The 2014 MacArthur “Genius” Fellows class has been announced, and among the 21 thinkers is mathematician Yitang Zhang.
In spring 2013, Zhang made an amazing breakthrough in a classic problem that has eluded mathematicians for centuries.
The Distance Between Prime Numbers
The prime numbers — numbers that are divisible only by themselves and one — have always been a major subject of study in mathematics. One particular aspect of the primes that has fascinated mathematicians throughout the centuries is their distribution — where primes fall on the number line.
We know that the prime numbers overall get rarer as numbers get larger, but do primes sometimes cluster together, or do the gaps between consecutive prime numbers get larger and larger?
The twin primes conjecture asserts that there are infinitely many pairs of twin primes — prime numbers that have only one number between them, like 11 and 13, or 17 and 19.
This is a simple statement, but mathematicians do not know whether or not it’s actually true. Most mathematicians have a sense that the twin primes conjecture should be true — the positioning of the prime numbers appear to be more or less random, even though on average the gaps between primes get larger, and if one has an infinitely long list of random odd numbers, we should have an infinite collection of twin pairs in our list. If at some point, prime numbers are always more than two numbers away from each other, then something is wrong with this intuition about numbers, and it would mean we understand the primes far less than we had thought.
In May 2013, Zhang made a huge step towards proving the conjecture. While we still don’t know if there are an infinite number of pairs of primes whose difference is 2, he showed that there are infinitely many pairs of consecutive primes whose difference is less than the somewhat awkwardly large (but helpful in Zhang’s proof) bound of 70,000,000.
While 70,000,000 is a pretty large and weak bound, Zhang’s discovery was the first time anyone had found any such number. Before this result, it was still considered possible that as we consider larger and larger numbers, the distance between consecutive primes would also get larger and larger. Zhang showed that, in fact, as we keep looking at ever more enormous primes, we will find pairs of consecutive primes that are relatively “near” each other.
Building on Zhang’s work, a number of mathematicians collaborated to reduce that huge bound, and that collaboration resulted in showing that there are infinitely many pairs of primes whose difference is no more than 246.
While much work still needs to be done to prove the twin primes conjecture, Zhang’s discovery and the work that followed are a huge step forward.
Erica Klarreich at Quanta magazine has a wonderful write up on Zhang’s work and the later developments.