Primes, those numbers (as some middle school teacher probably tried to make you memorize some years ago) whose only factors are themselves and one (2, 3, 5, 7, 11 etc.), recently made the news. This is usually a big deal, because the low hanging fruit in the number theory of primes, like most mathematical objects under study for at least a couple thousand years, was consumed long ago.
It’s been known since at least Euclid that there are infinitely many primes (for an accessible proof why, see here). More tantalizing has been the conjecture that there are infinitely many so-called twin primes—consecutive prime numbers spaced only two apart from one another (3 and 5, 5 and 7, 11 and 13, 17 and 19 etc.). As is often the case with conjectures of this sort in number theory, computers continuously search for larger and larger candidate pairs satisfying this simple yet maddeningly elusive condition, for while every once in a while a new twin prime pair is found (the largest being the 18,000 digit long monster 3756801695685 · 2666669 ± 1), no one has yet proved that there should be infinitely many.
Even worse, no one had proved the existence of infinitely many consecutive primes with any bounded distance between them—until about three weeks ago.
On May 14, Yitang Zhang’s paper came out in the Annals of Mathematics proving that there are infinitely many pairs of consecutive primes spaced by at most 70,000,000. A flurry of research activity has followed, organized and catalogued on the collaborative research wiki-style platform Polymath, with other discussion on the mathematical giant Terry Tao’s blog. At the time of this writing, that gap has fallen from 70,000,000 to 285,728, with new bounds occurring daily.
Unfortunately, the elusive Twin Prime Conjecture may yet resist this most recent attempt—it’s believed that the furthest far the gap can be pushed with Zhang’s methods is 16. Regardless, I’ll be checking the polymath page over the coming weeks, as the number of mathematicians openly collaborating on the project is inspirational in and of itself.