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Thursday, May 23, 2013

Why do prime numbers clump together, anyway? (More on that UNH professor's proof)

UNH has gotten more varied publicity from the proof by professor "Tom" Zhang of a weak version of the twin-prime conjecture than anything I can think of. Google him (first name Yitang) and you'll find coverage from Wired to Mother Jones to Time magazine.

Like my original blog item (May 14) and subsequent column, however, these don't talk much about the actual mathematics, aside from a laymen's-level discussion of prime numbers, because it's difficult stuff to distill.

So I enjoyed Slate's excellent disussion in their Do The Math column today (read it here) which skips quickly over the personal (he worked at Subway! He's old to make a math breakthrough!) and talks about the math:

If you’re looking at an even number, you never have to travel farther than 2 numbers forward to encounter the next even; in fact, the gaps between the even numbers are always exactly of size 2. For the powers of 2, it’s a different story. The gaps between successive powers of 2 grow exponentially, and there are finitely many gaps of any given size; once you get past 16, for instance, you will never again see two powers of 2 separated by a gap of size 15 or less.

Those two problems are easy, but the question of gaps between consecutive primes is harder. It’s so hard that, even after Zhang’s breakthrough, it remains a mystery in many respects.

The article, by Jordan Ellenberg in a column that discusses the mathematics of everyday life, delves into the difficult meaning of "random":

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.

And a lot of twin primes is exactly what number theorists expect to find no matter how big the numbers get—not because we think there’s a deep, miraculous structure hidden in the primes, but precisely because we don’t think so. We expect the primes to be tossed around at random like dirt. If the twin primes conjecture were false, that would be a miracle, requiring that some hitherto unknown force be pushing the primes apart.

Not to pull back the curtain too much, but a lot of famous conjectures in number theory are like this. The Goldbach conjecture that every even number is the sum of two primes? The ABC conjecture, for which Shin Mochizuki controversially claimed a proof last fall? The conjecture that the primes contain arbitrarily long arithmetic progressions, whose resolution by Ben Green and Terry Tao in 2004 helped win Tao a Fields Medal? All are immensely difficult, but they are all exactly what one is guided to believe by the example of random numbers.

It’s one thing to know what to expect and quite another to prove one’s expectation is correct.