But it was not clear. They would have to analyze a special set of functions, called Type I and Type II sums, for each version of their problem, and then show that the sums were equivalent regardless of which constraint they used. Only then would Green and Sawhney know that they could substitute rough primes in their proof without losing information.
They soon came to a realization: They could show that the sums were equivalent to a tool each of them had encountered independently 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 an entirely 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 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 had to use Gowers norms to show that their two sets of primes—the set built with rough primes, and the set built with real primes—were sufficiently similar.
As it turned out, Sawhney knew how to do it. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using Gowers norms. To his surprise, the technique was only 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 prime numbers that can be written as p2 + 4q2. Finally, they were able to extend their result to prove that there are also infinitely many prime numbers that belong to other kinds of families. The result marks a significant breakthrough on a type of problem where progress is usually very rare.
More importantly, 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’s 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 to use it to solve problems in number theory other than counting prime numbers.
“It’s a lot of fun for me to see things that I thought about some time ago have unexpected new applications,” Ziegler said. “It’s like as a parent, when you set your child 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 improve public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.