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Brun's constant

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 B2:

$B_2 = \left(\frac{1}{3} + \frac{1}{5}\right) + \left(\frac{1}{5} + \frac{1}{7}\right) + \left(\frac{1}{11} + \frac{1}{13}\right) + \left(\frac{1}{17} + \frac{1}{19}\right) + \left(\frac{1}{29} + \frac{1}{31}\right) + \cdots$

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 1014 (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

B2 = 1.90216 05823 ± 0.00000 00008 ,

by using the twins up to 1.6×1015.

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 B4, is the sum or the reciprocals of all prime quadruplets:

$B_4 = \left(\frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13}\right) + \left(\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19}\right) + \left(\frac{1}{101} + \frac{1}{103} + \frac{1}{107} + \frac{1}{109}\right) + \cdots$

with value:

B4 = 0.87058 83800 ± 0.00000 00005.

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