Brun's constant ()
Brun's constant: determine or improve its bounds; the repository records 1.840503 (lower) and 2.288513 (upper).
Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.
Problem statement
, Brun's Constant, is the sum of the reciprocals of the twin primes.
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
- The upper bounds here are of two kinds. An unconditional bound follows from a Crandall--Pomerance style estimate; smaller bounds are available under the Generalised Riemann Hypothesis. The README table records the best unconditional bound, which is why it reads rather than the smaller conditional in the table above.
- The conditional value quoted in some secondary sources, including the Wikipedia page below, is from the first arXiv version of [D2025]. The revised and published version gives , which is the figure recorded here.
- Similar constants exist for other prime families, e.g., cousin primes.
- Wikipedia page on Brun's theorem
References (from the repository page)
- [CP2005] Crandall, Richard and Pomerance, Carl. Prime Numbers: A Computational Perspective, second edition, Springer, New York, 2005.
- [K2007] Klyve, Dominic. Explicit Bounds on Twin Primes and Brun’s Constant. PhD thesis, Dartmouth College.
- [N1995] Nicely, Thomas. Enumeration to 1e14 of the twin primes and Brun's constant. (The Lynchburg pages return 404 as of 17 August 2026; the author died in 2019 and the trnicely.net domain has since been taken over by an unrelated site, so it is not a substitute.)
- [N2010] Nicely, Thomas. Enumeration of the twin-prime pairs from 1e16 to 2e16. (Also 404 as of 17 August 2026.)
- [PT2018] Platt, Dave and Trudgian, Tim. Improved bounds on Brun's constant.
- [D2025] Dunn, Lachlan. Improved upper bound on Brun's constant under GRH. Bulletin of the Australian Mathematical Society 113 (2025), no. 2, 293–303; arXiv:2504.15658.
Progress & source assessment
Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.840503; best known upper bound 2.288513. Values marked with an asterisk have minimal verification.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| 2.347 | [CP2005] | The first rigorous upper bound, from the Crandall--Pomerance argument; attributed as such in [PT2018, §1]. |
| 2.175398 | [K2007] | Conditional on the Generalised Riemann Hypothesis. [PT2018, §1] records this as "under the assumption of the Generalised Riemann Hypothesis we have ", so it is not comparable with the unconditional rows. Superseded under the same hypothesis by [D2025], which notes that Klyve "never published his result, and utilized numerical integration to calculate his subsequent bound". |
| 2.288513 | [PT2018] | Unconditional, and the value carried in the README table. Sharpens the Crandall--Pomerance bound by about 13%. |
| 2.1594 | [D2025] | Conditional on the Generalised Riemann Hypothesis, and the best conditional bound known. Theorem 1.1: "Assume GRH. Then, ." Dunn describes this as "the first mathematically rigorous upper bound on assuming GRH", the earlier [K2007] bound being unpublished and resting on numerical integration. |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| 1.8267324395006 | [N1995] | A lower bound is just a partial sum, since is a sum of positive terms; this one comes from the enumeration to of that reference. The much-quoted from the same computations is an extrapolation, not a bound. This enumeration led to the discovery of the Pentium FDIV bug. |
| 1.83180806343237901198 | [N2010] | Continues the same enumeration through the twin-prime pairs from to . |
| 1.840503 | [PT2018] | Unconditional, and the value carried in the README table. |
- [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Brun's constant ($C_{81}$) 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 recordedAttempts 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 notesNotes 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.