But it wasn’t clear. They must analyze a set of special functions called Type I and Type II sums for each version of the problem, and then show that the sums are the same no matter what constraints are used. Only then do Green and Sawhney realize that they can replace approximate decimals with evidence without losing information.
They soon realized. They were able to show that the sums were equal, using tools they had each independently encountered in previous work. This tool, known as the Gowers standard, was developed by mathematicians decades ago. Timothy Gowers It measures how random or structured a function or set of numbers is. On the surface, Gowers norms seemed to belong to a completely different area of mathematics. “It’s almost impossible to tell as an outsider how these things are related,” Sawhney said.
However, using the groundbreaking results proven by mathematicians in 2018, terrence tao and Tamar ZieglerGreen and Sawhney found a way to link Gowers norms with Type I and II sums. Essentially, they had to use Gowers norm to show that two sets of primes (one set using rough prime numbers and the other set using real prime numbers) are sufficiently similar.
As it turns out, Sawhney knew how to do it. Earlier this year he developed a technique for comparing sets using Gowers norms to solve unrelated problems. Surprisingly, this technique was good enough to show that the type I and II sums of the two sets were identical.
With this in hand, Green and Sawhney proved Friedlander and Iwaniec’s conjecture: that there are infinitely many prime numbers that can be written as blood2 + 4cue2. Ultimately, they were able to extend their results to prove that there are infinitely many prime numbers that also belong to other kinds of families. The results represent a major breakthrough on a type of problem for which progress has typically been very rare.
More importantly, this work demonstrated that Gowers norms can serve as powerful tools in new domains. “It’s so new, at least in this part of number theory, that there’s the potential to do a lot of different things with it,” Friedlander said. Now mathematicians are trying to further expand the scope of Gowers’ norm to solve other problems in number theory beyond calculating prime numbers.
“It’s very exciting for me to see things I was thinking about a while ago being applied to unexpected new applications,” Ziegler said. “As a parent, if you let your children run free, they will grow up and do mysterious and unexpected things.”
original story Reprinted with permission. Quanta MagazineEditorially independent publication Simmons Foundation Our mission is to enhance public understanding of science by covering research developments and trends in mathematics, physical sciences, and life sciences.
