A couple weeks ago rumors were circulating that Yitang Zhang was claiming a proof of a longstanding open conjecture in number theory, the “no Landau-Siegel zeros” conjecture. Such a proof would be a very major new result. Zhang was a little-known mathematician back in 2013 when he announced a proof of another very major result, on the twin prime conjecture. Before that, he had a 2007 arXiv preprint claiming a proof of the Landau-Siegel zeros conjecture, but this was never published and known to experts to have problems such that at best the argument was incomplete.
Zhang now has a new paper on the arXiv, claiming a complete proof. The strategy of the proof is the same as in the earlier paper, but he now believes that he has a complete argument. At 110 pages the argument in the paper is quite long and intricate. It may take experts a while to go through it carefully and check it. Note that this is a very different story than the Mochizuki/abc conjecture story: Zhang’s argument use conventional methods and is written out carefully in a manner that should allow experts to readily follow it and check it.
For an explanation of what the conjecture says and what its significance is, I’m not competent to do much more than refer you to the relevant Wikipedia article. For a MathOverflow discussion of the problems with the earlier proof, see here, for consequences of the new proof, see here.
Update: I’m hearing that the above is not quite right, that what Zhang proves is weaker than the conjecture, although strong enough for many of its interesting implications. Perhaps someone better informed can explain the difference…
Update: Davide Castelvecchi at Nature has a news story here.