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. |
| $240$ | [Sta26] | Stadlmann: “the Bombieri–Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli”, the latter being the exponent of distribution $\tfrac12 + \tfrac1{40}$ of [Sta25]. 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
-
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.
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.
- [Sta26] Stadlmann, Julia. Bounded gaps between primes. Preprint, 31 August 2026. arXiv:2608.31126.
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.