Everyone by now has heard about Zhang’s landmark result showing that there are infinitely many pairs of primes at most 70000000 apart.
His core result is that if a set of 3.5 * 10^6 (corrected, thanks to comment #2) numbers is admissible (see below), then there are infinitely many so that contains at least two primes. He then easily constructs an admissible set wherein the largest difference is 7 * 10^7, obtaining the stated result.
A set is admissible if there is no prime so occupies every residue class. For a given this is clearly a checkable condition; there’s no need to look at primes larger than .
(While Zhang went for a nice round number, Mark Lewko found his argument in fact gives 63374611, if you’re being overly specific about these things, which we are right now. :-)
In a short note on the arXiv yesterday, Tim Trudgian (whose office is not far from mine) pointed out another way to build an admissible set, giving a smaller largest difference, obtaining the result that there are infinitely many pairs of primes at most 59874594 apart. He considers sets of the form (where is Zhang’s constant 3.5 * 10^7). These aren’t necessarily admissible, but they are for some values of , and both Zhang and Tim noticed certain values for which this is easy to prove. Zhang used with , while Tim’s observation is that also works. (Comment #2 below points out this isn’t right, and Zhang also intended , and the slack in his estimate is coming from just looking at the largest element of , rather than the largest difference.) Thus the bound in his result is .
It turns out that checking admissibility for a given isn’t that hard; it takes about an hour to check a single value for (but if you find a prime witnessing not being admissible, it very often gives you a fast proof that is not admissible either, so searching is much faster).
I haven’t looked exhaustively, but one can check that gives an admissible , and hence there are infinitely many pairs of primes at most . (Sadly, it’s impossible to get below 59 million with this trick; no below 27000 works; all witnessed by or .)
I just couldn’t resist momentarily “claiming the crown” for the smallest upper bound on gap size. :-) Of course the actual progress, that’s surely coming soon from people who actually understand Zhang’s work, is going to be in reducing his 3.5 * 10^6. You can read more about prospects for that in the answers to this MathOverflow question.