In 1919
Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of
prime numbers which differ by two) converges to a sum now called
Brun's constant for twin primes and usually denoted by
B_{2} (
Sloane's
A065421):
in stark opposite 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