Courtesy of Not Even Wrong, we have news of further progress on the size of prime gaps. The polymath8 project is readying its paper that proves that gaps no bigger than 4680 occur infinitely often, but apparently they’ve been scooped by James Maynard, who has a paper on the arXiv preprint server that brings that number down to 600. There’s a very good expository article in the Simons Foundation‘s Quanta Magazine about the whole story.
If the number 600 holds up, I think that’s a nice improvement, since gaps of size 600 start to show up (as unusually large gaps, admitted) for numbers in the trillions, which are rather small numbers by the standards of number theory or, say, cryptography. And a gap of 600 becomes average for numbers with around 250 digits, well within the usual bounds of cryptography. So one could definitely say that they have proved that “the typical size gap for practical applications of prime numbers recurs infinitely often.” Not the twin prime conjecture, to be sure, but very nice.