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Number theory · primes · difference dynamics

Gilbreath's Conjecture

Collaboration beta

Do repeated absolute differences of consecutive primes always begin with one?

A(N-1,1)=1for everyN≥2
Known results and sources
A text-free mathematical cover representing Gilbreath's Conjecture.
Do repeated absolute differences of consecutive primes always begin with one?

Research problem

Exact mathematical statement

Let p1=2,p2=3,p3=5,…p_1=2,p_2=3,p_3=5,… be the primes and form the absolute-difference triangle

A(0,s)=ps,A(r+1,s)=|A(r,s+1)-A(r,s)|.A(0,s)=p_s,\qquad A(r+1,s)=|A(r,s+1)-A(r,s)|.

Gilbreath's Conjecture asks whether

A(N-1,1)=1for everyN≥2.A(N-1,1)=1\qquad\text{for every }N\ge2.

Problem infographic

Problem at a glance

A text-free scientific explainer showing the objects, constraints, and unresolved route for Gilbreath's Conjecture.
The packet advances a least-counterexample extension program for Gilbreath's Conjecture. Exact fold geometry divides every failure into three branches and supplies strong endpoint structure, but each branch still lacks the arithmetic contradiction needed to close the conjecture.

Current mathematical picture

Where work on Gilbreath's Conjecture stands

Open conjecture

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

Useful failureUnconditional odd-index endpoint decimation

The source explicitly marks unconditional odd-index endpoint decimation false. Use the source-reported two-branch pair compression.

Route status · Narrowed route
Main reductionCurrent reduction

A least counterexample is exhaustively reduced to the three extension branches E0, E1a, and E1b.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeExclude the low endpoint branch, beginning with defects 2, 4, and 6.Task status · Ready to work on

Work mapped so far

Gilbreath's Conjecture in numbers

999retained lines of mathematical investigation999 in the current working snapshot
Argument development
749 · 75%
Explored or eliminated routes
62 · 6%
Computational analysis
24 · 2%
Open obligations
124 · 12%
Definitions and setup
40 · 4%
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 Gilbreath'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.Repeated absolute differences of consecutive primes should always end at one on the left edge. — Depends on missing premiseRepeated absolutedifferences of consecutiveprimes…Current reduction — Depends on missing premiseCurrent reductionPrime difference triangle — Depends on missing premisePrime difference triangleClosing target — Depends on missing premiseClosing targetExhaustive E0/E1 split — Depends on missing premiseExhaustive E0/E1 splitLeast-extension fold — Depends on missing premiseLeast-extension foldPositive endpoint defect — Depends on missing premisePositive endpoint defectUnconditional odd-index endpoint decimation — stoppedUnconditional odd-indexendpoint decimationMultiplying raw wheel spectral-radius decrements — stoppedMultiplying raw wheelspectral-radius decrementsExclude the low endpoint branch, beginning with defects 2, 4, and 6. — OpenExclude the low endpointbranch, beginning withdefects…Exclude the large-charge endpoint alternative. — OpenExclude the large-chargeendpoint alternative.Prove conditional all-time killing for entropy-bearing growing-wheel components. — OpenProve conditional all-timekilling for entropy-bearinggrowing-wheel…Low endpoint exclusion — OpenLow endpoint exclusionInternal wall exclusions — OpenInternal wall exclusions
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 routeUnconditional odd-index endpoint decimation

The source explicitly marks unconditional odd-index endpoint decimation false. Use the source-reported two-branch pair compression.

Route status · Narrowed route
Narrowed routeMultiplying raw wheel spectral-radius decrements

The source explicitly marks uniform multiplicative decrement of the full wheel spectral radius false. Pursue conditional all-time killing for entropy-bearing growing-wheel components.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Exclude the low endpoint branch, beginning with defects 2, 4, and 6.Suggested move: Combine the placed impulse with its companion collar for the low endpoint-defect branch.
Ready to work on
02
Exclude the large-charge endpoint alternative.Suggested move: Merge e_j at least 4L-4 with the marked-spike, source-cone, and multiplicative excess budgets.
Ready to work on
03
Low endpoint exclusion

The placed impulse and companion collar must be excluded arithmetically.

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

The E1a zero block and E1b localized wall remain open.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Prove conditional all-time killing for entropy-bearing growing-wheel components.Suggested move: Classify and arithmetically exclude the quantitative coboundary exceptions.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 28, 2026
Current statusOpen conjecture

The cited preprint continues to state the prime absolute-difference assertion as a conjecture. Its random analogue and conditional obstruction analysis are meaningful partial progress but not a proof of the deterministic claim.

[1]
External progress

What the literature has established

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

  1. PreprintChase, Hunter, and Tao proved the random-model analogue and a conditional inverse theorem isolating deterministic obstructions, not the prime-sequence conjecture itself.[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 the prime sequence, the iterated absolute-difference triangle, and a proof that every left-edge entry after the initial row equals one.
  • Formalization targetThe random-model theorem and conditional inverse theorem must remain distinct from the deterministic prime-sequence conjecture.

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
  • lemma4 of 74
  • equivalence1 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 statementRepeated absolute differences of consecutive primes should always end at one on the left edge.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementPrime difference triangleintermediate
  • retained route statementLeast-extension foldintermediate
  • retained route statementExhaustive E0/E1 splitintermediate
  • retained route statementPositive endpoint defectintermediate
  • 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 failureUnconditional odd-index endpoint decimationreported failure
  • Useful failureMultiplying raw wheel spectral-radius decrementsreported failure
  • Research targetExclude the low endpoint branch, beginning with defects 2, 4, and 6.open
  • Research targetExclude the large-charge endpoint alternative.open
  • Research targetProve conditional all-time killing for entropy-bearing growing-wheel components.open
  • Research targetLow endpoint exclusionopen
  • Research targetInternal wall exclusionsopen
  • ComputationThe source reports finite calculations for algebra, indexing, exhaustive small profiles, finite-field kernels, and graph/period identities.The source says those calculations do not prove asymptotic prime distribution, a uniform component-lift entropy loss, or a prime-specific contradiction for any least-extension leaf. Intake did not execute them. · reported unreproduced
  • Narrowed routeUnconditional odd-index endpoint decimationThe source explicitly marks unconditional odd-index endpoint decimation false. Use the source-reported two-branch pair compression.
  • Narrowed routeMultiplying raw wheel spectral-radius decrementsThe source explicitly marks uniform multiplicative decrement of the full wheel spectral radius false. Pursue conditional all-time killing for entropy-bearing growing-wheel components.
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 bridgeExclude the low endpoint branch, beginning with defects 2, 4, and 6.

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 pointExclude the low endpoint branch, beginning with defects 2, 4, and 6.

Gilbreath'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

Do repeated absolute differences of consecutive primes always begin with one?

  • 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 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
    Gilbreath's conjecture: a Cramér random model and a deterministic analysispreprint · Zachary Chase, Zach Hunter, Terence Tao · arXiv · 2026-07-09 · ARXIV 2607.08712 · 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.

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