Mathematicians discover a new way to count prime numbers

But that was not obvious. They must analyze a special set of functions, called type I and type II sums, for each version of their problem, then show that the sums are equivalent regardless of which constraint they use. Only then would Green and Sonny have realized 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 using a tool they had each come across independently in previous work. The tool known as Gowers’ norm was developed decades earlier than mathematics Timothy Gowers to measure how random or structured a function or set of numbers is. At first glance, 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 connected,” Sawney said.

But using a remarkable result proven in 2018. by mathematicians Terence Tao and Tamar ZieglerGreen and Sonny found a way to make a connection between Gowers norms and type I and type II sums. Essentially, they had 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 turns out, Sawney knew how to do that. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using Gowers’ norms. To his surprise, the technique was good enough to show that the two sets had the same type I and type II sums.

With this in hand, Green and Sawney proved Friedlander and Ivanec’s conjecture: There are infinitely many primes that can be written as p² + 4r². Eventually, they were able to extend their result to prove that there are infinitely many primes belonging to other types of families as well. The result marks a significant breakthrough for 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 the potential to do a bunch of other things with it,” Friedlander said. Mathematicians now hope to extend the scope of Gowers’ norm even further—to try to use it to solve other problems in number theory beyond counting primes.

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

Original story reprinted with permission from Quanta Magazineeditorially independent publication of Simons Foundation whose mission is to improve public understanding of science by covering scientific developments and trends in mathematics and physics and the life sciences.