The source labels their shared involution and product residues as leads only and prohibits claiming full-problem progress from a one-point finite enlargement. Use the finite supports as fixtures and structural clues while proving a separate prime-uniform arithmetic support theorem.
Route status · Narrowed routeCombinatorial geometry
No-three-in-line problem
Collaboration betaHow many points can be selected from an n by n integer grid without ever placing three selected points on one affine line?
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Known results and sources 
Research problem
Exact mathematical statement
For each positive integer , define
Determine the true growth of . The retained source studies the concrete open target
for some fixed and infinitely many ; it does not present an asymptotic solution.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on No-three-in-line problem stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
The source reduces its target to a fixed eta enlargement of the modular construction for infinitely many primes.
Evidence posture · Source-reported route statement · dependencies incompleteWe 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 We removed a duplicate or outdated task or route step.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 unchangedWork mapped so far
No-three-in-line problem in numbers
7.6kretained lines of mathematical investigation7,610 in the current working snapshot
- Argument development
- 6,570 · 86%
- Explored or eliminated routes
- 63 · 1%
- Computational analysis
- 375 · 5%
- Open obligations
- 190 · 2%
- Definitions and setup
- 412 · 5%
6selected 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.
Recommended next task
Construct one fixed family of candidate classes that is arithmetically valid for infinitely many primes.
Suggested move: Formalize the observed involution and product-residue patterns as prime-uniform class definitions before performing any further optimization.
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
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
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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
The source labels their shared involution and product residues as leads only and prohibits claiming full-problem progress from a one-point finite enlargement. Use the finite supports as fixtures and structural clues while proving a separate prime-uniform arithmetic support theorem.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Construct one fixed family of candidate classes that is arithmetically valid for infinitely many primes.Suggested move: Formalize the observed involution and product-residue patterns as prime-uniform class definitions before performing any further optimization.
Ready to work on
Prove density bounds for source mass, pair incidence, and mixed incidence that make the thinning objective exceed 0.42.Suggested move: Derive exact symbolic density formulas for the proposed class family and certify their maximum over the unit thinning interval.
Ready to work on
Translate the prime-indexed certified enlargements into a fixed positive linear margin for infinitely many grid sizes.Suggested move: Bind one constant eta independent of p and apply n=2p exactly, keeping every quantifier explicit.
Ready to work on
Sourced mathematical context
The known mathematical landscape
External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintVoutier documented the error in the Guy-Kelly heuristic and the resulting correction to its conjectured upper bound.[2] PreprintPrellberg exhibited 2n-point configurations for every n through 60, moving the smallest unresolved equality case to n=61.[1] PreprintA bounded computational comparison reported provably optimal ILP solutions through 19 by 19 and smaller-range heuristic results; it did not address asymptotic growth.[3]
Mathematical neighborhood
Related results and reusable starting points
Current focusNo-three-in-line problem
The equality D(n)=2n is computationally realized for every n up to 60; this is a finite range, not an asymptotic solution.
[1]The corrected Guy-Kelly proposal concerns a heuristic upper-bound profile related to, but not identical with, determining the exact function t(n).
[2]Formal and computational footholds
Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.
- computation · not independently reproducedConstraint-satisfaction configurations through n=60
The preprint reports 2n-point configurations for n at most 60; no code or configuration set was independently run in this intake.
[1]
Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA formal definition of affine collinearity and maximum collinearity-free subsets of finite integer grids.
- Formalization targetCertificate formats and checkers for finite grid configurations, with a theorem connecting certificates to lower bounds for t(n).
- Formalization targetA statement-aligned asymptotic framework that keeps finite equality cases separate from growth-rate claims.
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.
Correction detailsCorrected the research recordCorrection note
Research-record correctionWe 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.
Correction detailsCorrected the research recordCorrection note
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 statements2 proposed statements3 open questions1 narrowed routes
- theorem candidate
1 of 6 1 - reduction
2 of 6 2 - lemma
2 of 6 2 - computational claim
1 of 6 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
- retained route statementHow large can a no-three-collinear subset of an n by n grid be?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementPrime-indexed enlargement reductionintermediate
- retained route statementPair-only line-excess certificateintermediate
- retained route statementFinite bound at n=38intermediate
- 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 failureExtrapolating from two optimized finite supportsreported failure
- Research targetConstruct one fixed family of candidate classes that is arithmetically valid for infinitely many primes.open
- Research targetProve density bounds for source mass, pair incidence, and mixed incidence that make the thinning objective exceed 0.42.open
- Research targetTranslate the prime-indexed certified enlargements into a fixed positive linear margin for infinitely many grid sizes.open
- Research targetTrue growth of t(n)superseded
- Research targetFixed positive linear marginsuperseded
- Research targetPrime-uniform arithmetic supportsuperseded
- Narrowed routeExtrapolating from two optimized finite supportsThe source labels their shared involution and product residues as leads only and prohibits claiming full-problem progress from a one-point finite enlargement. Use the finite supports as fixtures and structural clues while proving a separate prime-uniform arithmetic support 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
1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.
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 pointConstruct one fixed family of candidate classes that is arithmetically valid for infinitely many primes.
No-three-in-line problem · ready to start
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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
How many points can be selected from an n by n integer grid without ever placing three selected points on one affine line?
- 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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Sources and references3 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.
- 1Constraint Satisfaction Programming for the No-three-in-line Problempreprint · Thomas Prellberg · arXiv · 2026-02-08 · ARXIV 2602.07751 · accessed Aug 15, 2026
- 2On the Guy-Kelly Conjecture for the No-Three-In-Line Problempreprint · Paul M. Voutier · arXiv · 2026-02-27 · ARXIV 2603.00215 · accessed Aug 15, 2026
- 3Three methods, one problem: Classical and AI approaches to no-three-in-linepreprint · Pranav Ramanathan, Thomas Prellberg, Matthew Lewis, Prathamesh Dinesh Joshi, Raj Abhijit Dandekar, Rajat Dandekar, Sreedath Panat · arXiv · 2025-12-12 · ARXIV 2512.11469 · accessed Aug 15, 2026
Important qualifications
- The exact ProofAtlas statement asks for the true growth of t(n), not only whether t(n)=2n for each finite n.
- The 2026 constraint-satisfaction result is a bounded construction through n=60 and is not generalized asymptotically.
- The Guy-Kelly paper corrects a heuristic upper-bound argument and is not treated as a proof of a sharp asymptotic law.
- No packet URL or attachment was fetched, executed, rendered, or treated as external evidence.
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