But that wasn’t obvious. 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 no matter which constraint they used. Only then would Green and Sawhney know that they could substitute approximate prime numbers into their test without losing information.
They soon realized: they could prove that the sums were equivalent using a tool that each of them had found independently in previous work. The tool, known as Gowers’ norm, was developed decades earlier by the mathematician Timothy Gowers to measure how random or structured a function or set of numbers is. At first glance, Gowers’ norm seemed to belong to a completely different area of mathematics. “It’s almost impossible, from the outside, to say that these things are related,” Sawhney said.
But using a historical result demonstrated in 2018 by mathematicians terence tao and Tamara ZieglerGreen and Sawhney found a way to establish the connection between Gowers’ norms and Type I and II sums. Essentially, they needed to use Gowers’ rules to show that their two sets of primes (the set constructed from approximate primes and the set constructed from real primes) were sufficiently similar.
As it turned out, Sawhney knew how to do this. Earlier this year, to solve an unrelated problem, I had developed a technique for comparing sets using Gowers’ norms. To their surprise, the technique was good enough to show that the two sets had the same Type I and II sums.
With this in hand, Green and Sawhney proved the Friedlander and Iwaniec conjecture: there are infinitely many prime numbers that can be written as p2 + 4q2. In the end, they were able to extend their result to show that there are infinitely many prime numbers that also belong to other types of families. The result marks significant progress on a type of problem where progress is often very rare.
Even more importantly, the work demonstrates that Gowers’ norm can act as a powerful tool in a new realm. “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 expand the scope of Gowers’ norm even further: to try to use it to solve other problems in number theory beyond counting prime numbers.
“It’s really fun for me to see things I thought about a while ago have new, unexpected applications,” Ziegler said. “It’s like when a father sets his son free and he grows up and does mysterious and unexpected things.”
original story reprinted with permission of Quanta Magazinean editorially independent publication of the Simons Foundation whose mission is to improve public understanding of science by covering developments and trends in research in mathematics and the physical and biological sciences.