But that was not clear. They had to analyze a particular set of functions, called Type I and Type II sums, for each version of their problem, then show that the sums were equivalent no matter what constraints they used. Only then would Green and Sawhney discover that they could substitute raw primes in their proof without losing information.
They soon realized: They could show that the sums were equal using a tool that each of them had encountered independently in previous work. This tool, known as the Gowers Norm, was developed decades ago by mathematician timothy govers To measure how random or structured a function or set of numbers is. At first glance, the Gowers criterion seemed to belong to an entirely different area of mathematics. “It's almost impossible as an outsider to tell if 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 Gowers criteria and type I and II sums. Essentially, they needed to use Gowers' criteria 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 turned out, Sahni knew how to do it. Earlier this year, to solve an unrelated problem, he had developed a technique for comparing sets using Gowers criteria. To their surprise, the technique was good enough to show that Type I and II totals were the same in both sets.
With this in hand, Green and Sawhney proved Friedlander and Iwaniek's conjecture: there are infinitely many primes that can be written as P2 +4Why2Ultimately, they were able to extend their result to prove that there are also an infinite number of primes belonging to other type families. The result marks a significant breakthrough on a type of problem where progress is usually very rare.
More importantly, this work shows that the Gowers criterion can serve as a powerful tool in a new domain. “Because it's so new, at least in this part of number theory, there are likely many other things to do with it,” Friedlander said. Mathematicians now hope to expand the scope of the Gowers criterion even further – attempting to use it to solve other problems in number theory beyond counting prime numbers.
“It's so fun for me to see that things I thought about a while ago have unexpected new applications,” Ziegler said. “It's like as 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 Its mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.