From 717e86888460b7ae3fe4c3e2ef92c10926ba42c9 Mon Sep 17 00:00:00 2001 From: Taksh Date: Mon, 17 Aug 2026 11:24:32 +0530 Subject: [PATCH 1/3] C_81: mark Klyve's upper bound as conditional on GRH The table showed 2.175398 [K2007] above 2.288513 [PT2018] with no comments, so the page reads as if the best known upper bound were 2.175398 while the README carries 2.288513. The two are not comparable: [PT2018, section 1] records Klyve's as "under the assumption of the Generalised Riemann Hypothesis we have B < 2.1754", and its own result 1.840503 < B < 2.288513 as unconditional. Both rows now say which they are, and the discrepancy with the README is explained in the comments section rather than left to be rediscovered. --- constants/81a.md | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/constants/81a.md b/constants/81a.md index d1060c38..1f4573a5 100644 --- a/constants/81a.md +++ b/constants/81a.md @@ -8,8 +8,8 @@ $C\_{81a}$, Brun's Constant, is the sum of the reciprocals of the twin primes. | Bound | Reference | Comments | | ---------------------- | -------------- | --------------------------------------------- | -| 2.175398 | [K2007] | | -| 2.288513 | [PT2018] | | +| 2.175398 | [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 @@ -23,6 +23,11 @@ $C\_{81a}$, Brun's Constant, is the sum of the reciprocals of the twin primes. ## 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) From dbb3a6020cdde155f7ffd91d7dc1190a3969773d Mon Sep 17 00:00:00 2001 From: Taksh Date: Mon, 17 Aug 2026 11:25:34 +0530 Subject: [PATCH 2/3] C_81: record the first upper bound on Brun's constant The bound tables are histories, so the Crandall--Pomerance bound B < 2.347 belongs in it: [PT2018, section 1] says "The first upper bound appears to be given by Crandall and Pomerance [5], who showed that B < 2.347", and the 13% improvement claimed in that paper's abstract is measured against it. --- constants/81a.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/constants/81a.md b/constants/81a.md index 1f4573a5..89543a38 100644 --- a/constants/81a.md +++ b/constants/81a.md @@ -8,6 +8,7 @@ $C\_{81a}$, Brun's Constant, is the sum of the reciprocals of the twin primes. | 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. | | 2.288513 | [PT2018] | Unconditional, and the value carried in the README table. Sharpens the Crandall--Pomerance bound by about 13%. | @@ -33,7 +34,8 @@ $C\_{81a}$, Brun's Constant, is the sum of the reciprocals of the twin primes. ## References -- [K2007] Klyve, Dominic. [Explicit Bounds on Twin Primes and Brun’s Constant](https://doi.org/10.1349/ddlp.153). +- [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). - [N2010] Nicely, Thomas. [Enumeration of the twin-prime pairs from 1e16 to 2e16](https://faculty.lynchburg.edu/~nicely/twins/t2_0001.html). - [PT2018] Platt, Dave and Trudgian, Tim. [Improved bounds on Brun's constant](https://doi.org/10.48550/arXiv.1803.01925). From 5afeb46c5a9a20e568b187d71b0068da79424978 Mon Sep 17 00:00:00 2001 From: Taksh Date: Mon, 17 Aug 2026 11:26:48 +0530 Subject: [PATCH 3/3] C_81: say what the lower bounds are, and that two links are dead The lower-bound rows carried no comments, so nothing distinguished the partial sums recorded here from the extrapolated B ~ 1.902 that is much more widely quoted and is not a bound. Both Nicely references 404 as of 17 August 2026 (the author died in 2019 and his Lynchburg pages are gone). trnicely.net resolves but is now an unrelated site, so it is not a replacement; noted inline so the next reader does not repeat the search. --- constants/81a.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/constants/81a.md b/constants/81a.md index 89543a38..4d6a3f4c 100644 --- a/constants/81a.md +++ b/constants/81a.md @@ -17,9 +17,9 @@ $C\_{81a}$, Brun's Constant, is the sum of the reciprocals of the twin primes. | 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 @@ -36,7 +36,7 @@ $C\_{81a}$, Brun's Constant, is the sum of the reciprocals of the twin primes. - [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). -- [N2010] Nicely, Thomas. [Enumeration of the twin-prime pairs from 1e16 to 2e16](https://faculty.lynchburg.edu/~nicely/twins/t2_0001.html). +- [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).