Brun's constant

Brun's constant is the sum of the reciprocals of twin primes, which converges despite the divergence of the sum of the reciprocals of all primes:

$B_{2}=\left({\frac {1}{3}}+{\frac {1}{5}}\right)+\left({\frac {1}{5}}+{\frac {1}{7}}\right)+\left({\frac {1}{11}}+{\frac {1}{13}}\right)+\left({\frac {1}{17}}+{\frac {1}{19}}\right)+\left({\frac {1}{29}}+{\frac {1}{31}}\right)+\left({\frac {1}{41}}+{\frac {1}{43}}\right)+\cdots$

Viggo Brun proved this convergence in 1919. The convergence of this series leaves it unclear whether there are an infinite number of twin primes, one of the great unsolved problems in mathematics. The scarcity of twin primes as they grow larger enables their reciprocal sum to converge.