This is a story about prime numbers and the man who took a giant step toward solving a puzzle that has vexed mathematicians for centuries. That man, Yitang "Tom" Zhang, was exiled to the Chinese countryside during Mao's Cultural Revolution as a teen and forced to quit his studies and perform hard labor. Later, after receiving advanced degrees in China and the US, he struggled to find an academic position and at one point worked behind the counter at a Subway sandwich shop.
"There's nothing wrong with working at a Subway, but normally these proofs, these breakthroughs, are achieved by those that are working at Princeton, Harvard, these kind of really elite places," Tony Padilla, a physics professor at the UK's University of Nottingham, says in the Numberphile podcast below. "And now we've got somebody who's literally come of nowhere, that no one expected to produce this kind of results, and has done something impressive that many great minds were unable to do."
Zhang himself says that "something impressive" probably has no practical application, though prime numbers do extend beyond the realm of pure math into more real-world uses.
"Fifty years ago no one would have dreamed that anything about primes had a practical application, and large, secret primes are now the basis of some of the cryptography that makes Internet commerce possible," David Eisenbud, former head of the Mathematical Sciences Research Institute (MSRI) in Berkeley, Calif., told Crave.
Those who follow math closely may have heard of Zhang last year when a paper on his findings was accepted by the preeminent journal Annals of Mathematics just a short three weeks after being submitted, and other mathematicians responded to the research with exclamations like "beautiful," "stunned," and "astounded."
Since then, the University of New Hampshire professor, who is in his late fifties, has been invited to speak at institutions like Harvard, Princeton, and UC Berkeley, and won two prestigious math prizes. His work has sparked a frenzy among mathematicians the world over, who have collaborated through the online Polymath project to further hone his proof.
Now, even artists are getting in on the story. San Francisco Bay Area playwrights belonging to SF PlayGround, which develops new theatrical works, recently attended a lecture on Zhang at the MSRI. The talk served as an inspiration for original math-themed short plays, some of which will be performed Monday night at the Berkeley Repertory Theater in Berkeley, Calif., at a program titled "A Passion for Primes."
Zhang's finding relates to the Twin Prime Conjecture, a number theory problem that many attribute to the Greek mathematician Euclid. The conjecture holds that there is an infinite number of prime numbers (numbers divisible only by 1 and themselves) that are only two numbers apart -- like 3 and 5 or 17 and 19. These so-called twin primes occur often at the start of the lower end of the number spectrum but become less frequent as numbers get higher.
Minding the gap
In his paper, titled "Bounded gaps between primes" and bearing his name alone, Zhang attacked the problem by proving that the number of primes that are less than 70 million units apart is infinite. While 70 million is a long, long way away from 2, Zhang's work marked the first time anyone was able to assign any specific proven number to the gaps between primes. (For a highly detailed but clear explanation of Zhang's approach and results, read this Quanta Magazine article by Erica Klarreich).
Zhang's work builds on a 2005 breakthrough by Daniel Goldston of California's San Jose State University; Janos Pintz of the Alfred Renyi Institute of Mathematics in Budapest, Hungary; and Cem Yildirim of Bogazici University in Istanbul, who together showed that there will always be pairs of primes that are much closer to each other than average spacing predicts.
They developed a "sieve" to filter for pairs of primes that are closer together than average, as did Zhang. When his paper got accepted, Zhang said he expected the formulas contained therein to form the basis for narrowing the 70 million gap. "We may reduce it," he said at the time.
And they have. Terence Tao, a UCLA professor of mathematics and winner of the esteemed Fields Medal, launched Polymath8 as a forum where mathematicians could work to reduce that gap between 70 million and 2, which they did to 4,680 within a few months of Zhang submitting his paper.
That phase of the project was Polymath8a. In November, James Maynard, a postdoctoral researcher at the University of Montreal, presented independent work that built on Zhang's to further shrink the gap -- to 600. The second phase of Polymath8, called Polymath8b, builds on Maynard's work.
"Given the substantial progress made so far, it looks like we are close to the point where we should declare victory and write up the results," Tao wrote on his blog Sunday.
"Right now the best bound on gaps between primes is 270," he told Crave, "although we can get it down to the remarkably low level of 6 if we assume a strong additional conjecture (the generalized Elliott-Halberstam conjecture)."
Polymath 8 has been one of the most active, visible Polymath projects to date, and Tao attributes the excitement surrounding it, in part, to Zhang's compelling personal story -- after suffering the hardships of the Cultural Revolution, Zhang earned a bachelor's and a master's degree from Peking University, and completed a doctorate from Purdue, but had difficulty obtaining a university job in the US. Tao also credits the relative simplicity of the result, "which can be explained to any decent high-school math student (unlike many recent advances in mathematics)."
In addition, he said, Zhang's proof came as something of a shock to the number theory community. "The approach Zhang tried had been considered and discarded by most other experts in the field."
Zhang himself, a self-described "shy person," said in a UNH statement that the proof came to him during a vacation in Colorado, when he was feeling particularly relaxed. "I didn't bring any notes, any books, any paper," he said. "And suddenly it came to me."