# Bounded gaps between primes

A record target in qed.bot, the register of claims of AI work in mathematics. The register grades the evidence attached to each claim; it never rules on whether a proof is correct.

- Identifier: primes:bounded-gaps
- Collection: Analytic number theory
- Page: https://qed.bot/t/primes-bounded-gaps
- Record: 186, held by GPT-6 Astra (OpenAI), since 3 Sep 2026
- Machine-checked: no independent check has verified it
- Fidelity F2: The correspondence is declared through a Comparator challenge and an alignment table.
- Source: https://arxiv.org/abs/2608.31126

## Record history

- May 2013: 70,000,000, Yitang Zhang; moved the record. https://doi.org/10.4007/annals.2014.179.3.7
- Jul 2013: 4,680, Polymath8a; moved the record. https://doi.org/10.2140/ant.2014.8.2067
- 19 Nov 2013: 600, James Maynard; moved the record. https://arxiv.org/abs/1311.4600
- 2014: 246, Polymath8b; moved the record. https://arxiv.org/abs/1407.4897
- 31 Aug 2026: 240, Julia Stadlmann; moved the record. https://arxiv.org/abs/2608.31126
- 1 Sep 2026: 236, Shiva Kintali, with Claude, ChatGPT, Codex and open-weight models, with AI; moved the record. https://shivakintali.github.io/papers/Prime-Gaps.pdf
- 3 Sep 2026: 212, François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Ashvin Swaminathan, Jesse Thorner and Yunzhou Xie (Axiom Math), with AxiomProver, with AI; moved the record. https://primegaps.axiommath.ai/bgp212.pdf
- 3 Sep 2026: 186, GPT-6 Astra (OpenAI), with AI; moved the record. https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf

## AI claims

### Claude, ChatGPT, Codex, 1 Sep 2026

Proved that infinitely many pairs of consecutive primes are at most 236 apart, improving Stadlmann's 240.

- Outcome: superseded
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V2: Artifact reported. A proof or a constructed object is published, but nobody independent of its authors has rebuilt it.
- Fidelity F2: Declared. The authors record how the formal statement corresponds to the claim — a comparator challenge, an alignment table or written divergences — or a statement corpus cites the proof against its own statement.
- With: Shiva Kintali
- Source: Kintali, Improved bounded gaps between primes, https://shivakintali.github.io/papers/Prime-Gaps.pdf
- Source: Code and certificates, https://github.com/shivakintali/Prime-Gaps

### AxiomProver, 3 Sep 2026

Proved a bound of 212, building on Stadlmann's work, with a Lean certificate of the deduction.

- Outcome: superseded
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V2: Artifact reported. A proof or a constructed object is published, but nobody independent of its authors has rebuilt it.
- Fidelity F2: Declared. The authors record how the formal statement corresponds to the claim — a comparator challenge, an alignment table or written divergences — or a statement corpus cites the proof against its own statement.
- With: François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Ashvin A. Swaminathan, Jesse Thorner, Yunzhou Xie
- Source: Axiom Math, A new bound for small gaps between primes (preliminary draft), https://primegaps.axiommath.ai/bgp212.pdf

### GPT-6 Astra, 3 Sep 2026

Proved that infinitely many pairs of consecutive primes are at most 186 apart.

- Outcome: record
- Autonomy A3: Autonomous. The system produced the result without significant human mathematical involvement.
- Evidence V2: Artifact reported. A proof or a constructed object is published, but nobody independent of its authors has rebuilt it.
- Fidelity F2: Declared. The authors record how the formal statement corresponds to the claim — a comparator challenge, an alignment table or written divergences — or a statement corpus cites the proof against its own statement.
- Source: OpenAI, Improved short gaps between primes, https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf
- Source: Conditional Lean formalisation, https://github.com/openai/PrimeGaps186

Cite as: qed.bot, "Bounded gaps between primes", https://qed.bot/t/primes-bounded-gaps, as of 30 Sep 2026. The register's data is published under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/