The Bounded Gaps Between Primes Theorem has been proved
There’s really exciting news in the world of number theory, my old field. I heard about it last month but it just hit the mainstream press.
Namely, mathematician Yitang Zhang just proved is that there are infinitely many pairs of primes that differ by at most 70,000,000. His proof is available here and, unlike Mochizuki’s claim of a proof of the ABC Conjecture, this has already been understood and confirmed by the mathematical community.
Also, my buddy and mathematical brother Jordan Ellenberg has an absolutely beautiful article in Slate explaining why mathematicians believed this theorem had to be true, due to the extent to which we can consider prime numbers to act as if they are “randomly distributed.” My favorite passage from Jordan’s article:
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.