But that is not clear. They had 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 the same no matter what constraint they used. Only then did Green and Sawhney know that they could replace the rough primes in their proof without losing information.

They soon realized: They could show that the values ​​were the same using a tool that each of them had independently encountered in previous work. The device, 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 the face of it, the Gowers’ behavior seems to belong to an entirely different field of mathematics. “It’s almost impossible to tell as an outsider that these things are related,” Sawhney said.

But the use of a landmark result was proven in 2018 by mathematicians Terence Tao and Tamar ZieglerGreen and Sawhney found a way to make the connection between Gowers’ rules and Type I and II sums. Essentially, they had to use Gowers’ rules to show that their two sets of primes—the set constructed using rough primes, and the set constructed using real primes—were sufficiently similar.

As it turns out, Sawhney knows how to do it. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using Gowers’ rules. To his surprise, the technique simply showed that the two sets had the same Type I and II sum.

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. Finally, they were able to extend their result to prove that there are infinitely many primes that also belong to other types of families. The result marks a significant breakthrough in a type of problem where progress is often very rare.

More importantly, the work shows that Gowers’ behavior can act as a powerful tool in a new domain. “Because it’s so new, at least in this area of ​​number theory, there’s the potential to do a bunch of other things with it,” Friedlander said. Mathematicians now hope to expand the scope of the Gowers norm—to try to use it to solve other problems in number theory beyond counting primes.

“I really enjoy seeing things that I thought about before have unexpected new applications,” Ziegler said. “It’s like being a parent, when you let your child go and they grow up and do mysterious, unexpected things.”

Original story reprinted with permission from Quanta Magazinean editorially independent publication of 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.