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C_81: mark Klyve's upper bound as conditional on GRH #147

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C_81: mark Klyve's upper bound as conditional on GRH #147

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C_81: mark Klyve's upper bound as conditional on GRH
Chessing234 Aug 17, 2026
dbb3a60
C_81: record the first upper bound on Brun's constant
Chessing234 Aug 17, 2026
5afeb46
C_81: say what the lower bounds are, and that two links are dead
Chessing234 Aug 17, 2026
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23 changes: 15 additions & 8 deletions constants/81a.md
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Expand Up @@ -8,28 +8,35 @@ $C\_{81a}$, Brun's Constant, is the sum of the reciprocals of the twin primes.

| Bound | Reference | Comments |
| ---------------------- | -------------- | --------------------------------------------- |
| 2.175398 | [K2007] | |
| 2.288513 | [PT2018] | |
| 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. |
| 2.288513 | [PT2018] | Unconditional, and the value carried in the README table. Sharpens the Crandall--Pomerance bound by about 13%. |


## Known lower bounds

| Bound | Reference | Comments |
| ---------------------- | -------------- | --------------------------------------------- |
| 1.8267324395006 | [N1995] | Led to the discovery of the Pentium FDIV bug. |
| 1.83180806343237901198 | [N2010] | |
| 1.840503 | [PT2018] | |
| 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; a smaller bound is 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 $2.175398$ in the table above.
- Similar constants exist for other prime families, e.g., cousin primes.
- [Wikipedia page on Brun's theorem](https://en.wikipedia.org/wiki/Brun%27s_theorem)

## References

- [K2007] Klyve, Dominic. [Explicit Bounds on Twin Primes and Brun’s Constant](https://doi.org/10.1349/ddlp.153).
- [N1995] Nicely, Thomas. [Enumeration to 1e14 of the twin primes and Brun's constant](https://faculty.lynchburg.edu/~nicely/twins/twins.html).
- [N2010] Nicely, Thomas. [Enumeration of the twin-prime pairs from 1e16 to 2e16](https://faculty.lynchburg.edu/~nicely/twins/t2_0001.html).
- [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](https://doi.org/10.1349/ddlp.153). PhD thesis, Dartmouth College.
- [N1995] Nicely, Thomas. [Enumeration to 1e14 of the twin primes and Brun's constant](https://faculty.lynchburg.edu/~nicely/twins/twins.html). (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](https://faculty.lynchburg.edu/~nicely/twins/t2_0001.html). (Also 404 as of 17 August 2026.)
- [PT2018] Platt, Dave and Trudgian, Tim. [Improved bounds on Brun's constant](https://doi.org/10.48550/arXiv.1803.01925).