ProofAtlas

Number theory · prime divisors · matching theory

Grimm's Conjecture

Collaboration beta

Can distinct prime divisors always be assigned to a run of consecutive composite numbers?

∀n,k∈ℤ≥1,[∀i∈ℤ, 1≤i≤k⇒n+iis composite]⇒[∃ pairwise distinct primesp1,…,pk ∀i∈ℤ, 1≤i≤k⇒pi∣n+i]
Known results and sources
A text-free diagram of an abstract Hall-deficient incidence kernel for the sufficiently-large hypothetical-counterexample reduction, with five complete rectangular integer vertices above, four circular prime-neighbor vertices below, and every displayed incidence edge wholly inside one highlighted boundary.
Can distinct prime divisors always be assigned to a run of consecutive composite numbers?

Research problem

Exact mathematical statement

If n+1,n+2,…,n+kn+1,n+2,…,n+k are all composite, Grimm's Conjecture asks whether one can choose pairwise distinct primes p1,…,pkp_1,…,p_k with

pi∣n+i(1≤i≤k).p_i\mid n+i\qquad(1\le i\le k).

Equivalently, the divisibility bipartite graph should have a matching saturating every integer in the block.

Problem infographic

Problem at a glance

A text-free three-panel Grimm's Conjecture explainer: at left, five integer rectangles match one-to-one with five distinct prime circles; at center, one boundary encloses a five-integer, four-prime minimal Hall-deficient incidence kernel representing the sufficiently-large hypothetical-counterexample reduction; at right, four open prime circles connect by incomplete dashed lines to a disconnected unresolved motif beyond a visible gap.
For sufficiently large hypothetical counterexamples, the packet reports a reduction to a sparse Hall-deficient kernel and develops a detailed four-prime branch analysis. At that source-reported scope, several unique full-support graph classes survive, while the broader higher-support and finite-range completion tasks also remain open.

Current mathematical picture

Where work on Grimm's Conjecture stands

Open conjecture

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

Useful failureUsing a stronger scale inequality as the contradiction

The source shows a real exponent interval remains even under a stronger scale inequality. At the sufficiently-large hypothetical-counterexample scope, retain the minimal sparse Hall kernel and close its exact branch ledger rather than replacing it by the full interval.

Route status · Narrowed route
Main reductionCurrent reduction

For sufficiently large X, the source reports a canonical minimal Hall-deficient kernel with prime support equal to its neighborhood.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeEliminate the general K ≥ 2 and negative K = 1 cases in the unique three-prime-core exact regime.Task status · Ready to work on

Work mapped so far

Grimm's Conjecture in numbers

1.1kretained lines of mathematical investigation1,125 in the current working snapshot
Argument development
785 · 70%
Explored or eliminated routes
89 · 8%
Computational analysis
60 · 5%
Open obligations
75 · 7%
Definitions and setup
116 · 10%
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 Grimm's ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Consecutive composites should always have a system of distinct prime representatives. — Depends on missing premiseConsecutive compositesshould always have a systemof…Current reduction — Depends on missing premiseCurrent reductionHall matching target — Depends on missing premiseHall matching targetClosing target — Depends on missing premiseClosing targetMinimal deficient kernel — Depends on missing premiseMinimal deficient kernelPositive unit branch closed — Depends on missing premisePositive unit branch closedSupport at least four — Depends on missing premiseSupport at least fourUsing a stronger scale inequality as the contradiction — stoppedUsing a stronger scaleinequality as thecontradictionAssuming bounded primitive-relation degree — stoppedAssuming boundedprimitive-relation degreeEliminate the general K ≥ 2 and negative K = 1 cases in the unique three-prime-core exact regime. — OpenEliminate the general K ≥ 2and negative K = 1 cases inthe…Eliminate the surviving four-prime full-support graph types. — OpenEliminate the survivingfour-prime full-supportgraph…Control arbitrary higher support and the uncovered finite initial range. — OpenControl arbitrary highersupport and the uncoveredfinite…Unique full-support branch — OpenUnique full-support branchHigher support — OpenHigher support
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 routeUsing a stronger scale inequality as the contradiction

The source shows a real exponent interval remains even under a stronger scale inequality. At the sufficiently-large hypothetical-counterexample scope, retain the minimal sparse Hall kernel and close its exact branch ledger rather than replacing it by the full interval.

Route status · Narrowed route
Narrowed routeAssuming bounded primitive-relation degree

The source explicitly limits that classification to relations already known to have degree at most three. Use the degree census only within its hypothesis and seek independent degree or height control elsewhere.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Eliminate the general K ≥ 2 and negative K = 1 cases in the unique three-prime-core exact regime.Suggested move: Close those exact-regime cases while preserving every stated hypothesis.
Ready to work on
02
Eliminate the surviving four-prime full-support graph types.Suggested move: Determine whether the surviving unique full-support graph classes can occur.
Ready to work on
03
Unique full-support branch

Several graph classes survive in the unique full-support four-prime branch.

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

A full proof must address arbitrary higher support or reduce it validly.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Control arbitrary higher support and the uncovered finite initial range.Suggested move: Prove a valid reduction for arbitrary support or extend the kernel machinery beyond four primes, then close the finite range.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 28, 2026
Current statusOpen conjecture

The cited paper states the full distinct-prime-divisor assertion as a conjecture and proves related bounds and conditional implications. It does not prove the universal matching statement.

[1]
External progress

What the literature has established

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

  1. PreprintLaishram and Murty connected Grimm's conjecture to smooth numbers and proved unconditional and conditional bounds for the associated function g(n).[1]
1 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 consecutive composite blocks, prime divisibility, and a system of distinct representatives for their prime-divisor sets.
  • Formalization targetSmooth-number consequences and finite verification ranges do not establish the universal matching assertion.

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
  • lemma3 of 73
  • equivalence1 of 71
  • special case1 of 71
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 statementConsecutive composites should always have a system of distinct prime representatives.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementHall matching targetintermediate
  • retained route statementMinimal deficient kernelintermediate
  • retained route statementSupport at least fourintermediate
  • retained route statementPositive unit branch closedintermediate
  • 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 failureUsing a stronger scale inequality as the contradictionreported failure
  • Useful failureAssuming bounded primitive-relation degreereported failure
  • Research targetEliminate the general K ≥ 2 and negative K = 1 cases in the unique three-prime-core exact regime.open
  • Research targetEliminate the surviving four-prime full-support graph types.open
  • Research targetControl arbitrary higher support and the uncovered finite initial range.open
  • Research targetUnique full-support branchopen
  • Research targetHigher supportopen
  • ComputationThe current work states that scripts certify only their finite universes and identities.Any script result therefore remains limited to its stated finite universe and identity. Intake did not execute the scripts. · reported unreproduced
  • Narrowed routeUsing a stronger scale inequality as the contradictionThe source shows a real exponent interval remains even under a stronger scale inequality. At the sufficiently-large hypothetical-counterexample scope, retain the minimal sparse Hall kernel and close its exact branch ledger rather than replacing it by the full interval.
  • Narrowed routeAssuming bounded primitive-relation degreeThe source explicitly limits that classification to relations already known to have degree at most three. Use the degree census only within its hypothesis and seek independent degree or height control elsewhere.
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 the general K ≥ 2 and negative K = 1 cases in the unique three-prime-core exact regime.

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 pointEliminate the general K ≥ 2 and negative K = 1 cases in the unique three-prime-core exact regime.

Grimm's 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

Can distinct prime divisors always be assigned to a run of consecutive composite numbers?

  • 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
Hosted agentRun this task with a hosted agent

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

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references1 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
    Grimm's Conjecture and Smooth Numberspreprint · Shanta Laishram, M. Ram Murty · arXiv · 2013-06-04 · ARXIV 1306.0765 · 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.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image