In mathematics, Brun's theorem is a result of Viggo Brun in number theory. It has historical importance in the introduction of sieve methods. It was proved by Brun in 1919. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Viggo Brun (October 13, 1882 - August 15, 1978) was a Norwegian mathematician. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. ...

Let P(x) denote the number of primes p ≤ x for which p + 2 is also prime. Then, for x ≥ 3, we have

for some positive constant c.

This result shows that the sum of the reciprocals of the twin primes converges; in other words the p involved are a small set. In explicit terms the sum In combinatorics, a small set of positive integers is one such that the infinite sum converges. ...

converges, and its value is known as 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 2) converges to a mathematical constant now called Bruns constant for twin primes and usually denoted by B2 (sequence A065421 in OEIS): in stark contrast to the...

It is impossible to determine whether there are infinite twin primes or not by considering the sum of their reciprocals, as we can in the case of usual prime numbers.