Brun's theorem is a result in number theory pertaining to the distribution of prime numbers. Specifically, it relates to the sum of the reciprocals of the prime numbers. The theorem states that the sum of the reciprocals of the twin primes (pairs of primes that differ by two, such as (3, 5) and (11, 13)) converges to a finite value.
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The sum of reciprocals of twin primes converges, whether or not there are infinitely many twin primes. The twin-prime upper bound from a quadratic sieve and reciprocal-sum convergence from a counting bound prove it. This contrasts with the Euler proof that the sum of reciprocals of primes diverges.