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 (http://en.wikipedia.org/wiki/Isolated_prime # 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?