But it wasn’t easy. They would have to analyze a special set of functions, called Type I and Type II sums, for each version of their problem and then show that the sums were equivalent regardless of the constraint used. Only then would Green and Sawhney know that they could substitute approximate primes into their proof without losing information.
They soon realized that they could show that the sums were equivalent using a tool that each of them had encountered independently in previous work. This tool, known as the Gowers norm, was developed decades earlier by the mathematician Timothy Gowers to measure the randomness or structure of a function or set of numbers. At first glance, Gowers’ standard seemed to belong to a completely different mathematical domain. “It’s almost impossible to tell as an outsider that these things are related,” Sawhney said.
But using a historical result proven in 2018 by mathematicians Terence Tao And Tamar ZieglerGreen and Sawhney found a way to relate Gowers’ norms to Type I and Type II sums. Essentially, they had to use Gowers’ norms to show that their two sets of primes – the set constructed using coarse primes and the set constructed using real primes – were sufficiently similar.
It turned out that Sawhney knew how to do it. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using Gowers norms. To his surprise, the technique was just good enough to show that the two sets had the same type I and II sums.
That being said, Green and Sawhney proved Friedlander and Iwaniec’s conjecture: there are infinitely many prime numbers that can be written in the form p2 +4q2. Ultimately, they were able to extend their result to prove that there are infinitely many prime numbers belonging to other types of families as well. This result marks significant progress on a type of problem where progress is generally very rare.
More importantly, the work demonstrates that the Gowers standard can act as a powerful tool in a new domain. “Because it’s so new, at least in this part of number theory, it’s possible to do a lot of other things with it,” Friedlander said. Mathematicians now hope to expand the scope of the Gowers norm even further, trying to use it to solve other problems in number theory, beyond counting prime numbers.
“It’s a lot of fun for me to see things that I thought of a while ago have new, unexpected applications,” Ziegler said. “It’s like as a parent, when you release your child and they grow up and do mysterious and unexpected things.”
Original story reprinted with permission from Quanta Magazinean editorially independent publication of Simons Foundation whose mission is to improve public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.