Bounded gaps between primes
: the smallest for which infinitely many pairs of consecutive primes are at most apart.
- steps
- 6
- by AI
- 2
- since
- 2013
Not on the chart: Zhang's (2013). It is more than a hundred times off the scale of everything else, and an axis that reached it would flatten the rest of the history into a few pixels. The table below has every step.
The line is the frontier over time. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line. Grey dots along the bottom edge are results from before the quantity had a number, placed there because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.
About this frontier
The twin prime conjecture says . Every approach descends from Goldston, Pintz and Yildirim, who in 2005 showed the gap is infinitely often smaller than any fixed multiple of without producing a finite bound. Zhang gave the first finite one in 2013, and the Polymath and Maynard work brought it to 246 within eighteen months. There it stayed for twelve years. Between 31 August and 3 September 2026 it moved four times, three of them with AI in the loop.
Every step, newest first
| Date | Value | Who | Model | Status | Source |
|---|---|---|---|---|---|
| 3 Sept 2026 | best Builds on Stadlmann. AxiomProver generated the Lean certificate of the deduction; the mathematics is the authors'. | Charton, Hong, Lau, Ono, Remy, Siu, Swaminathan, Thorner and Xie (Axiom Math) | AxiomProver | AI step lean-checked | entry |
| 1 Sept 2026 | Announced on X, building on Stadlmann's work. Held the record for two days. | Kintali, with AI | unspecified AI agents | AI step unreviewed | entry |
| 31 Aug 2026 | No AI involved. Combines Bombieri-Vinogradov with newer equidistribution estimates for smooth moduli. The row that ended twelve years of stasis. | Stadlmann | – | historical | source ↗ |
| 30 Aug 2026 | Not counted as the record here, pending anyone outside OpenAI reading it. Dated 30 August, before Stadlmann's 240, but the Lean development appeared on 2 September and the announcement later. Its formal proof is conditional on three project axioms. As of 5 September no named number theorist has assessed it, erdosproblems.com and Wikipedia have not recorded it, and the press calls 212 the record. | GPT 6 Astra (OpenAI) | GPT 6 Astra | candidate lean-checked | entry |
| 2014 | Stood as the record for twelve years. AxiomProver's Lean formalisation of this bound was completed in August 2026. | Polymath8b | – | historical | source ↗ |
| 2013 | The first finite bound. Seventy million, and the point was that it was finite at all. | Zhang | – | historical | source ↗ |
| Dec 2013 | A new multidimensional refinement of the GPY sieve, independent of Zhang's equidistribution work. | Maynard | – | historical | source ↗ |
| 20 Jul 2013 | A collaborative optimisation of Zhang's argument, in under two months. | Polymath8a | – | historical | source ↗ |
Rows through 2014 follow the Wikipedia article on prime gaps, with Maynard and Polymath8b cited to their own preprints. Stadlmann is arXiv 2608.31126. The 236 row is an announcement on X with no preprint identified. The 212 row is the Axiom Math paper and its formalisation site, and is the current record as the field reckons it. The 186 row is OpenAI's paper of 30 August 2026: a serious claim, catalogued as an entry, but drawn as a candidate because as of 5 September nobody outside OpenAI has assessed it and the field's record-keepers have not moved.
Changelog3 changes
- Curatorchanged 186 row status from “published” to “candidate”
- Curatorchanged name from “Smallest proved bound on infinitely recurring prime gaps” to “Bounded gaps between primes”
- Curatoradded this entry