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Brun's Theorem given in 1919 ensures that the sum of the reciprocals of the twin primes converges.

Do you know a different proof of this same result?

Moreover, you know if the "generalization" of it is true, I mean specifically, The sum of the reciprocals of the primes spaced by 2k converges

Furthermore, recalling the isolated primes ( # Isolated_prime) associated with the twin primes. Seeing that the series of the reciprocals of the primes diverges and that of the reciprocals of the twin primes converges, then the sum of the reciprocals of the isolated primes diverges!

This mean is there more number of cousins ​​cousins ​​isolates belonging to a pair of twin primes?

Will this be true for isolated cousins ​​to cousins ​​distanced by 2k?

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closed as off-topic by Eric Naslund, Stefan Kohl, Lucia, Micah Milinovich, Boris Bukh Dec 7 '13 at 22:44

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All these results follow from any of the standard sieves (e.g. Brun's sieve, Selberg's sieve, large sieve). In particular, there are much fewer cousins distanced by $2k$ than non-cousins: up to $x$ there are only $O_k(x/\log^2 x)$ of them, and this implies that the sum of their reciprocals converges. – GH from MO Dec 7 '13 at 7:26