Brun’s Constant

Description of constant

$C_{81a}$, Brun’s Constant, is the sum of the reciprocals of the twin primes.

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 $B < 2.1754$”, 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 Superseded under the same hypothesis by [D2025], which notes that Klyve “never published his result, and utilized numerical integration to calculate his subsequent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, $B < 2.1594$.” Dunn describes this as “the first mathematically rigorous upper bound on $B$ assuming GRH”, the earlier [K2007] bound being unpublished and resting on numerical integration.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 $B$ is a sum of positive terms; this one comes from the enumeration to $10^{14}$ of that reference. The much-quoted $B \approx 1.902$ 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 $10^{16}$ to $2\cdot 10^{16}$.
1.840503	[PT2018]	Unconditional, and the value carried in the README table.

Additional comments and links

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 $2.288513$ rather than the smaller conditional $2.1594$ in the table above.
The conditional value $2.1609$ quoted in some secondary sources, including the Wikipedia page below, is from the first arXiv version of [D2025]. The revised and published version gives $2.1594$, which is the figure recorded here.
Similar constants exist for other prime families, e.g., cousin primes.
Wikipedia page on Brun’s theorem

References

[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.

Contribution notes

The [D2025] row, and the notes distinguishing the conditional bounds from the unconditional ones, were prepared with assistance from Claude Opus 5, which read the arXiv full text and the published abstract of [D2025]. The quoted sentences are verbatim from those sources; the journal details were checked against Crossref, and the $2.1594$ figure against the published version rather than the first arXiv version.