Brun's Theorem

Brun's theorem states that the sum of the reciprocals of all twin primes converges. Let denote the set of primes and define

(1)

so is the set of smaller members of the twin-prime pairs. Define its counting function by

(2)

Brun's sieve gives the big-O notation estimate

(3)

This estimate is strong enough to force convergence. In fact, summation by parts gives

(4)

and the bound makes the integral converge as . Since , it follows that

(5)

Brun proved the result using a new form of sieve argument (Brun 1919). The conclusion holds whether is finite or infinite, so the theorem does not decide whether there are infinitely many twin primes. That assertion is the unproved twin prime conjecture.