OpenAI Repos Show GPT-6-Astra Proving Prime Gaps ≤186 in Lean
On September 4, OpenAI open-sourced a GitHub repo named PrimeGaps186, containing a Lean-formalized proof produced by a model codenamed GPT-6-Astra: there are infinitely many pairs of consecutive primes with gaps of at most 186, i.e., liminf(p{n+1}-pn) ≤ 186, corresponding to the conditional result DHL[40,2]. The proof was verified with the Lean proof assistant, marking another output from a frontier model in formal mathematical proof.
Confirmed
- The repo is open-sourced; the proof was completed by GPT-6-Astra and formally verified in Lean 4
- The project derives the conditional conclusion DHL[40,2] from given inputs, i.e., a prime gap upper bound of 186
- According to @scaling01 and @johnowhitaker, the previous best bound of 240 was published just days earlier on August 31 and was immediately surpassed by this result
- @AcerFur, cited by @scaling01, added context: the result is conditional on 186 because it relies on a Deligne-related condition
Why it matters
- This is a number theory result independently produced by a frontier LLM and validated through rigorous formal verification, demonstrating AI's capability in formal mathematical proof
- It broke through to 186 just days after human researchers set a new record (240), a pace worth noting
- The open-source Lean repo makes the result verifiable and reproducible by the community
2026-09-04 ~ 2026-09-04 · 8 related posts
Primary sources
- [source] OpenAI repo claims GPT-6-Astra Lean-formalized a prime-gaps bound of 186 — scaling01 · 2026-09-04
- GPT-6-Astra Lean proof cuts prime gap bound from 240 to 186, days after previous record — scaling01 · 2026-09-04
- [source] OpenAI open-sources PrimeGaps186, a Lean formalization of a prime-gap bound of 186 — johnowhitaker · 2026-09-04
- OpenAI quietly publishes Lean proof repos, seen as warm-up for Astra release — NoFaithlessness951 · 2026-09-04
- Elementary proof with 4.5k-line Lean formalization beats SOTA on Erdős problem 4 — teortaxesTex · 2026-09-04
3 near-duplicate retellings: scaling01 · scaling01 · scaling01