 2013 · DOI 10.1017/S1446788712000468 · accessed Aug 14, 2026
3
The Erdős–Straus conjecture: new modular equations and checking up to N = 10^17software or dataset · Yannick Salez · arXiv · 2014 · ARXIV 1406.6307 · accessed Aug 14, 2026
4
Further verification and empirical evidence for the Erdős-Straus conjecturepreprint · Spiridon Mihnea, Dumitru C. Bogdan · arXiv · 2025-08-29 · ARXIV 2509.00128 · DOI 10.48550/arXiv.2509.00128 · accessed Aug 14, 2026
5
The Erdős–Straus Conjecture and the Structure of Primespeer reviewed result · Marc Chamberland · INTEGERS · 2026-04-03 · DOI 10.5281/zenodo.19403738 · accessed Aug 14, 2026
6
A Divisor Parametrization for the Erdős–Straus Conjecturepreprint · M. Bello-Hernández, M. Benito, E. Fernández · arXiv · 2026-06-09 · ARXIV 2606.10922 · accessed Aug 14, 2026

Important qualifications

The maintained Erdős Problems page explicitly labels its open-status field as the site owner’s current belief; it was corroborated with a peer-reviewed 2026 article that calls the problem unsolved.
Recent 2026 preprints and unreviewed proof claims were not treated as resolution evidence.
The current primary and maintained records report verification through 10^18. ProofAtlas did not independently rerun that computation, and any finite verification remains finite evidence only, not a proof of the universal conjecture.
Scoped searches of current mathlib and Isabelle public documentation found no end-to-end formal proof; that does not establish absence.

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