26.5.13

THE VALUE OF TENURE.

Prime numbers greater than 2 cannot be even, and larger prime numbers cannot be divisible by ever-larger lists of candidate prime factors.  Intuitively, it would seem as though prime number twins, such as 3,5; 5,7; 11,13; 17,19; 29,31; 41,43 ought to become fewer, farther between, and perhaps beyond some value, nonexistent.  Not so fast.
Last week, Yitang “Tom” Zhang, a popular math professor at the University of New Hampshire, stunned the world of pure mathematics when he announced that he had proven the “bounded gaps” conjecture about the distribution of prime numbers—a crucial milestone on the way to the even more elusive twin primes conjecture, and a major achievement in itself.

The stereotype, outmoded though it is, is that new mathematical discoveries emerge from the minds of dewy young geniuses. But Zhang is over 50. What’s more, he hasn’t published a paper since 2001. Some of the world’s most prominent number theorists have been hammering on the bounded gaps problem for decades now, so the sudden resolution of the problem by a seemingly inactive mathematician far from the action at Harvard, Princeton, and Stanford came as a tremendous surprise.
Sometimes, being able to reflect on a problem without feeling the obligation to deliver a workshop or write a grant or race another colleague into print can be a blessing.  Too much academic effort goes into producing minimal publishable units.

The proof has an intuitive explanation.
You might think that, because prime numbers get rarer and rarer as numbers get bigger, that they also get farther and farther apart. On average, that’s indeed the case. But what Yitang Zhang just proved is that there are infinitely many pairs of primes that differ by at most 70,000,000. In other words, that the gap between one prime and the next is bounded by 70,000,000 infinitely often—thus, the “bounded gaps” conjecture.

On first glance, this might seem a miraculous phenomenon. If the primes are tending to be farther and farther apart, what’s causing there to be so many pairs that are close together? Is it some kind of prime gravity?

Nothing of the kind. If you strew numbers at random, it’s very likely that some pairs will, by chance, land very close together.
I'm struck with the frequency with which winning Power Ball tickets, including the big jackpot ones that are rational for even modest risk-averters to play, have a pair of adjacent numbers in them. (It's crazy, though, to guess which pair. Might be simpler, if you want to roll your own, to have a random number generator give you four numbers, then construct a set of tickets by pairing up one number at a time, above and below.)  The theorem, which establishes that the set of primes that differ by at most 70m is countable, asserts the existence of prime twins anywhere along the natural number line, but finding them is harder than finding winning Power Ball tickets.

Via 11-D.  The article has a number of references for the reader who would like to study further.  The Simons Foundation release includes a reference to a proof of the ternary Goldbach conjecture.