GPT-5.6 improves the Erdős–Rankin / Ford–Green–Konyagin–Maynard–Tao bound for large prime gaps
Awaiting review
Importance: major (4 of 5)The takeaway
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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean proof, not yet reviewed
- Importance
- Major (4 of 5)
- Last verified
- 15 October 2026
Your AI and this story
- GPT-6 Astra118 days after its cutoff
- Claude Opus 5.557 days after its cutoff
- Gemini 4 Hydrogen148 days after its cutoff
- Grok 4.787 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 57 days before it.
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
- Follow-ups by Tristan Freiberg: an improvement to long runs of smooth integers and the divisor function of n!, whose ‘core new argument … was generated during a private interaction with Claude Fable 5.1’ on 8 Sep 2026 (arXiv 2609.15597); and an elementary exposition of the ‘tilted sieve introduced by GPT-5.6 Sol’ (arXiv 2609.28253, which notes the Lean formalisation by B. Alexeev)
- Downstream (Sep–Oct 2026): Artyom Radomskii (HSE) used the preprint’s weight / truncated divisor-sum construction to improve the Kalmynin–Konyagin bound for gaps between Romanoff numbers (arXiv 2609.38408) and then to answer the $50 O(log N) part of Erdős Problem #32 on additive complements to the primes (arXiv 2610.14127); no AI-use statement in either paper
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.
What is disputed or not yet verified
| Verification | Lean formalization of GPT-6 Astra’s version reported; human expert review ongoing |
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Sources
6 sources from 4 sites. Numbers match the chips in the text.
6 sources: 2 primary, 1 press, 3 reactions
Primary
- arXiv 2609.15597: Long runs of integers with small prime factors and the divisor function of n! (Freiberg; Claude Fable 5.1)arxiv.org, paper
- arXiv 2609.28253: A tyro’s approach to the tilted sieve (Freiberg)arxiv.org, paper
Press
- Traictory: GPT-5.6 claims a prime-gap record. Who checks the proof?traictory.com, press
Reactions
- Erdős problem #4erdosproblems.com, discussion
- erdosproblems.com forum: problem #4 proof claimserdosproblems.com, discussion
- Wikipedia: List of mathematical discoveries by artificial intelligenceen.wikipedia.org, discussion
Changes
- Added Radomskii’s downstream papers (Romanoff gaps, Erdős #32) that reuse the construction
- Added Freiberg follow-ups (arXiv 2609.15597, 2609.28253)
- Filed