In addition to what the sibling commenters wrote --
The basic strategy of Zhang, as well as Goldston-Pintz-Yildirim, Maynard, and others is as follows: consider the numbers
n, n+2, n+4, n+6, n+8, ... n+70,000,000
and prove that "on average" at least two of them are prime. Prove this, and the result follows.
Now this is not literally true, the average spacing between primes is unbounded. The main technical work is in coming up with a weighting function, which is biased towards those n for which the "average" claim is likely to be true. Prove this, and you're done.
One then has to compute "the weighted average number of primes in n, n+2, ..., n+70,000,000", subject to the weighting function you chose. It's relatively easy to do in principle, but it requires various estimates for counting prime numbers, all of which come with error terms, and the hard part is to prove that the error terms don't accumulate enough to completely drown out the main term.
Work on this subject has proceeded along two lines: (1) come up with a more efficient weighting function, that does a better job of picking out the n we really want to count; (2) improving the error estimates in the various counts that come up. Both of these are intrinsically quantitative, and inevitably less precise than one expects to be true -- and the upshot is that if you aim for 2, you might prove 70,000,000 instead, which is still an amazing achievement.
The basic strategy of Zhang, as well as Goldston-Pintz-Yildirim, Maynard, and others is as follows: consider the numbers
n, n+2, n+4, n+6, n+8, ... n+70,000,000
and prove that "on average" at least two of them are prime. Prove this, and the result follows.
Now this is not literally true, the average spacing between primes is unbounded. The main technical work is in coming up with a weighting function, which is biased towards those n for which the "average" claim is likely to be true. Prove this, and you're done.
One then has to compute "the weighted average number of primes in n, n+2, ..., n+70,000,000", subject to the weighting function you chose. It's relatively easy to do in principle, but it requires various estimates for counting prime numbers, all of which come with error terms, and the hard part is to prove that the error terms don't accumulate enough to completely drown out the main term.
Work on this subject has proceeded along two lines: (1) come up with a more efficient weighting function, that does a better job of picking out the n we really want to count; (2) improving the error estimates in the various counts that come up. Both of these are intrinsically quantitative, and inevitably less precise than one expects to be true -- and the upshot is that if you aim for 2, you might prove 70,000,000 instead, which is still an amazing achievement.