The method doesn't really yield "exactly 70,000,000". If you traced through his method, and worked out each step in more detail, you'd probably get a bound of (say) 64,189,288 -- or some random number of like that. If you read through it still more carefully, and tried to introduce genuine improvements, you'd improve this further.

Indeed, the bound has been improved to 246. You can see the(long!) proof here:

https://www.dropbox.com/s/85pt6mvzf5ghukw/newergap-submitted...

Zhang stated 70 million because it's easy to remember.

Anyway, the basic strategy of the proof is to attempt to prove the twin prime conjecture, and not fall too short.

Where do these numbers come from? You end up needing to do lots of random computations in the course of the proof. For example, can you compute a constant C such that the bound

e^(.1x^2 + .4x) + 5cos(pi * x) - sqrt(x^2 + 1) < Cx + 4

holds for all x between 3 and 5? If this appeared in the course of an analytic number theory proof, you probably wouldn't try to compute the absolute best value C. It would be tedious and boring, and nobody would want to read it. You'd just compute something that was good enough, and move on.

https://www.wolframalpha.com/input/?i=maximize+%7B+%28e%5E%2...

-> You can find C by imposing equality for x = 5.

One way to think about this, without any number theory requirements: if we keep throwing a coin with probability of heads p which is around 1/2 at the beginning, but then keeps decreasing on the rate of log n / n, we would expect (using undergraduate probability theory) that two heads indeed come up very close infinitely many times.

As primes up to n have the frequency of log n / n, this would be enough if the primes behaved like coin flip results. However, that is not the case, as primes are leftovers of some sieve, rather than totally independent and random.

The goal of some of the current methods is to establish a good ``probabilistic'' model of the primes, so that this intuition can be transformed to reality.