What about the gaps between consecutive primes? 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. (The left-hand picture on this page is a nice illustration of how this works in the plane; the points are chosen independently and completely randomly, but you see some clumps and clusters all the same.)
It’s not hard to compute that, if prime numbers behaved like random numbers, you’d see precisely the behavior that Zhang demonstrated. Even more: You’d expect to see infinitely many pairs of primes that are separated by only 2, as the twin primes conjecture claims.
(The one computation in this article follows. If you’re not onboard, avert your eyes and rejoin the text where it says “And a lot of twin primes …”)
Among the first N numbers, about N/log N of them are primes. If these were distributed randomly, each number n would have a 1/log N chance of being prime. The chance that n and n+2 are both prime should thus be about (1/log N)^2. So how many pairs of primes separated by 2 should we expect to see? There are about N pairs (n, n+2) in the range of interest, and each one has a (1/log N)^2 chance of being a twin prime, so one should expect to find about N/(log N)^2 twin primes in the interval.
There are some deviations from pure randomness whose small effects number theorists know how to handle; a more refined analysis taking these into account suggests that the number of twin primes should in fact be about 32 percent greater than N/(log N)^2. This better approximation gives a prediction that the number of twin primes less than a quadrillion should be about 1.1 trillion; the actual figure is 1,177,209,242,304. That’s a lot of twin primes.
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. Despite the apparent simplicity of the bounded gaps conjecture, Zhang’s proof requires some of the deepest theorems of modern mathematics, like Pierre Deligne’s results relating averages of number-theoretic functions with the geometry of high-dimensional spaces. (More classically minded analytic number theorists are already wondering whether Zhang’s proof can be modified to avoid such abstruse stuff.)
Building on the work of many predecessors, Zhang is able to show in a rather precise sense that the prime numbers look random in the first way we mentioned, concerning the remainders obtained after division by many different integers. From this (following a path laid out by Goldston, Pintz, and Yıldırım, the last people to make any progress on prime gaps) he can show that the prime numbers look random in a totally different sense, having to do with the sizes of the gaps between them. Random is random!
Zhang’s success (along with the work of Green and Tao) points to a prospect even more exciting than any individual result about primes—that we might, in the end, be on our way to developing a richer theory of randomness. How wonderfully paradoxical: What helps us break down the final mysteries about prime numbers may be new mathematical ideas that structure the concept of structurelessness itself.
(A few suggestions for further reading for those with more technical tastes: Number theorist Emmanuel Kowalski offers a first report on Zhang’s paper. And here’s Terry Tao on the dichotomy between structure and randomness.)