GPT-5.6 improves the Erdős–Rankin / Ford–Green–Konyagin–Maynard–Tao bound for large prime gaps
On 26 Aug 2026 the user "DottedCalculator" posted to erdosproblems.com (problem #4) a proof, generated with GPT-5.6, that there are infinitely many prime gaps larger than C·log n·log log n / log log log log n. This removes a log log log n factor from the 2018 Ford–Green–Konyagin–Maynard–Tao bound. Thomas Bloom wrote an exposition calling the ideas elementary. A fuller proof by GPT-6 Astra with a Lean formalization followed on 4 Sep 2026.
Key facts
- New bound: p_{n+1} − p_n > C·log n·log log n / log log log log n for infinitely many n
- Previous record: FGKMT 2018 (Ford, Green, Konyagin, Maynard, Tao), which had an extra log log log n factor in the denominator
- Method: a new weighting function to filter residue subsets, combined with the FGKMT18 machinery; Bloom notes neither ingredient alone improves the record
- Model naming differs: erdosproblems.com says 'GPT 5.6 Pro (prompted by DottedCalculator)', while Wikipedia's AI-discoveries list says GPT-5.6 Sol
- Follow-up: GPT-6 Astra full proof submitted 4 Sep 2026 with a Lean formalization (openai/LongGapsBetweenPrimes)
- Traictory (1 Sep 2026): no independent human verification yet at that point
Science result
- Field
- mathematics / analytic number theory
- Problem
- Large gaps between consecutive primes (Erdős–Rankin; Erdős problem #4)
- Result
- Improved lower bound for infinitely many large prime gaps, saving a log log log n factor over FGKMT 2018.
- AI system
- GPT-5.6 Pro/Sol, GPT-6 Astra
- Human role
- AI-assisted: pseudonymous user DottedCalculator prompted the model; Thomas Bloom wrote the exposition
- Verification
- Lean formalization of GPT-6 Astra's version reported; human expert review ongoing
- Status
- pending
What happened
A pseudonymous user got a GPT-5.6 model to combine new sieve weights with the Ford–Green–Konyagin–Maynard–Tao construction. The result improved the long-standing record for how large prime gaps can be. Thomas Bloom wrote it up on erdosproblems.com (last edited 31 Aug 2026). OpenAI's GPT-6 Astra then produced a complete proof with a Lean formalization.
Why it matters
Large prime gaps were famously advanced by Maynard and by Ford–Green–Konyagin–Tao in 2014–2018, and experts treated the FGKMT bound as hard to beat. This came four days before GPT-6 Astra's bounded-gaps record (246 → 186), so both ends of the prime-gap problem moved within a week.
Changelog
- 2026-09-29: created
Related events
- GPT-6 Astra lowers the bounded prime gaps record from 246 to 186 ★★★★
- GPT-6 Astra's Epoch AI run adds more Lean-checked results: Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved ★★★
Sources (4)
- discussionErdős problem #4
- discussionerdosproblems.com forum: problem #4 proof claims
- pressTraictory: GPT-5.6 claims a prime-gap record. Who checks the proof?
- discussionWikipedia: List of mathematical discoveries by artificial intelligence
id: 2026-08-26-erdos-rankin-large-prime-gaps-improved · updated 2026-09-29 · open in the interactive timeline