Mathematicians Uncover a New Way to Count Prime Numbers

But that was not clear. They had to analyze a special set of functions called type I and type II sums for each version of the problem, then show that the sums were equivalent regardless of the constraints they used. Only then will Green and Sawhney know that they can substitute rough primes in their proof without losing information.

They soon came to a realization: they could show that the equations were equivalent using a tool that each of them had encountered independently in previous work. The tool, known as a Gowers norm, was developed decades ago by mathematicians Timothy Gowers To measure how random or structured a function or set of numbers is. On the face of it, Gower’s ideals seem to belong to an entirely different realm of mathematics. “As an outsider it’s almost impossible to say that these things are related,” Sawhney said.

But using a breakthrough result proved by mathematicians in 2018 Terence too And Tamar ZieglerGreen and Sawhney found a way to connect Gowers rule and type I and II sums. Basically, they needed to use Gower’s 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 turned out, Sawhney knew how to do it. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using Gower’s rules. To his surprise, the technique was good enough to show that the two sets had the same type I and II sums.

Taking this in hand, Green and Sawhney proved the Friedlander and Iwaniek conjecture: there are infinitely many primes that can be written P² + 4q². Ultimately, they were able to extend their results to prove that other types of families also have infinitely many primes. The result marks a significant advance in a problem where progress is usually very rare.

More importantly, the work shows that Gowers ideology can serve 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 are now hoping to broaden the scope of Gowers ideal – to try to use it to solve other problems in number theory beyond prime counting.

“It’s fun for me to see things that I thought about a while ago,” Ziegler said. “It’s like parenting, when you free your child and they grow up and do mysterious, unexpected things.”

original story Reprinted with permission from Quanta MagazineAn editorially independent publication Simons Foundation It aims to improve public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.