Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge? Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that  $\prod_i (1+\frac{1}{p_i})$ diverges ?

Based on André Nicolas hint I realized :
$\prod_{i=1}^k (1+\dfrac{1}{p_i}) < (\sum_{i=1}^k (1+\dfrac{1}{p_i}))^2$
And by Brun's theorem it follows the product converges.
Q.E.D.
mick
