But that's not clear. They will have to analyze a special set of functions, called Type I and Type II sums, for each type of their problem, then show that the sums are equivalent regardless of which constraints they use. Only then did Green and Sawhney know that they could substitute raw prime numbers into their proof without losing information.
They soon realized: They could prove that these sums were equivalent by using a tool that each of them had encountered in previous work. This tool, known as the 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 surface, the Gowers norm seems to belong to a completely different area of mathematics. “It's almost impossible to say as an outsider that these things are related,” Sawhney said.
But using a landmark result proven by mathematicians in 2018 Terence Tao And Tamar ZieglerGreen and Sawhney found a way to make a connection between the Gowers norm and the sum of Types I and II. Essentially, they need to use Gowers norms to show that their two sets of prime numbers—the set constructed with rough primes and the set constructed with real primes—are sufficiently similar. .
Turns out, Sawhney knows how to do this. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using the Gowers norm. To his surprise, this 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 Sawhney 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 result to show that there are infinitely many primes that also belong to other families. The results mark a significant breakthrough on a class of problems where progress is often rare.
More importantly, this work demonstrates that the Gowers benchmark can act as a powerful tool in a new field. “Because it is so new, at least in this part of number theory, there is the potential to do a lot of other things with it,” Friedlander said. Mathematicians now hope to expand the scope of the Gowers norm even further – to try using it to solve other problems in number theory beyond counting prime numbers.
“I'm excited to see that things I thought about a while ago have surprising new applications,” Ziegler said. “It's like parents, when you release your children and they grow up and do mysterious, unexpected things.”
Original story reprinted with permission from Quanta Magazinean editorially independent publication of Simons Foundation has a mission to advance public understanding of science by addressing research developments and trends in mathematics, physics and the life sciences.