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. More recently he has improved this estimate to
by using the twins up to 1.6×10^{15}.
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:
See also : twin prime, twin prime constant, twin prime conjecture
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