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In his paper "Bounded gaps between primes", Yitang Zhang proves that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Which is currently the best improvement on this result? Also, where can I find a list of all the results that have been obtained as regards Zhang's theorem (I guess we have to check the Polymath Project)?

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  • $\begingroup$ A general comment: The Polymath project has essentially ended, and they're just focusing on getting their papers to print. I don't expect much progress beyond 246 from that group (at least until some other discovery catalyzes them to further action). $\endgroup$ – Charles May 1 '14 at 23:13
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Here is the current Polymath page, which puts the bound at $246$. If additional unproven hypotheses are assumed, the gap falls to $6$ or $12$. This is also cited by Wikipedia (search the page for "$246$").

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I don't follow the precise status. But I think you can do worse than keeping an eye on the blog of Terence tao http://terrytao.wordpress.com/ and his posts on the Polymath 8b project.

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This is the page which keeps the current best bounds, with a timeline of the bounds being linked to.

http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes

I don't know where you can find a comprehensive list of all the results obtained from it but the papers by Pintz and Maynard are the big ones.

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