But it 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 are equivalent regardless of the constraints used. Only then will Green and Sawhney know that they can replace approximate prime numbers 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 of them had independently encountered in previous work. This tool, known as the Gowers norm, had been developed decades earlier by a mathematician Timothy Gowers to measure the randomness or structure of a function or set of numbers. On the surface, the Gowers standard seemed to belong to a completely different field of mathematics. “From the outside, it’s almost impossible to tell that these things are related,” Sawhney said.
But using a breakthrough result proven in 2018 by mathematicians Terence Tao AND Tamar ZieglerGreen and Sawhney found a way to relate Gowers norms to Type I and II sums. Essentially, they had to employ Gowers’ norms to show that their two sets of prime numbers – the set built using approximate prime numbers and the set built using real prime numbers – were sufficiently similar.
As it turns 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, the technique was good enough to show that both sets had the same Type I and II totals.
With this in hand, Green and Sawhney proved the Friedlander and Izaniec conjecture: There are infinitely many prime numbers that can be written as P2 +4Q2. Eventually, they were able to extend their result to prove that there are infinitely many prime numbers that also belong to other types of families. The result represents a significant breakthrough in solving a problem where progress is usually very sporadic.
More importantly, the work shows that the Gowers norm can act as a powerful tool in a fresh field. “Because it’s something 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 expand the Gowers norm even further and try to employ it to solve other number theory problems beyond counting prime numbers.
“It’s great fun for me when I see that things I thought about a while ago have unexpected new applications,” Ziegler said. “It’s like being a parent: you give your child freedom, and they grow up and do mysterious, unexpected things.”
Original story reprinted with permission Quanta Magazineeditorially independent publication Simons Foundation whose mission is to escalate society’s understanding of science by incorporating research developments and trends in mathematics and the physical and life sciences.