Bounded gaps between primes occur when there is a finite number 
such that infinitely many pairs of consecutive prime
numbers differ by at most 
. If 
denotes the 
th prime, define the infimum limit
 |
The assertion that bounded gaps between primes exist is the statement 
.
Zhang (2014) first proved this unconditionally, obtaining 
. In 2013, the Polymath8 project reduced the
bound to 
(Polymath 2014a), Maynard (2015) reduced it to 
, and further work of the Polymath8 project established
the unconditional bound 
(Polymath 2014b). Tao independently developed the
multidimensional sieve at about the same time, obtaining a slightly weaker result
(Maynard 2015, Charton et al. 2026). Maynard also proved that 
is finite for every positive integer 
(Maynard 2015). Consequently, for every fixed 
, infinitely many intervals of some fixed length contain at
least 
primes.
Stronger conditional bounds are known. Assuming the Elliott-Halberstam conjecture, Maynard (2015) proved 
and 
. The Polymath8 project improved the latter bound
to 
under the same conjecture. Under the stronger generalized form of the Elliott-Halberstam
conjecture, it proved 
and 
(Polymath 2014b). These are conditional bounds,
unlike the unconditional bound 
.
Except for the gap following 2, every prime gap is even. Therefore, 
implies that at least one even gap among 2, 4, ...,
246 occurs infinitely often between consecutive primes. It does not specify which
one. In particular, none of the unconditional or conditional bounds proves the twin prime conjecture, which is equivalent to
.
Several reports in August and September 2026 claimed improvements on 
. The following table lists these reports by decreasing
bound. The dates identify the cited papers or versions, rather than establish priority
of announcement.
bound on  | report date | reference | disclosed AI role |
| 240 | Aug. 31, 2026 | Stadlmann (2026) | no AI disclosed |
| 240 | Sep. 2, 2026 | H. Zhang (2026) | proof attributed to GPT-5.6 Sol |
| 236 | Sep. 1, 2026 | Kintali (2026) | human-directed work with multiple AI systems |
| 212 | Sep. 3, 2026 | Charton et al. (2026) | AxiomProver generated the Lean certificate |
| 186 | Aug. 30, 2026 | OpenAI (2026ab) | proof attributed to GPT-6 Astra |
As of Sep. 7, 2026, the reported checks had the scopes summarized below.
| reported bound | reported verification | scope and status |
| 240 (Stadlmann) | mathematical argument and matrix computation | no proof-assistant verification reported |
| 240 (H. Zhang) | exact arithmetic programs | finite calculations checked; analytic argument outside the checker |
| 236 (Kintali) | exact arithmetic programs, checked by the author | finite calculations checked; analytic inputs imported from the literature |
| 212 | Lean certificate generated by AxiomProver | analytic equidistribution estimates, other analytic inputs, and a separately checked variational certificate assumed |
| 186 | Lean kernel, Nanoda, Comparator, and Python/FLINT | conditional Lean proof; two established character-sum estimates and separately checked numerical integral bounds assumed |
Kintali (2026) disclosed the use of Claude, ChatGPT, and two open-weight models for proof development, with Codex writing the code and Kintali checking it. For the 186 bound, Comparator checked the correspondence with the specified formal statements, while Lean's kernel and Nanoda checked proofs conditional on the listed inputs (OpenAI 2026b). Neither the 212 nor the 186 development was an end-to-end Lean proof of the unconditional bound. The papers supplied mathematical arguments for the external inputs, but independent verification of the complete new bounds had not been reported as of Sep. 7, 2026.
