OpenFrontiers
Join
← Back to the index
OP-12105OpenNumber theory

Exponent for bounded gaps between many primes (C88bC_{88b})

Exponent for bounded gaps between many primes: determine C88bC_{88b} or improve its bounds; the repository records 0 (lower) and 3.8075 (upper).

MSC 11N05 · Distribution of primes

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let pnp_n denote the nn-th prime and, for m≥1m \ge 1, write

Hm:=lim inf⁡n→∞(pn+m−pn)H_m := \liminf_{n \to \infty} (p_{n+m} - p_n)

for the least limit point of the gaps between primes mm apart. Maynard [May15] proved that HmH_m is finite for every mm, with Hm≪m3e4mH_m \ll m^3 e^{4m}; the growth rate in mm is exponential in all bounds known, and the exponent for bounded gaps between many primes is

C88b:=lim sup⁡m→∞log⁡Hmm,C_{88b} := \limsup_{m \to \infty} \frac{\log H_m}{m},

equivalently the least cc for which Hm≪εexp⁡((c+ε)m)H_m \ll_\varepsilon \exp((c+\varepsilon)m) holds for every ε>0\varepsilon > 0.

The case m=1m = 1 is recorded separately as C88aC_{88a}; note that C88bC_{88b} is insensitive to any finite number of HmH_m, so the two are genuinely independent quantities.

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

  • The expected value is 00. The prime kk-tuples conjecture predicts Hm≍mlog⁡mH_m \asymp m \log m, which is sub-exponential, so conjecturally C88b=0C_{88b} = 0. The interest is therefore entirely in the upper bound, as for C62aC_{62a} and other exponents in this repository whose conjectural value is 00.

  • Conditional bound. Assuming the Elliott–Halberstam conjecture, Maynard obtained Hm≪m3e2mH_m \ll m^3 e^{2m} [May15], as reported in the abstract of [Pol14b], giving C88b≤2C_{88b} \le 2 under EH.

  • Caution: this does not improve C66C_{66}. The exponents of distribution appearing here — Polymath's 12+7300\tfrac12 + \tfrac7{300} and Stadlmann's 12+140=0.525\tfrac12 + \tfrac1{40} = 0.525 — are established only for smooth (friable) moduli, and with well-factorable weights. C66C_{66} as defined on its own page is the supremum of admissible levels for the full sum over all moduli q≤Qq \le Q, for which 12\tfrac12 (Bombieri–Vinogradov) remains the record. The two must not be conflated: 12+140\tfrac12 + \tfrac1{40} is not a lower bound for C66C_{66}.

  • Relation to the other entries. C88aC_{88a} is H1H_1 itself. Both this constant and C88aC_{88a} improved as a direct consequence of the same equidistribution input, [Sta25], which is what [Sta26] then combined with Bombieri–Vinogradov to reach H1≤240H_1 \le 240.

References (from the repository page)

  • [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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0; best known upper bound 3.8075. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
44 [May15] Hm≪m3e4mH_m \ll m^3 e^{4m}, the first finiteness result for m≥2m \ge 2, from the multidimensional Selberg sieve.
4−28157=3.821656…4 - \frac{28}{157} = 3.821656\dots [Pol14b] Hm≪me(4−28/157)mH_m \ll m e^{(4-28/157)m}. Polymath8b, quoted in this form by [Sta26].
3.8153.815 [BI17] Baker–Irving.
3.80753.8075 [Sta25] Stadlmann: Hm=O(exp⁡(3.8075 m))H_m = O(\exp(3.8075\,m)), deduced from an exponent of distribution 12+140\tfrac12 + \tfrac1{40} for the primes in arithmetic progressions to smooth moduli, improving Polymath's 12+7300\tfrac12 + \tfrac7{300}. The new ingredient is a modification of the qq-van der Corput process.

Known lower bounds

Bound Reference Comments
00 Trivial No positive lower bound is known, and none is expected: see below.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Exponent for bounded gaps between many primes ($C_{88b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.