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But that was not clear. They would have to analyze a special set of functions, called the sum of type I and type II, for each version of their problem, and then show that the sum was equivalent no matter which constraint they used. Only then will Green and Sonny know that they can substitute approximate primes in their proof without losing information.
They soon came to a realization: they could prove that the two sums were equal using a tool each had encountered independently in previous work. This tool, known as Gowers’ criterion, was developed decades ago by a mathematician Timothy Gowers To measure the randomness or organization of a function or set of numbers. On the face of it, Gower’s criterion seems to belong to a completely different world of mathematics. “It’s almost impossible to tell as an outsider that these things are related,” Sawhney said.
But using a historical result that was proven in 2018 by mathematicians Terence Tao and Tamar ZieglerGreen and Soni found a way to relate Gower’s criteria to sums of the first and second kind. Essentially, they needed to use Gowers’ criteria to show that the two sets of primes—the set built using approximate primes, and the set built using real primes—were similar enough.
As it turned out, Sawhney knew how to do it. Earlier this year, in order to solve an unrelated problem, he developed a technique for comparing groups using Gower’s criteria. To his surprise, the technique was good enough to show that the two groups had the same type I and II sums.
Using this, Greene and Soni proved the Friedlander and Iwanyk conjecture: there are an infinite number of prime numbers that can be written as follows: p2 + 4S2. Eventually, they were able to extend their results to show that there are an infinite number of prime numbers that belong to other types of families as well. The result represents significant progress on a type of problem where progress is usually very rare.
Most importantly, this work demonstrates that Gowers’ criterion can serve as a powerful tool in a new field. “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 Gowers’ rule even further, to try to use it to solve other problems in number theory beyond counting prime numbers.
“It’s very fun for me to see things that I’ve thought about for some time have new, unexpected applications,” Ziegler said. “It’s like when you’re a parent, when you let your child go and they grow up and do mysterious and unexpected things.”
Original story Reprinted with permission from Quanta Magazinean editorially independent publication of Simmons Foundation Its mission is to enhance public understanding of science by covering developments and research trends in mathematics, physical and life sciences.
