But it's not obvious. They must analyze a special set of functions, called type I and type II sums, for each version of the problem, and then prove that the sums are equivalent no matter which constraints they use. Only then would Green and Soni learn that they could substitute rough prime numbers into their proof without losing information.
They soon realized: they could prove that these sums were equivalent using tools that each of them had encountered independently in previous work. The tool, known as the Gaussian norm, was developed decades ago by mathematician Timothy Gowers to measure how random or structured a function or set of numbers is. On the surface, the Gauss norm seems to belong to an entirely different area of mathematics. "As an outsider, it's almost impossible to tell whether these things are related," Soni said.
But using landmark results demonstrated in 2018 by mathematicians Terence Tao and Tamar Ziegler, Green and Soni found a way to connect the Gaussian norm to the sum of Type I and Type II method. Essentially, they needed to use the Gaussian norm to prove that their two sets of primes (the set built using crude primes and the set built using real primes) were sufficiently similar.
It turns out Sony knows how to do it. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using the Gaussian norm. To his surprise, the technique was enough to show that both groups had the same sum of Type I and Type II.
With this, Green and Soni proved Friedlander and Iwanik's conjecture: there are infinitely many prime numbers that can be written as p2 + 4q2. Eventually, they were able to extend their results to show that there are infinitely many primes belonging to other types of families as well. The results mark a major breakthrough on a class of problems where progress is often difficult to achieve.
More importantly, this work shows that the Gaussian norm can serve as a powerful tool in new fields. "Because it's so new, at least in this part of number theory, there's potential to do a lot of other things with it," Friedlander said. Mathematicians now hope to further expand the scope of the Gaussian norm and try to use it to solve other problems in number theory besides counting prime numbers.
“It’s fun for me to see unexpected new applications for things I thought of a while ago,” Ziegler said. "It's like as a parent, when you let your children be free, they grow up and do mysterious, unexpected things."
ability Reprinted with permission from Quanta Magazine, an editorially independent publication Simons Foundation Its mission is to enhance the public's understanding of science by reporting on research advances and trends in the mathematical, physical, and life sciences.