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But this was not obvious. They had to analyze a special set of functions, called Types I and Types II, for each version of their problem, then show that the sums were equivalent, no matter which constraint they used. Only then did Green and Sawhney know that they could replace the primes in their test without losing information.
They soon came to a realization: They could show that the sums were equivalent with a tool that each of them had encountered independently in previous work. The tool, known as a Gowers norm, was developed decades earlier by the mathematician Timothy Gowers to measure how random or structured a function or set of numbers is. On the face of it, Gowers’ norm seemed to belong in an entirely different realm of mathematics. “It’s almost impossible to tell as an outsider that these things are related,” Sawhney said.
But using a reference result proven in 2018 by mathematicians Terence Tao and Tamar ZieglerGreen and Sawhney found a way to make the connection between Gowers norms and sums of Type I and II. Essentially, they needed to use Gowers’ norms to show that their two sets of primes—the set constructed with gross primes, and the set constructed with real primes—were sufficiently similar.
As it turned out, Sawhney knew how to do this. Earlier this year, to solve an unrelated problem, he had developed a technique to compare sets with Gowers standards. To his surprise, the technique was good enough to show that both sets had the same sum of Type I and II.
With this in hand, Green and Sawhney proved Friedlander and Iwaniec’s conjecture: There are infinitely many primes that can be written as p2 + 4Q2. Ultimately, they were able to extend their result to prove that there are infinitely many primes that also belong to other types of families. The result marks a significant advance in a type of problem where progress is usually very rare.
Even more importantly, the work shows that Gowers’ norm can be 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 scope of Gowers’ norm even further – to try to use it to solve other problems in number theory besides counting primes.
“It’s a lot of fun for me to see things that I thought about a while ago have unexpected new applications,” Ziegler said. “It’s like a parent, when you let your child free and grow up and do mysterious, unexpected things.”
Original story reprinted with permission from Quanta Magazinean independent editorial publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
