But it was 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 equal no matter 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 equal using a tool that each of them had encountered independently in previous work. The tool, known as Gower’s ideal, 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 ideal seemed to belong to an entirely different field of mathematics. “It’s almost impossible as an outsider to tell that these things are related,” Sawhney said.
But using a historical result proved in 2018 by mathematicians Terence Tao And Tamar ZieglerGreen and Sawhney found a way to relate Gower’s Laws and type I and II sums. Essentially, they needed to use Gower’s Law 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 Gowers’ rules. To his surprise, the technique was good enough to show that the two sets contained the same types I and II joints.
Taking this in hand, Green and Sawhney proved the Friedlander and Ivanyk conjecture: there are infinitely many primes that can be written this way. P2 + 4q2. Eventually, they were able to extend their result to prove that there are infinitely many primes belonging to other types of families. The result marks a significant breakthrough on a type of problem where progress is usually slow.
More importantly, the work demonstrates that the Gowers ideal 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 now hope to expand the scope of Gowers’ ideal even further—to use it to solve other problems in number theory beyond the counting of primes.
“It’s fun for me to see things that I thought a while ago suddenly have new applications,” Ziegler said. “It’s like a parent, when you set your child free and they grow up and do mysterious, unexpected things.”
Original story Reprinted with permission Quanta MagazineAn editorially independent publication of Simons Foundation It aims to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
