TITLE: Triplet prime reciprocal series
QUESTION [5 upvotes]: Does any body know if the series of reciprocals of triplet primes of form $p, p+2, p+6$ or $p, p+4, p+6$ converges or diverges? Could this be used as a proof of infinity of twin primes?

REPLY [6 votes]: It is a well-known result that through Brun's sieve it is possible to prove that the series of the reciprocals of the twin primes converges, hence your series are converging, too. As a reference, see Halberstam, Sieve methods.
However, the fact that:
$$\sum_{\substack{p\in\mathcal{P}\\p+2\in\mathcal{P}}}\frac{1}{p}<+\infty$$
does not imply anything about the infinitude of twin primes. It just implies that they cannot be too "dense" in $\mathbb{N}$.