How small can prime gaps be?

Overview

Known

Finite recurring-gap bounds are established in the cited literature, while H₁ = 2 remains unresolved. The selected 2026 preliminary draft states an unconditional bound of 212; the 186 formal result has explicit input assumptions.

Gaps at most 70,000,000, infinitely oftenProvedCharton et al. 2026
Gaps at most 600, infinitely oftenProvedCharton et al. 2026
Gaps at most 246, infinitely oftenProvedPolymath 2014

Open questions

Establish recurring gaps of two, or independently validate and strengthen the selected bounded-gap results and their formal evidence.

Latest

A preliminary draft reaches 212Sep 3, 2026
A conditional formal deduction for 186Sep 2, 2026

Precise statement

The primes pₙ are listed in increasing order. $H_1=\liminf_{n\to\infty}(p_{n+1}-p_n)$ measures the smallest gap recurring arbitrarily far out. A bound H₁ ≤ B does not bound every gap or prove twin primes unless B = 2.

Question paraphrased by OpenMath. Original wording is available in the sources.

Landscape

Figures and explorable examples. Selecting a statement opens its exact wording.

Visual setup

A small gap is not yet an infinite family

The consecutive primes 101, 103, 107 and 109 have gaps 2, 4 and 2. These finite examples explain the quantity being bounded.

Reading boundary. A finite list cannot establish infinitely many twin primes or any asymptotic bound. The reported bounds are presented with their own assumptions below.

Read the infinite recurrence targetQuestion scope

Select a statement below for its scope and evidence. The text view describes the complete drawing.

Text version

On a number line, consecutive primes 101, 103, 107 and 109 are marked, with gaps 2, 4 and 2; the universal question asks for such small gaps arbitrarily far out.

Coordinates in the drawing are illustrative unless the caption specifies an exact example.

Read the infinite recurrence target
Open
Gaps of two infinitely often
The twin-prime target. Finite upper bounds larger than two leave it unresolved.
Gaps of two infinitely often →

Sources & review

Authored by OpenMath source curation on 2026-10-13 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.

A new bound for small gaps between primesTheorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProverPreprintOriginal passage ↗
Variants of the Selberg sieve, and bounded intervals containing many primesMain bounded-gap results and introductionPublishedOriginal passage ↗

Results

The question 1

Open
Gaps of two infinitely often
The twin-prime target. Finite upper bounds larger than two leave it unresolved.
Conjecture · Primary formulation
Open
Gaps of two infinitely often
For every real bound X there are consecutive primes p < q with p ≥ X and q − p = 2; equivalently $H_1=2$ .
Assumptions
- Consecutive positive primes in increasing order.
In this problem
The twin-prime target. Finite upper bounds larger than two leave it unresolved.
Sources
- Variants of the Selberg sieve, and bounded intervals containing many primes →
- A new bound for small gaps between primes →
Open statement page →

Bounds 5

Proved
Gaps at most 70,000,000, infinitely often
The cited mathematical result bounds recurring consecutive-prime gaps by 70,000,000.
Charton et al. 2026
Bound
ProvedCharton et al. 2026
Gaps at most 70,000,000, infinitely often
For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 70000000; in other words $H_1\le70{,}000{,}000$ .
Assumptions
- Consecutive positive primes.
- No additional conjectural hypothesis in the cited mathematical result.
In this problem
The cited mathematical result bounds recurring consecutive-prime gaps by 70,000,000.
An unconditional upper bound on the recurring gap, attributed to Zhang; it does not imply every gap is this small.
Does not cover: The target H₁ = 2 is not established.
Sources
- A new bound for small gaps between primes →
Open statement page →
Proved
Gaps at most 600, infinitely often
The cited mathematical result bounds recurring consecutive-prime gaps by 600.
Charton et al. 2026
Bound
ProvedCharton et al. 2026
Gaps at most 600, infinitely often
For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 600; in other words $H_1\le600$ .
Assumptions
- Consecutive positive primes.
- No additional conjectural hypothesis in the cited mathematical result.
In this problem
The cited mathematical result bounds recurring consecutive-prime gaps by 600.
An unconditional upper bound on the recurring gap, attributed to Maynard; it does not imply every gap is this small.
Does not cover: The target H₁ = 2 is not established.
Sources
- A new bound for small gaps between primes →
Open statement page →
Proved
Gaps at most 246, infinitely often
The cited mathematical result bounds recurring consecutive-prime gaps by 246.
Polymath 2014
Bound
ProvedPolymath 2014
Gaps at most 246, infinitely often
For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 246; in other words $H_1\le246$ .
Assumptions
- Consecutive positive primes.
- No additional conjectural hypothesis in the cited mathematical result.
In this problem
The cited mathematical result bounds recurring consecutive-prime gaps by 246.
An unconditional upper bound on the recurring gap, attributed to Polymath8b; it does not imply every gap is this small.
Does not cover: The target H₁ = 2 is not established.
Sources
- Variants of the Selberg sieve, and bounded intervals containing many primes →
Open statement page →
Proved
Gaps at most 240, infinitely often
The cited mathematical result bounds recurring consecutive-prime gaps by 240.
Charton et al. 2026
Bound
ProvedCharton et al. 2026
Gaps at most 240, infinitely often
For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 240; in other words $H_1\le240$ .
Assumptions
- Consecutive positive primes.
- No additional conjectural hypothesis in the cited mathematical result.
In this problem
The cited mathematical result bounds recurring consecutive-prime gaps by 240.
An unconditional upper bound on the recurring gap, attributed to Stadlmann; it does not imply every gap is this small.
Does not cover: The target H₁ = 2 is not established.
Sources
- A new bound for small gaps between primes →
Open statement page →
Proved
Gaps at most 212, infinitely often
The cited mathematical result bounds recurring consecutive-prime gaps by 212.
Charton et al. 2026
Bound
ProvedCharton et al. 2026
Gaps at most 212, infinitely often
For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 212; in other words $H_1\le212$ .
Assumptions
- Consecutive positive primes.
- No additional conjectural hypothesis in the cited mathematical result.
In this problem
The cited mathematical result bounds recurring consecutive-prime gaps by 212.
An unconditional upper bound on the recurring gap, attributed to the 2026 preliminary draft; it does not imply every gap is this small.
Does not cover: The target H₁ = 2 is not established.
Sources
- A new bound for small gaps between primes →
Open statement page →

Conditional results 2

Connections

Consequences 1

Assuming Gaps at most 212, infinitely oftenProved

Smaller recurring gaps, with the twin-prime target still open
Under stated assumptions
The 212 paper’s assertion would improve the unconditional recurring-gap bound in the selected literature. It gives arbitrarily large prime pairs at that distance, without controlling all successive gaps.
Limit The manuscript is a preliminary draft; independent OpenMath review is pending. A bound larger than two does not prove twin primes.
Sources & review
Supporting statements
- Gaps at most 212, infinitely often →
- A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver
Source-grounded explanation · independent explanation review: pending.

Approaches

How the problem is attacked, and what each line of attack cannot yet reach.

Sieve weights, distribution estimates and exact certificates

The 212 argument improves a sieve construction descended from Stadlmann’s work and verifies rational inequalities. Different formal artifacts certify different deductions and inputs.

Cannot yet reach The public 246 repository cannot be substituted for the 212 paper’s reported certificate.

Gaps at most 212, infinitely often (Proved)
Bombieri–Vinogradov implies recurring gaps at most 246 (Proved)
A gap bound of 186 from explicit input assumptions (Proved)

Sources & review

Supporting statements

Variants of the Selberg sieve, and bounded intervals containing many primes ↗Main bounded-gap results and introduction
A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver
PrimeGapsLib: a formal deduction of the 246 bound ↗README main results; PrimeGaps/Bounded246.lean
Prime gaps at most 186: a conditional Lean development ↗README input axioms and verification; PrimeGaps186.lean

Source-grounded explanation · independent explanation review: pending.

Timeline

Sep 3, 2026A preliminary draft reaches 212Bound established
The cited source records H₁ ≤ 212.
This is a bound recurring infinitely often, not a resolution of twin primes.
- Gaps at most 212, infinitely oftenProved
- A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver
Also on the Frontier: Bounded gaps between consecutive primes
Sep 2, 2026A conditional formal deduction for 186Formalization
The source endpoint uses three named mathematical input axioms and a separate numerical certificate.
This is not ranked as a new unconditional bound.
- A gap bound of 186 from explicit input assumptionsProved
- Prime gaps at most 186: a conditional Lean development ↗README input axioms and verification; PrimeGaps186.lean
Aug 18, 2026A public formal deduction for 246Formalization
PrimeGapsLib exposes the 246 conclusion under a Bombieri–Vinogradov premise.
The formal development and the unconditional mathematical result have different completeness boundaries.
- Bombieri–Vinogradov implies recurring gaps at most 246Proved
- PrimeGapsLib: a formal deduction of the 246 bound ↗README main results; PrimeGaps/Bounded246.lean
2026Stadlmann lowers the mathematical bound to 240Bound established
The cited source records H₁ ≤ 240.
This is a bound recurring infinitely often, not a resolution of twin primes.
- Gaps at most 240, infinitely oftenProved
- A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver
2014Polymath8b reaches 246Bound established
The cited source records H₁ ≤ 246.
This is a bound recurring infinitely often, not a resolution of twin primes.
- Gaps at most 246, infinitely oftenProved
- Variants of the Selberg sieve, and bounded intervals containing many primes ↗Main bounded-gap results and introduction
2013Zhang establishes bounded gapsBound established
The cited source records H₁ ≤ 70,000,000.
This is a bound recurring infinitely often, not a resolution of twin primes.
- Gaps at most 70,000,000, infinitely oftenProved
- A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver
2013Maynard introduces a multidimensional sieveBound established
The cited source records H₁ ≤ 600.
This is a bound recurring infinitely often, not a resolution of twin primes.
- Gaps at most 600, infinitely oftenProved
- A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver

Full history with sources →

Evidence

Evidence by kind
Literature	1 published paper · 1 preprint · 2 projects Status OpenMath assessment · Oct 13, 2026. Partly resolved: Variants of the Selberg sieve, and bounded intervals containing many primes · Bounded gaps established; twin primes remain unresolved ↗ · source date not recorded · checked Oct 13, 2026H₁ = 2 remains unresolved despite finite upper bounds.
Formal	2 complete proofs 2 formalizations below ↓ Reported in sources: Public Lean evidence covers a 246 deduction from Bombieri–Vinogradov and a 186 deduction from three input axioms. The 212 paper reports a separate certificate with analytic and variational hypotheses; its exact public source revision was not identified. No OpenMath build is recorded.
Computation	None recorded
Review	Not yet reviewed OpenMath assessment: Finite recurring-gap bounds are established in the cited literature, while H₁ = 2 remains unresolved. The selected 2026 preliminary draft states an unconditional bound of 212; the 186 formal result has explicit input assumptions. Source comparison and page curation only; independent mathematical review is pending. OpenMath pilot collection · OpenMath source curation · assessed Oct 13, 2026 · review due Jan 13, 2027 Collection last checked Oct 13, 2026.

Formalizations

Bombieri–Vinogradov implies recurring gaps at most 246 →

Lean: 246 from Bombieri–VinogradovComplete formal proof reported externally

Open formal source ↗

Complete formal proof reported externally

Complete as an implication with the stated premise. This is the public 246 artifact, not the later reported 212 certificate.

Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026

Declaration & revision

bombieriVinogradov_implies_frequently_prime_gap_le_246

Commit 1faa7b14e82ddebc2772dfb9153922f01b106477

PrimeGapsLib: a formal deduction of the 246 bound: README main results; PrimeGaps/Bounded246.lean →

A gap bound of 186 from explicit input assumptions →

Lean: 186 with three input axiomsComplete formal proof reported externally

Open formal source ↗

Complete formal proof reported externally

Reported complete as a conditional theorem. Its numerical receipt does not turn the input axioms into Lean proofs.

Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026

Declaration & revision

PrimeGap186.primeGapLiminf_le_186

Commit 61340d0b74163003b32756bb16e91d9209a5e330

Prime gaps at most 186: a conditional Lean development: README input axioms and verification; PrimeGaps186.lean →

External proof reports do not establish an OpenMath run, independent statement correspondence, or independent mathematical review.

Sources

Choose a starting point below. Each result links to its exact statement, cited passages and checks.

Start here

Introduction: How small can prime gaps be? →

Bibliography 4

Variants of the Selberg sieve, and bounded intervals containing many primes →
D. H. J. Polymath · Published · 2014
Main bounded-gap results and introduction
A new bound for small gaps between primes →
François Charton · Letong Hong · Kenny Lau · Ken Ono · Guillaume Remy · Ho Chung Siu · Ashvin A. Swaminathan · Jesse Thorner · Yunzhou Xie · Preprint · Sep 3, 2026
Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver
PrimeGapsLib: a formal deduction of the 246 bound →
Axiom Math · PrimeGapsLib contributors · Project documentation · Aug 18, 2026
README main results; PrimeGaps/Bounded246.lean
Prime gaps at most 186: a conditional Lean development →
OpenAI · PrimeGaps186 contributors · Project documentation · Sep 2, 2026
README input axioms and verification; PrimeGaps186.lean

Research activity · 2 entries

Contributions

The 212 mathematical paper credits nine authors and builds on Julia Stadlmann. AxiomProver is credited for its formal certificate, not independently discovering that bound.

Formalization

The 212 certificate is reported in the paper

The appendix credits AxiomProver with a Lean deduction that assumes five equidistribution estimates, bilinear Bombieri–Vinogradov, Harman decomposition and the variational certificate.

No exact public commit or declaration for that 212 certificate was established by this source check; it is shown as a paper-reported artifact, not a fabricated formal-code link.

Related statements · context, not additional proved progress

Gaps at most 212, infinitely often →

A new bound for small gaps between primes · sources & artifacts →

Computation

The 186 numerical certificate and Lean assumptions

The numerical program and the conditional Lean development are distinct artifacts with different checking boundaries.

A passing numerical receipt does not discharge the three Lean input axioms.

Related statements · context, not additional proved progress

A gap bound of 186 from explicit input assumptions →

Prime gaps at most 186: a conditional Lean development · sources & artifacts →

Selected literature; source observations and coverage limits are retained in the publication details.

Record details

Record version: 3
Classification: MSC2020 11N05 / 11N36
Prerequisite level: graduate

Source observations & version history →

Overview

Known

Open questions

Latest

Landscape

A small gap is not yet an infinite family

Gaps of two infinitely often

Results

The question 1

Assumptions

In this problem

Sources

Bounds 5

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

Conditional results 2

Assumptions

In this problem

Formal proofs

Sources

Assumptions

In this problem

Formal proofs

Sources

Connections

Consequences 1

Related problems

Approaches

Sieve weights, distribution estimates and exact certificates

Timeline

Evidence

Formalizations

Bombieri–Vinogradov implies recurring gaps at most 246 →

A gap bound of 186 from explicit input assumptions →

Sources

Start here

Bibliography 4

Contributions

The 212 certificate is reported in the paper

The 186 numerical certificate and Lean assumptions