GPT-6 Astra lowers the bounded prime gaps record from 246 to 186
An OpenAI preprint (30 Aug 2026) claims lim inf (p_{n+1} − p_n) ≤ 186, improving Polymath8b's bound of 246, which had stood since 2014. It uses 'triply densely divisible' conditions feeding a multidimensional Selberg sieve and was announced with a Lean formalisation. Julia Stadlmann independently reached 240 at about the same time.
Key facts
- Previous record: 246 (Polymath8b, 2014), building on Zhang (2013) and Maynard (2013)
- New claimed bound: 186
- Lean formalisation announced (Weijie Su); independent human verification not complete
- Human counterpart: Julia Stadlmann (UIUC), arXiv 2608.31126 (submitted 31 Aug 2026), proves 240 alone, 'with the assistance of traditional numerical computation, but not modern AI tools' (Tao); key idea: Motohashi–Pintz–Zhang estimates for only 'partly smooth' moduli
Science result
- Field
- mathematics / analytic number theory
- Problem
- Bounded gaps between primes (toward the twin prime conjecture) (open since 2014)
- Result
- Claimed proof that infinitely many pairs of primes differ by at most 186.
- AI system
- GPT-6 Astra
- Human role
- Largely AI-generated per OpenAI
- Verification
- Formal proof in Lean (announced); not yet independently peer-reviewed
- Status
- pending
- Why surprising
- A record that a large Polymath collaboration of top number theorists could not push for 12 years moved by 60 in one AI result.
What happened
OpenAI's model found a refinement of the Maynard–Tao sieve set-up that substantially improves the gap bound.
Why it matters
Bounded prime gaps were one of the celebrated stories of 2013–14. An AI improving the collaborative record is a striking, if still pending, result.
Changelog
- 2026-09-29: corrected arXiv 2608.31126 label (it is Stadlmann's human paper, not OpenAI's); added Tao's Mathstodon post on it
- 2026-09-29: created
Related posts (2)
- Tao: open problems have become a non-renewable resource Terence Tao @tao@mathstodon.xyz · other · 2026-09-03
Tao's influential threads arguing that AI labs racing to 'solve' famous problems use up a non-renewable resource, with Navier–Stokes as the example. They set the terms of the September 2026 debate. - Weijie Su Weijie Su @weijie444 · x · 2026-09-03
Cited as a source by: 2026-08-30-bounded-prime-gaps-186
Related events
- OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs ★★★★★
- OpenAI releases GPT-6 Astra, its first GPT-6 model ★★★★★
- GPT-5.6 improves the Erdős–Rankin / Ford–Green–Konyagin–Maynard–Tao bound for large prime gaps ★★★★
- GPT-6 Astra's Epoch AI run adds more Lean-checked results: Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved ★★★
Sources (4)
- paperOpenAI: short gaps between primes (PDF)
- paperJulia Stadlmann: Bounded gaps between primes (arXiv 2608.31126; human-only, bound 240)
- discussionTerence Tao on Mathstodon: Stadlmann shaves 246 to 240 without modern AI tools
- discussionWeijie Su on X (Lean formalisation)
id: 2026-08-30-bounded-prime-gaps-186 · updated 2026-09-29 · open in the interactive timeline