GPT-6 Astra lowers the bounded prime gaps record from 246 to 186
Awaiting review
Importance: major (4 of 5)The takeaway
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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean proof, not yet reviewed
- Importance
- Major (4 of 5)
- Last verified
- 29 September 2026
Your AI and this story
- GPT-6 Astra122 days after its cutoff
- Claude Opus 5.561 days after its cutoff
- Gemini 4 Hydrogen152 days after its cutoff
- Grok 4.791 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 61 days before it.
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
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.
What is disputed or not yet verified
| Verification | Formal proof in Lean (announced); not yet independently peer-reviewed |
|---|
Sources
4 sources from 4 sites. Numbers match the chips in the text.
4 sources: 2 primary, 2 reactions
Primary
- OpenAI: short gaps between primes (PDF)cdn.openai.com, paper
- Julia Stadlmann: Bounded gaps between primes (arXiv 2608.31126; human-only, bound 240)arxiv.org, paper
Reactions
- Terence Tao on Mathstodon: Stadlmann shaves 246 to 240 without modern AI toolsmathstodon.xyz, discussion
- Weijie Su on X (Lean formalisation)x.com, discussion
Changes
- Corrected arXiv 2608.31126 label (it is Stadlmann’s human paper, not OpenAI’s); added Tao’s Mathstodon post on it
- Filed