A famous conjecture in number theory has stood unproven for more than 150 years, but for the second time this year, mathematicians have gotten dramatically closer to proving it. With a strategy others had abandoned, a young mathematician has narrowed the gap between primes, in hopes of ultimately proving the twin prime conjecture. His work has also shown that prime numbers bunch together in clusters as well as in pairs.
The twin prime conjecture asserts that infinitely many pairs of prime numbers are separated by only two, as are 3 and 5 or 1997 and 1999. Prime numbers are divisible only by themselves and 1. Number theorists know that as numbers get larger, primes gradually get sparser. Nonetheless, if the twin prime conjecture is true, pairs of primes spaced as closely together as possible continue to pop up forever.
Number theorists have been revved up since May, when Yitang “Tom” Zhang, a University of New Hampshire mathematician, announced a partial solution to the twin prime problem, which will appear in Annals of Mathematics. Researchers have since been refining his methods, getting closer to solving the problem. In October, James Maynard, a postdoctoral researcher at the University of Montreal, announced at a workshop in Germany that he had not only improved Zhang’s estimate for prime pairs, but had also resuscitated a previously discredited technique to prove that primes also occur in clusters. Maynard posted his results November 19 at arXiv.org.
Zhang had shown that pairs of primes with gaps no larger than 70 million keep occurring forever. He was the first to demonstrate a finite cap on the minimum gaps between primes. Even though 70 million is a long way from 2, it’s an even longer way from infinity. “He didn’t waste time trying to get a smaller number,” says John Friedlander of the University of Toronto, because he knew that just getting any finite number would be a sensational result.
But other mathematicians enthusiastically pounced on the problem. In June, hardly a day passed without a new world record for the smallest prime gap that repeats infinitely often. By the end of July, the record stood at 4,680. The researchers made these gains using tweaks and refinements of Zhang’s argument, not by breaking new mathematical ground.
Meanwhile, as he was finishing up his doctorate at the University of Oxford, Maynard pondered a related question: Do primes occur in pairs like cherries or in bunches like grapes?
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In addition to twin primes with their gaps of 2, triplets may occur in baskets of width 6. Such a basket catches 7, 11 and 13 or 2707, 2711 and 2713, and the harvest presumably continues forever. This statement is the prime triplets conjecture, and there is an analogous conjecture for prime quadruplets and larger.
Zhang’s work says nothing about prime triplets or other multiples. It uses a prime detection tool called the one-dimensional Selberg sieve that, for reasons not completely understood, can detect only pairs of primes. The sieve is a theoretical function (not an actual device or program) that weights numbers roughly according to their probability of being prime.
The original multidimensional Selberg sieve, discovered in the 1940s by the Norwegian mathematician Atle Selberg, does not have the pair limitation. Zhang had borrowed the simpler one-dimensional version from 2005 work on prime gaps by mathematicians Daniel Goldston, János Pintz and Cem Yıldırım. That paper was itself a modification of an argument from a 2003 paper by Goldston and Yıldırım, which had used the multidimensional sieve to address the twin prime conjecture but had to be retracted because of a fatal error. Because of this history, many number theorists considered the multidimensional sieve inherently flawed. “I think the issue was perhaps more psychological than technical,” says Terence Tao of UCLA.
But Maynard, who had nothing to lose because he was essentially done with his doctoral dissertation, decided to play around with the discredited sieve. And nearly on the eve of his dissertation defense, he figured out how to make it work.
Using the sieve, Maynard found a list of 105 numbers (0, 10, 12, 24, … 594, 598, 600) that serves as a template for prime pairs. This means that infinitely many numbers, when added to the numbers in the template, produce at least two primes. (For example, 3 works because 3+0 and 3+10 are both prime.) The template produces pairs of primes that are separated by 600 at most — a major improvement over the record of 4,680. Even more importantly, Maynard showed that longer templates exist for prime triplets, quadruplets and higher order prime multiples, thereby establishing world records that had not even existed. (Tao independently used the same ideas to reach a similar, but slightly weaker result.)
“Maynard’s proof is much shorter than Zhang’s and much more elementary, and it produces stronger results,” Friedlander says. “But the proofs are quite different, so at some point in time Zhang’s ideas and Maynard’s could be incorporated together to get results stronger than either one got alone. It’s a wonderful situation.”