Brun's theorem
= Brun's theorem
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{title2=$\sum_{p,p+2\text{ prime}}(1/p+1/(p+2))<\infty$}
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= Brun theorem
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{synonym}
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>.