# Brun's constant

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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 B2 (sequence A065421 in OEIS):

in stark contrast to the fact that the sum of the reciprocals of all primes is divergent. If the series diverged, this would give a proof of the twin prime conjecture. But since it converges, we cannot conclude that there are infinitely many twin primes. Similarly, if it were ever to be proved that Brun's constant was irrational, the twin primes conjecture would follow immediately, whereas a proof that it is rational would not decide it either way.

Brun's sieve was refined by J.B. Rosser, G. Ricci and others.

By calculating the twin primes up to 1014 (and discovering the Pentium FDIV bug along the way), Thomas R. Nicely heuristically estimated Brun's constant to be 1.902160578.[1] Nicely has extended his computation to 1.6×1015 as of 18 January 2010 but this is not the largest computation of its type.

In 2002 Pascal Sebah and Patrick Demichel used all twin primes up to 1016 to give the estimate:

It is based on extrapolation from the sum 1.830484424658... for the twin primes below 1016. Richard Crandall and Carl Pomerance reported that it is known rigorously that 1.83 < B2 < 2.347.[2]. Dominic Klyve showed conditionally [3] that B2 < 2.1754 (assuming the Extended Riemann Hypothesis.

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 of the reciprocals of all prime quadruplets:

with value:

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 B4. Wolf derived an estimate for the Brun-type sums Bn of 4/n.