Mathematicians have found a new way to count prime numbers

But it was not clear. They will 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 are equivalent regardless of which constraint they use. Only then will Green and Sawhney know that they can substitute rough simplexes in their proofs without losing information.

They soon figured it out: They could show that the sums were equivalent using a tool they had each come across independently in their previous work. A tool known as the Gowers norm was developed decades ago by a mathematician Timothy Gowers to measure whether a function or set of numbers is random or structured. As it turns out, Gowers’ norm belongs to a completely different area of ​​mathematics. “It’s almost impossible as an outsider to tell that these things are related,” Sawhney said.

But using a remarkable result proved by mathematicians in 2018 Terence Tao and Tamar ZieglerGreen and Sawhney found a way to relate Gowers norms to type I and type II sums. Essentially, they had to use Gowers’ norms such that their two basis sets—the set constructed using rough bases and the set constructed using true bases—were sufficiently similar.

Turns 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. Surprisingly, the technique was good enough to show that the two sets had the same amounts of type I and type II.

Green and Sawhney thereby 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 results to the fact that there are infinitely many prime numbers for other types of families. The result represents significant progress 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 the potential to do a lot of other things with it,” Friedlander said. Mathematicians now hope to extend the reach of Gowers’ norm even further—trying to use it to solve other problems in number theory besides counting honeys.

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

Original story reprinted with permission Quanta magazineeditorially independent publication Simons Foundation mission is to increase public understanding of science by covering research and trends in mathematics, physics, and the life sciences.