Analytic number theory·primes:bounded-gaps
Bounded gaps between primes
No independent check recorded yet. A formal artifact, declaration or published object is attached, but no rebuild of it is recorded here.
Fidelity F2: The correspondence is declared through a Comparator challenge and an alignment table.
The smallest H for which infinitely many pairs of consecutive primes are at most H apart. The twin prime conjecture says H is 2; before 2013 no finite bound was known.
Record history
8 steps·gap between consecutive primes, lower is betterZhang gave the first finite bound in 2013, and within a year the Polymath projects and Maynard's multidimensional sieve brought it to 246, where it stayed for twelve years. On 31 August 2026 Julia Stadlmann posted 240. Within three days Shiva Kintali, directing several AI models, reached 236; Axiom Math reached 212, with a Lean certificate of the deduction produced by AxiomProver; and OpenAI released a report reaching 186, which it says is due to GPT-6 Astra. Science News reports that OpenAI's release came within two hours of Axiom's announcement, on 3 September, and the two steps are ordered that way here.
The report, dated 30 August, says the proof is due to GPT 6 Astra; its Lean formalisation is conditional on stated numerical bounds and exponential-sum estimates. Science News reports it was released with the model on 3 September, within two hours of Axiom's announcement.
Builds on Stadlmann's work. AxiomProver generated a Lean certificate of the deduction from natural-language specifications, taking the equidistribution estimates as hypotheses, as Appendix A of the preliminary draft says.
Extends Stadlmann's framework with exactly checked computer certificates. The paper's statement on AI: a harness the author built directed several models, and the author is responsible for the proofs and the code.
Combines the Bombieri–Vinogradov theorem with newer equidistribution estimates for smooth moduli.
Published in Research in the Mathematical Sciences in 2014; the record for twelve years.
The multidimensional sieve, needing only the Bombieri–Vinogradov theorem; published in the Annals of Mathematics in 2015.
Reached by the Polymath8a project in 2013, sharpening Zhang's method; published in Algebra & Number Theory in 2014.
The first finite bound; published in the Annals of Mathematics in 2014.
AI activity
How grades workProved that infinitely many pairs of consecutive primes are at most 236 apart, improving Stadlmann's 240.
Reasoning and sources
Autonomy
The author ran a harness directing several models, corrected and redirected them, and takes responsibility for the proofs and the code, as the paper's statement on AI use says.
Proved a bound of 212, building on Stadlmann's work, with a Lean certificate of the deduction.
Reasoning and sources
Autonomy
The mathematics is the authors'; AxiomProver played a supporting role, producing the Lean certificate of the deduction from their natural-language specifications (Appendix A).
Details
formalisation: A Lean certificate of the deduction, taking the equidistribution estimates as hypotheses
Proved that infinitely many pairs of consecutive primes are at most 186 apart.
Reasoning and sources
Autonomy
OpenAI's report says the proof is due to GPT 6 Astra and describes no human contribution to it.
Details
formalisation: Lean 4, conditional on stated numerical integral and cap bounds and on exponential-sum estimates
Fidelity
How fidelity is gradedF2 declared. The correspondence is declared through a Comparator challenge and an alignment table.
Declared by the projects
1As each project's formalization.yaml states it.
PrimeGaps186
- authors
- OpenAI
- method
- agent — GPT 6 Astra
- review
- self-assessed
- axioms
- Classical.choice, PrimeGap186.kloosterman2_correlation_bound, PrimeGap186.kloosterman3_bound, PrimeGap186.physical_integral_bounds, Quot.sound, propext
- results
- 3 main results named, checked with Comparator, with an alignment table
- sources
- Improved Gaps Between Primes — formalizes; Numerical certificate for prime gaps at most 186 — adapts
- checked by
- nobody independent of its authors yet
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qed.bot, “Bounded gaps between primes”, https://qed.bot/t/primes-bounded-gaps, as of 30 Sep 2026.