Three and five are prime numbers — that is, they are divisible only by 1 and by themselves. So are 5 and 7. And 11 and 13. And for each of these pairs of prime numbers, the difference is 2. Mathematicians have long believed that there are an infinite number of such pairs, called twin primes, meaning that there will always be a larger pair than the largest one found. This supposition, the so-called Twin Prime Conjecture, is not necessarily obvious. As numbers get larger, prime numbers become sparser among vast expanses of divisible numbers. Yet still — occasionally, rarely — two consecutive odd numbers will both be prime, the conjecture asserts. The proof has been elusive.
But last month, a paper from a little-known mathematician arrived “out of the blue” at the journal Annals of Mathematics, said Peter Sarnak, a professor of mathematics at Princeton University and the Institute for Advanced Study and a former editor at the journal, which plans to publish it. The paper, by Yitang Zhang of the University of New Hampshire, does not prove that there are an infinite number of twin primes, but it does show an infinite number of prime pairs whose separation is less than a finite upper limit — 70 million, for now.
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