In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a mathematical constant now called **Brun's constant for twin primes** and usually denoted by *B*_{2} (sequence A065421 in OEIS):

in stark contrast to the fact that the sum of the reciprocals of all primes is divergent. Had this series diverged, we would have a proof of the twin primes conjecture. But since it converges, we do not yet know if there are infinitely many twin primes. His sieve was refined by J.B. Rosser, G. Ricci and others.

By calculating the twin primes up to 10^{14} (and discovering the infamous Pentium FDIV bug along the way), Thomas R. Nicely heuristically estimated Brun's constant to be 1.902160578. The best estimate to date was given by Pascal Sebah in 2002, using all twin primes up to 10^{16}:

*B*_{2} ≈ 1.902160583104

There is also a **Brun's constant for prime quadruplets**. A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by *B*_{4}, is the sum or the reciprocals of all prime quadruplets:

with value:

*B*_{4} = 0.87058 83800 ± 0.00000 00005.

This constant should not be confused with the **Brun's constant for cousin primes**, prime pairs of the form (*p*, *p* + 4), which is also written as *B*_{4}.

*See also*: twin prime, twin prime constant, twin prime conjecture, Meissel-Mertens constant

## External link

- Nicely's article on twins enumeration and Brun's constant http://www.trnicely.net/twins/twins2.html
- Computation of Brun's constant http://numbers.computation.free.fr/Constants/Primes/twin.html

Last updated: 04-25-2005 03:06:01