But that was not visible. They would have to analyze a special set of functions, called type I and type II equations, for each version of their problem, and show that the equations were equivalent regardless of the reasoning they used. Only then will Green and Sawhney know that they can include strong primes in their proof without losing information.
They soon realized: They could show that the calculations were equivalent using a tool that each of them had come up with independently in previous work. The tool, known as the Gowers norm, was developed decades earlier by mathematician Timothy Gowers to measure how random or structured a function or set of numbers is. On the face of it, Gowers’ practice seemed to belong to an entirely different field of mathematics. “It’s almost impossible to tell as an outsider that these things are related,” Sawhney said.
But using a landmark result proven in 2018 by mathematicians Terence Tao and Tamar Ziegler, Green and Sawhney found a way to make a connection between Gowers’ behavior and type I and II equations. Essentially, they had to use Gowers’ theorem to show that their two sets of primes—the set built using negative primes, and the set built using real primes—were sufficiently similar.
As it turned out, Sawhney could do this. Earlier this year, to solve an unrelated problem, he developed a method for comparing sets using Gowers’ principles. To his surprise, this method was good enough to show that the two sets had the same statistics 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. Finally, they were able to extend their result to prove that there are infinitely many types of other types of families. The result marks a significant breakthrough in a type of problem where progress is often very rare.
More importantly, the work shows that Gowers’ practice can serve as a powerful tool in a new domain. “Because it’s so new, at least in this part of number theory, there’s an opportunity to do a lot of other things with it,” Friedlander said. Mathematicians now hope to expand the scope of Gowers’ technique even further—trying to use it to solve other problems in number theory beyond the first calculus.
“I’m very excited to see things that I’ve thought about in the past have new and unexpected applications,” said Ziegler. “It’s like a parent when you release your child and he grows up doing strange and unexpected things.”
The first story reprinted with permission from Quanta Magazine, the independent editorial publication of Simons Foundation whose mission is to improve public understanding of science by incorporating research developments and trends in mathematics and the natural and biological sciences.