ProofAtlas

Analytic number theory and prime gaps

Twin Prime Conjecture

Collaboration beta

Do infinitely many primes occur two apart? Bounded-gap theorems show infinitely many prime pairs within a fixed finite distance, but the exact gap two remains unproved.

#{pprime:p+2prime}=∞
Known results and sources
A dark number line with repeated prime pairs separated by two units, fading into an unresolved open horizon.
Twin primes are prime pairs exactly two apart; bounded gaps are known, but infinitely many gap-two pairs remain unproved.

Research problem

Exact mathematical statement

There are infinitely many primes p such that p+2 is also prime. Equivalently in the source's smooth squarefree von Mangoldt formulation, T_sf,w(X)>0 for arbitrarily large X.

#{pprime:p+2prime}=∞\#\{p\text{ prime}:p+2\text{ prime}\}=\infty

Problem infographic

Problem at a glance

A problem-first Twin Prime explainer lists example prime pairs exactly two apart, asks whether infinitely many primes p have p+2 prime, and distinguishes known bounded gaps from the still-open exact gap two.
Bounded prime gaps are known, but infinitely many prime pairs exactly two apart remain unproved; proposed detector routes belong in the argument map, not the problem explainer.

Current mathematical picture

Where work on Twin Prime Conjecture stands

Open conjecture

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

Useful failureSource-reported closed or invalid route

A polylogarithmic progression modulus does not decouple the Möbius factor from the shifted prime, so Siegel–Walfisz alone cannot prove the needed correlation. Prove a signed one-sided bound for the combined prime–composite and composite–composite false-pair modes that leaves a fixed positive reserve below the hyperbolic main term.

Route status · Narrowed route
Main reductionCurrent reduction

Split the signed false-pair term into prime–composite, composite–prime, and composite–composite parts; preserve Möbius signs and couple the principal modes before applying absolute-value inequalities.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeBound the signed prime–composite correlation.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 We removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Twin Prime Conjecture in numbers

5kretained lines of mathematical investigation5,020 in the current working snapshot
Argument development
4,394 · 88%
Explored or eliminated routes
109 · 2%
Computational analysis
54 · 1%
Open obligations
215 · 4%
Definitions and setup
248 · 5%
7selected mapped statements2routes 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.

Status checkedAug 15, 2026
Source recordedAug 15, 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 Twin Prime ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.There are infinitely many primes p such that p+2 is also prime. Equivalently in the source's smooth squarefree von Mangoldt formulation, T_sf,w(X)>0 for arbitrarily large X. — Depends on missing premiseThere are infinitely manyprimes p such that p+2 isalso…Current reduction — Depends on missing premiseCurrent reductionSigned Decomposition — Depends on missing premiseSigned DecompositionSmooth Detector — Depends on missing premiseSmooth DetectorThree False Pair Types — Depends on missing premiseThree False Pair TypesClosing target — Depends on missing premiseClosing targetNo Overclaim — Depends on missing premiseNo OverclaimSource-reported closed or invalid route — stoppedSource-reported closed orinvalid routeSecond source-reported limitation — stoppedSecond source-reportedlimitationBound the signed prime–composite correlation. — OpenBound the signedprime–composite correlation.Put the composite–composite tail in a common bilinear form. — OpenPut the composite–compositetail in a common bilinearform.Couple the two false-pair principal modes with a strict reserve. — OpenCouple the two false-pairprincipal modes with astrict…Exact Conjecture — OpenExact Conjecture
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 routeSource-reported closed or invalid route

A polylogarithmic progression modulus does not decouple the Möbius factor from the shifted prime, so Siegel–Walfisz alone cannot prove the needed correlation. Prove a signed one-sided bound for the combined prime–composite and composite–composite false-pair modes that leaves a fixed positive reserve below the hyperbolic main term.

Route status · Narrowed route
Narrowed routeSecond source-reported limitation

The composite–composite support product bound does not make either complementary factor polylogarithmic, and rank shells cannot be estimated separately before terminal Möbius summation. Prove a signed one-sided bound for the combined prime–composite and composite–composite false-pair modes that leaves a fixed positive reserve below the hyperbolic main term.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Bound the signed prime–composite correlation.Suggested move: Preserve the Möbius factor, coprimality, transition cutoff, shifted prime indicator, and both orientations through an explicit Type I/II decomposition.
Ready to work on
02
Put the composite–composite tail in a common bilinear form.Suggested move: Start from the corrected determinant tail, sum terminal ranks first, and expose coefficient norms, diagonal equations, and principal character.
Ready to work on
03
Exact Conjecture

The conjecture asserts that infinitely many primes p have p+2 prime.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Couple the two false-pair principal modes with a strict reserve.Suggested move: Compute their joint zero mode before Cauchy–Schwarz and prove a one-sided estimate strong enough for the exact completion criterion.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The exact gap-two conjecture remains open in the dated sources retained here. Bennett Chow's 2023 AMS monograph treats twin primes in Chapter 1 as an excursion into the unknown, and a 2025 heuristic arXiv preprint describes their quantitative distribution as a central open problem. The preprint is contemporary status context only. The peer-reviewed bounded-gap result H_1 <= 246 remains strictly weaker than H_1=2.

[4][5][2]
External progress

What the literature has established

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

  1. Peer reviewedMaynard proved bounded intervals containing multiple primes and an unconditional consecutive-gap bound of 600 in the published paper.[3]
  2. Peer reviewedPolymath8b obtained the unconditional bound H_1 <= 246; H_1=2 is the twin-prime target, so bounded gaps remain a weaker result.[2]
  3. Peer reviewedZhang proved that the liminf of consecutive prime gaps is finite, initially below 70,000,000; this does not establish gap 2.[1]
5 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusTwin Prime Conjecture
Weaker or relaxed formBounded gaps between primes

Bounded-gap theorems prove infinitely many prime pairs within some fixed finite interval; the Twin Prime Conjecture requires the exact gap 2.

[1][3]

Formalization opportunities

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

  • Formalization targetNo proof of infinitely many exact gap-two prime pairs was identified in this bounded dated-source correction; no separate official or primary 2026 status update was located.
  • Formalization targetAny use of the current work's signed detector route must independently verify its standard analytic interfaces and prove the two unresolved false-pair estimates.

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. 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

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.

5 standing statements2 proposed statements4 open questions2 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction3 of 73
  • lemma1 of 71
  • equivalence1 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.22 displayed rows · 2 routes included
  • retained route statementThere are infinitely many primes p such that p+2 is also prime. Equivalently in the source's smooth squarefree von Mangoldt formulation, T_sf,w(X)>0 for arbitrarily large X.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementSmooth Detectorintermediate
  • retained route statementSigned Decompositionintermediate
  • retained route statementThree False Pair Typesintermediate
  • retained route statementNo Overclaimintermediate
  • 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
  • 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 failureSource-reported closed or invalid routereported failure
  • Useful failureSecond source-reported limitationreported failure
  • Research targetBound the signed prime–composite correlation.open
  • Research targetPut the composite–composite tail in a common bilinear form.open
  • Research targetCouple the two false-pair principal modes with a strict reserve.open
  • Research targetExact Conjectureopen
  • Research targetOpen Correlationssuperseded
  • ComputationThe source proposes exact factorization experiments for B_Z and the two false-pair decompositions; ProofAtlas did not run any packet code or computation.No numerical experiment is treated as mathematical progress or as evidence for infinitely many twin primes. · reported unreproduced
  • Narrowed routeSource-reported closed or invalid routeA polylogarithmic progression modulus does not decouple the Möbius factor from the shifted prime, so Siegel–Walfisz alone cannot prove the needed correlation. Prove a signed one-sided bound for the combined prime–composite and composite–composite false-pair modes that leaves a fixed positive reserve below the hyperbolic main term.
  • Narrowed routeSecond source-reported limitationThe composite–composite support product bound does not make either complementary factor polylogarithmic, and rank shells cannot be estimated separately before terminal Möbius summation. Prove a signed one-sided bound for the combined prime–composite and composite–composite false-pair modes that leaves a fixed positive reserve below the hyperbolic main term.
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 bridgeBound the signed prime–composite correlation.

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.

Read-only beta · actions unavailable
Prepared starting pointBound the signed prime–composite correlation.

Twin Prime Conjecture · 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

Do infinitely many primes occur two apart? Bounded-gap theorems show infinitely many prime pairs within a fixed finite distance, but the exact gap two remains unproved.

  • 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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Proof attempt or partial resultSupporting notes or data
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Sources and references5 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Bounded gaps between primespeer reviewed result · Yitang Zhang · Annals of Mathematics · 2014-05-01 · DOI 10.4007/annals.2014.179.3.7 · MR MR3171761 · accessed Aug 15, 2026
  2. 2
    Variants of the Selberg sieve, and bounded intervals containing many primespeer reviewed result · D. H. J. Polymath · Research in the Mathematical Sciences · 2014-10-17 · ARXIV 1407.4897 · DOI 10.1186/s40687-014-0012-7 · MR MR3374754 · accessed Aug 15, 2026
  3. 3
    Small gaps between primespeer reviewed result · James Maynard · Annals of Mathematics · 2015-01-01 · DOI 10.4007/annals.2015.181.1.7 · MR MR3272929 · accessed Aug 15, 2026
  4. 4
    Introduction to Proof Through Number Theorysurvey or monograph · Bennett Chow · American Mathematical Society · 2023 · accessed Aug 15, 2026
  5. 5
    A Constructive Heuristic Sieve for the Twin Prime Problempreprint · Yuhang Shi · arXiv · 2025-07-10 · ARXIV 2507.03107 · DOI 10.48550/arXiv.2507.03107 · accessed Aug 15, 2026

Important qualifications

  • This was a bounded correction and current-status check, not an exhaustive priority or attribution review.
  • Bennett Chow's Introduction to Proof Through Number Theory is an AMS monograph published in 2023; Chapter 1 is the relevant source context, not a 2026 webpage.
  • The 2025 arXiv source is a heuristic preprint used only as contemporary primary-source status context; it is not peer-reviewed evidence and none of its mathematical claims were upgraded.
  • The bounded search did not identify a separate official or primary source published in 2026 that updates the exact conjecture's status; the as-of-2026 statement is therefore limited to the dated sources recorded here and does not claim an exhaustive literature review.
  • No source-package attachment was executed or rendered, and no URL copied from a submitted packet was fetched.
  • Metadata has no proof, review, acceptance, credit, publication, or deployment effect.

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