How small can prime gaps be?

7 selected milestones · Source coverage checked 2026-10-21 · Historical annotation and mathematical review remain separate.

1. 2013 preprint

   Bound established

   Maynard introduces a multidimensional sieve

   The cited source records H₁ ≤ 600.

   Statements & sources

   • Gaps at most 600, infinitely often →

   • A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

2. 2013 preprint

   Bound established

   Zhang establishes bounded gaps

   The cited source records H₁ ≤ 70,000,000.

   Statements & sources

   • Gaps at most 70,000,000, infinitely often →

   • A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

3. 2014 publication

   Bound established

   Polymath8b reaches 246

   The cited source records H₁ ≤ 246.

   Statements & sources

   • Gaps at most 246, infinitely often →

   • Variants of the Selberg sieve, and bounded intervals containing many primes ↗Main bounded-gap results and introduction

   Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

4. 2026 preprint

   Bound established

   Stadlmann lowers the mathematical bound to 240

   The cited source records H₁ ≤ 240.

   Statements & sources

   • Gaps at most 240, infinitely often →

   • A new bound for small gaps between primes ↗Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

5. Aug 18, 2026 formalization

   Formalization

   A public formal deduction for 246

   PrimeGapsLib exposes the 246 conclusion under a Bombieri–Vinogradov premise.

   Scope & review: The formal development and the unconditional mathematical result have different completeness boundaries.

   Statements & sources

   • Bombieri–Vinogradov implies recurring gaps at most 246 →

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

   Historical annotation review: pending. The date refers to the formalization, with the precision supported by the source.

6. Sep 2, 2026 formalization

   Formalization

   A conditional formal deduction for 186

   The source endpoint uses three named mathematical input axioms and a separate numerical certificate.

   Scope & review: This is not ranked as a new unconditional bound.

   Statements & sources

   • A gap bound of 186 from explicit input assumptions →

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

   Historical annotation review: pending. The date refers to the formalization, with the precision supported by the source.

7. Sep 3, 2026 preprint

   Proposal

   A preliminary draft reaches 212

   The preliminary draft states H₁ ≤ 212; it is shown as proposed.

   Statements & sources

   • 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

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

Scope & review: This is a bound recurring infinitely often, not a resolution of twin primes.