Bounded Gaps Between Primes, Finally!

It has been known since antiquity that there are infinitely many primes, and the proof is easy. But looking at certain patterns of primes, like showing that there are infinitely many pairs of primes that differ by 2, seems to make the problem intractable. For a couple of centuries mathematicians have been wondering about this, making some progress, but still unable to show that infinitely often there are pairs of primes a bounded distance apart. Finally in 2013 I was able to show this.  More specifically, infinitely often there is a pair of primes that differ by at most 70,000,000.