A 58-year-old lecturer who had published no original work in 12 years, and who had spent years keeping the books at a friend's Subway sandwich shop, mailed a paper to the Annals of Mathematics on April 17, 2013. Just over a month later it was accepted: pairs of consecutive primes less than 70 million apart occur infinitely often.

It sounds reasonable that the most reliable place to look for a solution to a famous math problem is a famous mathematician at a famous institution. That is where the money, the graduate students, and the reputations sit. So when one of the oldest open questions in number theory finally cracked in the spring of 2013, the surprise was not only that it cracked. It was who did the cracking, and from where.

The problem in the background is the twin primes conjecture. Primes are the numbers divisible only by themselves and one, and they get scarcer the higher you count. Among ten-digit numbers, only about 4 percent are prime. Yet every so often two of them sit right next to each other, separated by just two: 11 and 13, 17 and 19, 41 and 43. The conjecture says such pairs never stop appearing, no matter how far out you go. Nobody has proved it. Daniel Goldston, a number theorist at San Jose State, put the mood plainly. “It’s one of those problems you weren’t sure people would ever be able to solve,” he said.

Who Zhang was in April 2013

The paper landed in the inbox of the Annals of Mathematics, one of the most selective journals in the field. The author was Yitang Zhang, a lecturer at the University of New Hampshire, born in 1955, which made him 58. He had earned his doctorate from Purdue University in 1991. In the years after, without steady academic work, he took odd jobs. At one point he lived in Lexington, Kentucky for a few years and managed the finances at a local Subway franchise.

By the time he submitted, he had published no original research in more than a decade. To most of the field he was a name attached to nothing recent. Andrew Granville, a number theorist at the Université de Montréal, put the contrast plainly. “Basically, no one knows him,” Granville said. “Now, suddenly, he has proved one of the great results in the history of number theory.”

What stays with us is how ordinary the surface was. A lecturer, no grant deadlines pushing him, no lab to run, working on a problem the biggest names in the area had circled and left alone. Granville was candid about his own expectations. “The big experts in the area tried and failed,” he said. “I personally didn’t think anyone was going to be able to do it any time soon.”

What 70 million actually claimed

Zhang did not prove the twin primes conjecture. He proved something next to it that, for the field, was enormous. His paper, titled “Bounded Gaps Between Primes,” showed there is some fixed number smaller than 70 million such that infinitely many pairs of primes differ by exactly that amount. Not that pairs of primes always sit within 70 million of each other. That some single gap under that ceiling keeps recurring, all the way up the number line, forever.

The figure of 70 million is large next to the number two, which is what the twin primes conjecture is really about. But the distance from two to 70 million is the point, not the problem. The wall everyone had been stuck at was infinity. Before Zhang, no one had shown any finite bound at all. Showing that the recurring gap was a specific, countable number, rather than one that grows without limit, was the crack in the wall. Once the bound is finite, it can be pushed down.

Zhang built on earlier work: a 2005 paper by Goldston, Janos Pintz, and Cem Yildirim that number theorists call GPY. It introduced a method for finding primes very close together and came close, but could not close the last step. Zhang’s advance came from reworking their method.

Five weeks at a journal that usually takes far longer

Mathematics journals are not fast. A submission to a top journal can sit under review for a year or more, sometimes several, while referees work line by line through an argument. Zhang’s paper did not sit. Received in mid-April, it was reviewed and accepted weeks later, an almost unheard-of turnaround at that level. Granville, reading the manuscript, was struck by how carefully it was built. The experts who had tried and failed now had a proof they could check.

The speed says something about the paper. Referees do not rush a proof they distrust. They rush one they can follow, check, and cannot fault. That an unknown lecturer’s manuscript could clear one of the strictest review processes in about five weeks tells you the work was not merely interesting. On inspection, it was obviously right.

What we take from a result that arrived so late

Zhang had been working on the problem for years, and by his own account the decisive idea came to him in the summer of 2012, only months before he sent the paper off. Not a young prodigy’s sprint, but a long, quiet grind by someone the field had effectively filed away.

The tidy lesson would be that genius can come from anywhere, and there is truth in it. But the version we find more useful is smaller and harder. Deep results sometimes require a long stretch of unglamorous, uncredited persistence that no grant cycle rewards and no institution notices until it is done. Zhang put it in ordinary words. “There are a lot of chances in your career,” he said, “but the important thing is to keep thinking.”