On May 13, an obscure mathematician — one whose talents had gone so unrecognized that he had worked at a Subway restaurant to make ends meet — garnered worldwide attention and accolades from the mathematics community for settling a long-standing open question about prime numbers, those numbers divisible by only one and themselves. Yitang Zhang, a lecturer at the University of New Hampshire, showed that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes separated by at most 70 million. His finding was the first time anyone had managed to put a finite bound on the gaps between prime numbers, representing a major leap toward proving the centuries-old twin primes conjecture, which posits that there are infinitely many pairs of primes separated by only two (such as 11 and 13).
In the months that followed, Zhang found himself caught up in a whirlwind of activity and excitement: He has lectured on his work at many of the nation’s preeminent universities, has received offers of jobs from top institutions in China and Taiwan and a visiting position at the Institute for Advanced Study in Princeton, N.J., and has been told that he will be promoted to full professor at the University of New Hampshire.
Meanwhile, Zhang’s work raised a question: Why 70 million? There is nothing magical about that number — it served Zhang’s purposes and simplified his proof. Other mathematicians quickly realized that it should be possible to push this separation bound quite a bit lower.
By the end of May, mathematicians had uncovered simple tweaks to Zhang’s argument that brought the bound below 60 million. A May 30blog post by Scott Morrison of the Australian National University in Canberra ignited a firestorm of activity, as mathematicians vied to improve on this number, setting one record after another. By June 4, Terence Tao of the University of California, Los Angeles, a winner of the Fields Medal, mathematics’ highest honor, had created a “Polymath project,” an open, online collaboration to improve the bound that attracted dozens of participants.
For weeks, the project moved forward at a breathless pace. “At times, the bound was going down every thirty minutes,” Tao recalled. By July 27, the team had succeeded in reducing the proven bound on prime gaps from 70 million to 4,680. Now, a preprint posted to arXiv.org on November 19 by James Maynard has upped the ante. Just months after Zhang announced his result, Maynard has presented an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration’s techniques with Maynard’s approach to push this bound even lower.