Exponent for bounded gaps between many primes

Let p_n denote the n-th prime and, for m ≥ 1, write H_m := liminf_n → ∞ (p_n+m - p_n) for the least limit point of the gaps between primes m apart.

Current state (tracked automatically)

Best known bounds for constant 88b: 0 ≤ C ≤ 3.8075. Upper bound ([Sta25], Stadlmann: H\_m = O((3.8075\,m)), deduced from an exponent of distribution 12 + 140 for the primes in arithmetic progressions to smooth moduli, improving Polyma); lower bound (Trivial, No positive lower bound is known, and none is expected: see below).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 88b)	0 Trivial — No positive lower bound is known, and none is expected: see below.	Tao's optimization constants
Best upper bound with an explicit value (constant 88b)	3.8075 [Sta25] — Stadlmann: H\_m = O((3.8075\,m)), deduced from an exponent of distribution 12 + 140 for the primes in arithmetic progressions to smooth moduli, improving Polyma	Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.

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Cite

@misc{cairn-constant-88b-exponent-for-bounded-gaps-between-many-primes,
  title        = {Exponent for bounded gaps between many primes},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-88b-exponent-for-bounded-gaps-between-many-primes}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

Let $p_{n}$ denote the $n$ -th prime and, for $m \geq 1$ , write

$H_{m} := n \to \infty lim inf (p_{n + m} - p_{n})$

for the least limit point of the gaps between primes $m$ apart. Maynard [May15] proved that $H_{m}$ is finite for every $m$ , with $H_{m} ≪ m^{3} e^{4 m}$ ; the growth rate in $m$ is exponential in all bounds known, and the exponent for bounded gaps between many primes is

$C_{88 b} := m \to \infty lim sup \frac{lo g H _{m}}{m},$

equivalently the least $c$ for which $H_{m} ≪_{ε} exp ((c + ε) m)$ holds for every $ε > 0$ .

The case $m = 1$ is recorded separately as $C_{88 a}$ ; note that $C_{88 b}$ is insensitive to any finite number of $H_{m}$ , so the two are genuinely independent quantities.

Known upper bounds

Bound	Reference	Comments
$4$	[May15]	$H_{m} ≪ m^{3} e^{4 m}$ , the first finiteness result for $m \geq 2$ , from the multidimensional Selberg sieve.
$4 - \frac{28}{157} = 3.821656 \dots$	[Pol14b]	$H_{m} ≪ m e^{(4 - 28/157) m}$ . Polymath8b, quoted in this form by [Sta26].
$3.815$	[BI17]	Baker–Irving.
$3.8075$	[Sta25]	Stadlmann: $H_{m} = O (exp (3.8075 m))$ , deduced from an exponent of distribution $\frac{1}{2} + \frac{1}{40}$ for the primes in arithmetic progressions to smooth moduli, improving Polymath's $\frac{1}{2} + \frac{7}{300}$ . The new ingredient is a modification of the $q$ -van der Corput process.

Known lower bounds

Bound	Reference	Comments
$0$	Trivial	No positive lower bound is known, and none is expected: see below.

Additional comments and links

The expected value is $0$ . The prime $k$ -tuples conjecture predicts $H_{m} ≍ m lo g m$ , which is sub-exponential, so conjecturally $C_{88 b} = 0$ . The interest is therefore entirely in the upper bound, as for $C_{62 a}$ and other exponents in this repository whose conjectural value is $0$ .

Conditional bound. Assuming the Elliott–Halberstam conjecture, Maynard obtained $H_{m} ≪ m^{3} e^{2 m}$ [May15], as reported in the abstract of [Pol14b], giving $C_{88 b} \leq 2$ under EH.

Caution: this does not improve $C_{66}$ . The exponents of distribution appearing here — Polymath's $\frac{1}{2} + \frac{7}{300}$ and Stadlmann's $\frac{1}{2} + \frac{1}{40} = 0.525$ — are established only for smooth (friable) moduli, and with well-factorable weights. $C_{66}$ as defined on its own page is the supremum of admissible levels for the full sum over all moduli $q \leq Q$ , for which $\frac{1}{2}$ (Bombieri–Vinogradov) remains the record. The two must not be conflated: $\frac{1}{2} + \frac{1}{40}$ is not a lower bound for $C_{66}$ .

Relation to the other entries. $C_{88 a}$ is $H_{1}$ itself. Both this constant and $C_{88 a}$ improved as a direct consequence of the same equidistribution input, [Sta25], which is what [Sta26] then combined with Bombieri–Vinogradov to reach $H_{1} \leq 240$ .

References

[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.
[BI17] Baker, Roger C.; Irving, Andrew J. Bounded intervals containing many primes. Mathematische Zeitschrift 286 (2017), no. 3–4, 821–841. DOI: 10.1007/s00209-016-1786-7.
[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.

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.