Brun's constant

In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B₂ (sequence in the OEIS). Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods.

The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let ${\displaystyle \pi _{2}(x)}$ denote the number of primes p ≤ x for which p + 2 is also prime (i.e. ${\displaystyle \pi _{2}(x)}$ is the number of twin primes with the smaller at most x). Then, for x ≥ 3, we have

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