Yitang Zhang: Prime Gaps

Solution 1:

70 million is exactly what is mentioned in the abstract.

It is quite likely that this bound can be reduced; the author says so in the paper:

This result is, of course, not optimal. The condition $k_0 \ge 3.5 \times 10^6$ is also crude and there are certain ways to relax it. To replace the right side of (1.5) by a value as small as possible is an open problem that will not be discussed in this paper.

He seems to be holding his cards for the moment...

You can download a copy of the full accepted paper on the Annals page if your institution subscribes to the Annals.

Solution 2:

As to the "idea" of the proof, I would suggest looking at the following.

As mentioned by Mark Lewko, according to WolframAlpha, the proof gives a gap size of $63, 374, 610.$