---
id: "2026-08-26-erdos-rankin-large-prime-gaps-improved"
url: "https://postcutoff.com/e/2026-08-26-erdos-rankin-large-prime-gaps-improved/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-08-26"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: "Lean proof, not yet reviewed"
sources: 6
editor: human editor
human_review: null
version: "2026-10-15"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-08-26-erdos-rankin-large-prime-gaps-improved/

# 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
- 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 118 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 57 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 148 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 87 days after its cutoff

## Sources

1. [arXiv 2609.15597: Long runs of integers with small prime factors and the divisor function of n! (Freiberg; Claude Fable 5.1)](https://arxiv.org/abs/2609.15597) (arxiv.org, paper)
2. [arXiv 2609.28253: A tyro's approach to the tilted sieve (Freiberg)](https://arxiv.org/abs/2609.28253) (arxiv.org, paper)
3. [Traictory: GPT-5.6 claims a prime-gap record. Who checks the proof?](https://traictory.com/news/2026-09-01-gpt-5-6-prime-gap-math-proofs) (traictory.com, press)
4. [Erdős problem #4](https://www.erdosproblems.com/4) (erdosproblems.com, discussion)
5. [erdosproblems.com forum: problem #4 proof claims](https://www.erdosproblems.com/forum/thread/4/proof-claims) (erdosproblems.com, discussion)
6. [Wikipedia: List of mathematical discoveries by artificial intelligence](https://en.wikipedia.org/wiki/List_of_mathematical_discoveries_by_artificial_intelligence) (en.wikipedia.org, discussion)

## Changes

- 2026-10-15: Added Radomskii's downstream papers (Romanoff gaps, Erdős #32) that reuse the construction
- 2026-09-30: Added Freiberg follow-ups (arXiv 2609.15597, 2609.28253)
- 2026-09-29 (filed): Created

## Related

- 2026-10-09: [Erdős #32: no O(log N) complement to the primes (Radomskii)](https://postcutoff.com/e/2026-10-09-erdos-32-additive-complements-primes-radomskii/index.md)
- 2026-09-10: [GPT-6 Astra's Epoch AI run adds more Lean-checked results](https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/index.md)
- 2026-08-30: [GPT-6 Astra lowers the bounded prime gaps record from 246 to 186](https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/index.md)
- People: [Thomas Bloom](https://postcutoff.com/person/thomas-bloom/)