In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound for the number of primes $p$ below $x$ such that $p'p=g$, with $p'$ the smallest prime strictly greater than $p$?
Thanks in advance.
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Zhang's proof can be refined to show that the number of those primes is $\gg x/\log^k x$, where $k$ is the size of the tuple whose translates contain the relevant pairs, i.e. $k=3{,}500{,}000$ in the original proof and $k=50$ in the current record by PolyMath8b. For more details see the Main Theorem in Pintz's article here. 

