But it wasn’t clear. For each version of the problem, they had to analyze a special set of functions called Type I and Type II sums and show that the sums were equivalent regardless of the constraints they used. Only then did Greene and Sawhney realize that they could substitute rough prime numbers into their proof without losing information.
They noticed right away. Using tools each encountered separately in previous studies, we were able to show that the sums are equal. This tool, known as the Gowers norm, was developed by mathematicians decades ago. timothy gowers Measures how random or structured a function or set of numbers is. At first glance, the Gowers Standards seemed to belong to a completely different area of ​​mathematics. “As an outsider, it’s almost impossible to tell that these things are related,” Sawhney said.
However, using a breakthrough result proven by mathematicians in 2018, Terrence Tao and Tamer ZieglerGreen and Sawhney discovered a way to relate the Gowers criterion to Type I and II sums. Essentially, we use the Gowers standard to show that two sets of primes (one constructed using rough primes and one constructed using actual primes) are sufficiently similar. There was a need.
As it turns out, Sawhney knew how to do this. Earlier this year, to solve an unrelated problem, he developed a method to compare sets using the Gowers criterion. Surprisingly, this technique was sufficient to show that the Type I and Type II sums of the two sets were the same.
Using this, Green and Sawhney proved Friedlander and Iwaniec’s conjecture. There are an infinite number of prime numbers that can be written as follows. p2 +4q2. Eventually, they were able to extend their results to prove that there are an infinite number of prime numbers that belong to other kinds of families as well. This result represents significant progress in a type of problem where progress is typically very rare.
More importantly, this study demonstrates that Gowers’ norms can serve as a powerful tool in new areas. “Because this is so new, at least in this part of number theory, we could potentially do a lot of other things with it,” Friedlander says. Mathematicians are now looking to extend the scope of the Gowers norm even further and use it to solve other problems in number theory beyond counting prime numbers.
“It’s so fun to see something you thought up a while ago put into unexpected new uses,” Ziegler said. “It’s like when parents set their children free, they grow up to do strange and unexpected things.”
original story Reprinted with permission from Quanta Magazineis an editorially independent publication. simmons foundation Its mission is to enhance the public’s understanding of science by covering research developments and trends in mathematics, physical sciences, and life sciences.
