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Polymath8b project allowed, building on Zhang's 2013 breakthrough, to prove that there are infinitely prime gaps of size less or equal to 600. Under the generalized Elliott-Halberstam conjecture, one can reach the upper bound 6.

My question is: in early August 2016, what is the narrowest interval $I=[a,b]$ such that we know that there are infinitely many prime gaps whose size belongs to $I$?

I'm essentially interested in unconditional results, but those obtained under very plausible conjectures such as (G)EH and or (G)RH can be of some interest too.

Many thanks in advance.

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    $\begingroup$ The twin prime conjecture is "very plausible" too... $\endgroup$ – YCor Aug 3 '16 at 17:08
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You should find this wiki page useful. The current unconditional record is 246. Assuming Elliot Halberstam, the current record is 12, and assuming generalized Elliot Halberstam, the current record is 6.

Here is the polymath paper.

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