Brun's theorem

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 A065421 in the OEIS). Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods. 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 A065421 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 π 2 ( x ) {displaystyle pi _{2}(x)} denote the number of primes p ≤ x for which p + 2 is also prime (i.e. π 2 ( x ) {displaystyle pi _{2}(x)} is the number of twin primes with the smaller at most x). Then, for x ≥ 3, we have That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor.It follows from this bound that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form a small set. In explicit terms the sum