But this was not clear. They must analyze a specific set of functions, called type I and type II sums, for each version of their problem, then show that the sums are equivalent regardless of the restriction they use. Only then do Green and Sawhney know that they can substitute approximate primes in their proof without losing information.
They soon came to a conclusion: they could show that the sums were equivalent using a tool that each had encountered independently in previous work. Known as the Gowers Norm, this tool was developed decades ago by mathematician Timothy Gowers to measure the randomness or structure of a function or set of numbers. On the surface, Gowers’ norm seemed to belong to a completely different realm of mathematics. “It’s almost impossible as an outsider to say that these things are connected,” Sawhney said.
But using a landmark result proved in 2018 by mathematicians Terrence Tao and Tamar Ziegler, Green and Sawhney found a way to make a connection between Gowers norms and type I and II sums. Essentially, they needed to use Gowers’ norms to show that their two sets of primes—one constructed using rough primes and one constructed using real primes—were sufficiently are similar
As it turned out, 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, this technique was good enough to show that two sets had the same type I and type II sums.
With this in hand, Greene and Sawhney proved Friedlander and Ivanick’s conjecture: there are infinitely many primes that can be written like this. p2 + 4q2. Finally, they were able to extend their result to prove that there are infinitely many prime numbers belonging to other types of families. The result is a significant improvement in a type of problem where progress is usually very rare.
More importantly, the work shows that the Gowers norm 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 potential to do a bunch of other things with it,” Friedlander said. Mathematicians now hope to extend the scope of Gowers’ norm even further—trying to use it to solve other problems in number theory beyond counting primes.
“It’s really 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 out and he grows up and does mysterious and unexpected things.”
The original story Reprinted with permission from Quanta Magazine, an editorially independent publication Simmons Foundation whose mission is to increase the public understanding of science by covering advances and research trends in mathematics and the physical and biological sciences.
