There’s a New Way to Count Prime Numbers

But that wasn’t obvious. They’d have to analyze a special set of functions, called Type I and Type II sums, for each version of their problem, then show that the sums were equivalent no matter which constraint they used. Only then would Green and Sawhney know they could substitute rough primes into their proof without losing information.

They soon came to a realization: They could show that the sums were equivalent using a tool that each of them had independently encountered in previous work. The tool, known as a Gowers norm, was developed decades earlier by the mathematician Timothy Gowers to measure how random or structured a function or set of numbers is. On its face, the Gowers norm seemed to belong to a completely different realm of mathematics. “It’s almost impossible to tell as an outsider that these things are related,” Sawhney said.

But using a landmark result proved in 2018 by the mathematicians Terence Tao and Tamar Ziegler, Green and Sawhney found a way to make the connection between Gowers norms and Type I and II sums. Essentially, they needed to use Gowers norms to show that their two sets of primes—the set built using rough primes, and the set built using real primes—were sufficiently similar.

As it turned out, Sawhney knew how to do this. Earlier this year, in order to solve an unrelated problem, he had developed a technique for comparing sets using Gowers norms. To his surprise, the technique was just good enough to show that the two sets had the same Type I and II sums.

With this in hand, Green and Sawhney proved Friedlander and Iwaniec’s conjecture: There are infinitely many primes that can be written as p² + 4q². Ultimately, they were able to extend their result to prove that there are infinitely many primes belonging to other kinds of families as well. The result marks a significant breakthrough on a type of problem where progress is usually very rare.

Even more important, the work demonstrates that the Gowers norm can act as a powerful tool in a new domain. “Because it’s so new, at least in this part of number theory, there is potential to do a bunch of other things with it,” Friedlander said. Mathematicians now hope to broaden the scope of the Gowers norm even further—to try using it to solve other problems in number theory beyond counting primes.

“It’s a lot of fun for me to see things I thought about some time ago have unexpected new applications,” Ziegler said. “It’s like as a parent, when you set your kid free and they grow up and do mysterious, unexpected things.”

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.