Bounded prime gap constant

Description of constant

Let $p_n$ denote the $n$-th prime. The bounded prime gap constant is

\[C_{88a} \;=\; H_1 \;:=\; \liminf_{n \to \infty} (p_{n+1} - p_n),\]

the least limit point of the sequence of gaps between consecutive primes. That $H_1$ is finite — that some bounded gap recurs infinitely often — was proved by Zhang [Zha14] in 2013; the twin prime conjecture is exactly the assertion $H_1 = 2$.

More generally one writes $H_m := \liminf_{n\to\infty}(p_{n+m} - p_n)$ for $m \ge 1$. Maynard [May15] proved $H_m$ finite for every $m$; the growth rate of $H_m$ in $m$ is recorded separately as $C_{88b}$.

Known upper bounds

Bound	Reference	Comments
$70\,000\,000$	[Zha14]	The first finiteness proof, via a bounded-level equidistribution estimate for smooth moduli beyond the Bombieri–Vinogradov range.
$4680$	[Pol14a]	Polymath8a, by optimizing Zhang’s equidistribution estimates.
$600$	[May15]	Maynard, via a multidimensional generalization of the Selberg sieve, using only Bombieri–Vinogradov. Obtained independently by Tao.
$246$	[Pol14b]	Polymath8b, generalizing the Maynard sieve further together with extensive numerical work. Stood as the record for twelve years.
$240$	[Zha26]	Zhang: proof discovered with GPT-5.6 Sol. Preprint publicly deposited on Zenodo on 28 August 2026, with an exact-arithmetic verification code released separately.
$186$	[OAI26]	Establishes $\mathrm{DHL}[40,2]$: every admissible $40$-tuple has infinitely many translates containing at least two primes. Combines the equidistribution estimates of [Pol14a] and [Sta25] with factorization conditions making suitable least common multiples of divisor products triply densely divisible, which widens the support of the multidimensional Selberg sieve, together with an improved numerical optimization. Applied to the admissible $40$-tuple $\{0, 2, 6, \dots, 182, 186\}$ of diameter $186$. Preprint of 30 August 2026, one day before [Sta26], which it describes as independent concurrent work; the proof is attributed to the model GPT 6 Astra. See the note on its formalization below.
$240$	[Sta26]	Stadlmann: independent proof using the Bombieri–Vinogradov theorem combined with newer equidistribution estimates for smooth moduli. Preprint of 31 August 2026.

Known lower bounds

Bound	Reference	Comments
$2$	Trivial	$p_{n+1} - p_n \ge 2$ for every $n \ge 2$. Equality, $H_1 = 2$, is the twin prime conjecture.

Additional comments and links

Reading the table. Rows are in chronological order, not in order of strength, so the last row is not the record. The record is $H_1 \le 186$ [OAI26], of 30 August 2026; the two $240$ rows on either side of it are independent proofs of the previous record, deposited on 28 and 31 August. Three of the four came within four days of each other and none cites the others.
Conditional bounds. Assuming the Elliott–Halberstam conjecture, $H_1 \le 12$ [May15], improving the bound $H_1 \le 16$ of Goldston–Pintz–Yıldırım [GPY09]. Assuming the generalized Elliott–Halberstam conjecture, $H_1 \le 6$ [Pol14b]. Reaching $H_1 = 2$ by these methods is obstructed by the parity problem.
Where the numbers come from. Each upper bound is the diameter of an explicit admissible $k$-tuple, for a $k$ that the sieve analysis determines; narrow admissible tuples are catalogued at Sutherland’s tables. Improvements therefore come either from lowering the admissible $k$ (better sieve) or from finding a narrower tuple of that size.
Relation to $C_{66}$. The level of distribution of the primes is what drives all of these bounds, and $C_{66} = 1$ (Elliott–Halberstam) would give $H_1 \le 12$. Note however that the equidistribution estimates actually used here — Zhang’s, Polymath’s $\tfrac12+\tfrac7{300}$, and Stadlmann’s $\tfrac12+\tfrac1{40}$ — are for smooth (friable) moduli with well-factorable weights, and so do not lower-bound $C_{66}$, which is defined over all moduli $q \le Q$. See the caution on the $C_{88b}$ page.
Formalization status of the $186$ bound. [OAI26] is accompanied by a Lean 4 development, openai/PrimeGaps186, in which PrimeGap186.dhl_40_2 and PrimeGap186.primeGapLiminf_le_186 carry no sorry. The proofs are nonetheless conditional on three declared axioms: the rank-three hyper-Kloosterman bound $\lvert \mathrm{Kl}_3(c;p)\rvert \le 3$, which follows from Deligne’s theorem as stated in Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Theorem 4.1.1; the rank-two Kloosterman correlation bound of Fouvry–Kowalski–Michel, Proposition 2; and $104$ outer plus $45$ inner physical-integral bounds together with three cap bounds, which are checked by an accompanying Python/FLINT certificate rather than proved in Lean. The first two are established results quoted from the literature; the third is the numerical part of the argument. The repository reports that Comparator, Nanoda and the Lean kernel accepted all three results, and describes its own review status as self-assessed, with no independent human semantic review.
Wikipedia page on Polignac’s conjecture

References

[GPY09] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples I. Annals of Mathematics 170 (2009), no. 2, 819–862. DOI: 10.4007/annals.2009.170.819.
[Zha14] Zhang, Yitang. Bounded gaps between primes. Annals of Mathematics 179 (2014), no. 3, 1121–1174. DOI: 10.4007/annals.2014.179.3.7.
[Pol14a] D. H. J. Polymath. New equidistribution estimates of Zhang type. Algebra & Number Theory 8 (2014), no. 9, 2067–2199. DOI: 10.2140/ant.2014.8.2067. arXiv:1402.0811.
[Pol14b] D. H. J. Polymath. Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences 1 (2014), Art. 12. DOI: 10.1186/s40687-014-0012-7. Erratum: ibid. 2 (2015), Art. 15, DOI: 10.1186/s40687-015-0033-x. arXiv:1407.4897.
[May15] Maynard, James. Small gaps between primes. Annals of Mathematics 181 (2015), no. 1, 383–413. DOI: 10.4007/annals.2015.181.1.7.
[Sta25] Stadlmann, Julia. On primes in arithmetic progressions and bounded gaps between many primes. Advances in Mathematics 468 (2025), Paper No. 110190. DOI: 10.1016/j.aim.2025.110190. arXiv:2309.00425.
[Zha26] Zhang, Hanxin. A bound of 240 for gaps between primes. Preprint, 28 August 2026. Zenodo. DOI: 10.5281/zenodo.22135842. Computational certificate and verification code: https://github.com/hanxinzhang/prime-bound.
[Sta26] Stadlmann, Julia. Bounded gaps between primes. Preprint, 31 August 2026. arXiv:2608.31126.
[OAI26] OpenAI. Improved short gaps between primes. Preprint, 30 August 2026. PDF. Conditional Lean 4 formalization and Python numerical certificate: openai/PrimeGaps186 (Apache-2.0). A companion document, Numerical certificate for prime gaps at most 186, carries the coefficient tables and verification records.

Contribution notes

Prepared with assistance from Claude Opus 5, which read the arXiv abstract and HTML introduction of [Sta26], the abstracts of [Sta25] and [Pol14b], and confirmed the bibliographic details of every reference through Crossref. The conditional bounds and the $70\,000\,000 \to 4680 \to 600 \to 246$ chain are quoted from the abstract of [Pol14b]; the $240$ and its provenance from [Sta26]. No bound was independently verified.

The $186$ row was added in #166 by @andrew-yoo and filled out with assistance from Claude Opus 5, which read [OAI26] itself and the README.md and formalization.yaml of the accompanying Lean repository. The axiom list, the zero-sorry claim and the self-assessed review status are as reported there; neither the Lean build nor the numerical certificate was re-run here.

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