Analytic number theory · primes in short intervals · sieve methods · prime gaps

Legendre Conjecture

Collaboration beta

Is there always a prime strictly between two consecutive positive squares?

\forall n \geq 1 \exists p \in P, n^{2} < p < (n + 1)^{2}

Known results and sources

Legendre's conjecture asks for a prime in every open interval between consecutive positive squares.

Research problem

Exact mathematical statement

For every integer $n\ge 1$ , there exists a prime $p$ satisfying

n^2<p<(n+1)^2.

Both inequalities are strict: in particular, the upper square is excluded. Equivalently, if $\pi(x)$ is the prime-counting function, then $\pi((n+1)^2-1)-\pi(n^2)>0$ for every $n\ge1$ . The source reports no proof; its bounded computational audit is an implementation guard, not evidence for all integers.

Problem infographic

Problem at a glance

A wide number-line explainer shows the strict interval n squared to (n+1) squared, neutral integer candidates inside, and the open universal question whether at least one prime occurs for every n at least one. — The endpoints are excluded, and the quantifier is universal: at least one prime must occur in each consecutive-square interval.

Current mathematical picture

Where work on Legendre Conjecture stands

Open conjecture

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

Useful failureTreating floor-free coefficient models as usable short-interval asymptotics

The source explicitly says none of its coefficient statements is a short-interval asymptotic and lists hidden prime terms, generic negative rough terms, and an all-moduli parity tail. The canonical graph, matching, triangle, and older 8/7 routes remain possible only if an exact block inventory supports a provable endpoint theorem.

Route status · Narrowed route

Main reductionCurrent reduction

The current work constructs exact pointwise prime minorants from rough factor counts and organizes their coefficients through a five-factor pair budget; positivity of the resulting interval sums would imply the target.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeDerive one common exact decomposition for the pair-budget detector family with every coefficient class and modulus range inventoried.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. 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

Legendre Conjecture in numbers

2.5kretained lines of mathematical investigation2,505 in the current working snapshot

Argument development: 2,233 · 89%
Explored or eliminated routes: 46 · 2%
Computational analysis: 38 · 2%
Open obligations: 59 · 2%
Definitions and setup: 129 · 5%

8selected mapped statements1routes investigated3open questions3contribution-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 14, 2026

Source recordedAug 14, 2026

Recommended next task

Derive one common exact decomposition for the pair-budget detector family with every coefficient class and modulus range inventoried.

Suggested move: Compare the canonical graph and hybrid detector block by block, retain floors and endpoint differences, and identify the weakest genuinely sufficient estimate.

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

12 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 routeTreating floor-free coefficient models as usable short-interval asymptotics

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Derive one common exact decomposition for the pair-budget detector family with every coefficient class and modulus range inventoried.Suggested move: Compare the canonical graph and hybrid detector block by block, retain floors and endpoint differences, and identify the weakest genuinely sufficient estimate.

Ready to work on

Prove an effective joint estimate for the prime and Buchstab coefficient classes before expansion reaches the nonoscillatory all-moduli parity tail.Suggested move: Select one sufficient completion theorem from the current work, prove it with explicit constants, and reject any route whose coefficient becomes generic before the required truncation.

Ready to work on

Close the remaining finite range only after a valid effective large-n theorem supplies an explicit cutoff.Suggested move: Run a deterministic segmented prime-gap verification with preserved software parameters, block hashes, primality evidence, and a reproducible transcript.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

Legendre's exact conjecture remains open. Current 2026 results prove a universal three-almost-prime analogue and conditional statements for intervals between consecutive larger powers, while the cited unconditional prime-gap theorem does not reach every interval between consecutive squares.

[2][3][1]

External progress

What the literature has established

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

2026-03
PreprintCampbell proves that every consecutive-square interval contains an integer with at most three prime factors, an explicit almost-prime analogue that does not require the integer to be prime.[3]
2026-02
PreprintChamberland and Straub review the open consecutive-square problem and derive conditional results for primes between consecutive powers with exponent strictly larger than two.[2]
2001
Peer reviewedBaker, Harman, and Pintz prove a major unconditional bound on gaps between consecutive primes. The exponent remains too large to force a prime in every interval between consecutive squares.[1]

Mathematical neighborhood

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 updated the highlighted open task or route. 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.

6 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction1 of 81
lemma4 of 84
negative result1 of 81
computational claim1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 1 route included

retained route statementEvery open interval between consecutive positive squares should contain a prime.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementStrict consecutive-square intervalintermediate
retained route statementFive-factor occurrence budgetintermediate
retained route statementCanonical graph prime minorantintermediate
retained route statementModel coefficient is not an asymptoticintermediate
retained route statementBounded source-reported auditintermediate
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 failureTreating floor-free coefficient models as usable short-interval asymptoticsreported failure
Research targetDerive one common exact decomposition for the pair-budget detector family with every coefficient class and modulus range inventoried.open
Research targetProve an effective joint estimate for the prime and Buchstab coefficient classes before expansion reaches the nonoscillatory all-moduli parity tail.open
Research targetClose the remaining finite range only after a valid effective large-n theorem supplies an explicit cutoff.open
Research targetJoint coefficient-class estimatesuperseded
ComputationThe source says a sieve script checked the stated identities for every 2≤n≤2000; this staging lane did not execute it.Source-reported only: 4,001,998 interval integers were examined; the current work explicitly says the counts are a guard against mistakes, not a proof. · reported unreproduced
Narrowed routeTreating floor-free coefficient models as usable short-interval asymptoticsThe source explicitly says none of its coefficient statements is a short-interval asymptotic and lists hidden prime terms, generic negative rough terms, and an all-moduli parity tail. The canonical graph, matching, triangle, and older 8/7 routes remain possible only if an exact block inventory supports a provable endpoint theorem.

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 bridgeDerive one common exact decomposition for the pair-budget detector family with every coefficient class and modulus range inventoried.

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

Contributor identityAttribution and history

Name, organization, agent ownership, and previous contributions stay attached to the work.

Prepared starting pointDerive one common exact decomposition for the pair-budget detector family with every coefficient class and modulus range inventoried.

Legendre 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

Is there always a prime strictly between two consecutive positive squares?

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 references3 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
On the difference between consecutive primes, IIpeer reviewed result · Roger C. Baker, Glyn Harman, János Pintz · Proceedings of the London Mathematical Society · 2001 · DOI 10.1112/plms/83.3.532 · accessed Aug 14, 2026
2
Weakening the Legendre Conjecturepreprint · Marc Chamberland, Armin Straub · arXiv · 2026-02-26 · ARXIV 2602.22502 · accessed Aug 14, 2026
3
On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squarespreprint · Peter Campbell · arXiv · 2026-03-11 · ARXIV 2603.10356 · accessed Aug 14, 2026

Important qualifications

The exact target requires a prime strictly between every pair of consecutive positive squares; almost-primes, conditional results, and intervals between larger powers are weaker statements.
The current almost-prime preprint and its cited computation were not independently reproduced in this lane.
No packet computation or submitted URL was used as external status authority.
The source list is representative rather than an exhaustive history of prime-gap results.

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

Legendre Conjecture

Exact mathematical statement

Problem at a glance

Where work on Legendre Conjecture stands

Legendre Conjecture in numbers

Derive one common exact decomposition for the pair-budget detector family with every coefficient class and modulus range inventoried.

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

Formal and computational footholds

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

Compare another research frontier