But that is not obvious. He had to analyze a set of special functions, called Type I and Type II sums, for each version of his problem, and then show that the sums were the same regardless of the restrictions used. Only then did Green and Sawhney know they could change the hard primes to evidence without losing information. They soon came to a realization: They could show that the same amount was used by each of them independently found in previous work. The tool, known as the Gowers norm, was developed decades earlier by mathematician Timothy Gowers to measure how random or structured a function or set of numbers is. On the face of it, Gowers’ norm seems to belong to a very different field of mathematics. “It’s almost impossible to tell as an outsider that these things are related,” Sawhney said. Type I and II amounts. Essentially, he had to use the Gowers norm to show that two sets of primes—a set constructed using rough primes, and a set constructed using real primes—are sufficiently similar. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using the Gowers norm. To his surprise, the technique was just good enough to show that the two sets had equal amounts of Type I and II. With this, Green and Sawhney proved Friedlander and Iwaniec’s conjecture: There are many primes that can be written as p2. + 4q2. In the end, he was able to improve his results to prove that there are many primes in other families. The results mark a significant breakthrough on a type of problem where progress is typically rare. Even more importantly, the work shows that Gowers’ norms can act as powerful tools in new domains. “Because it’s so new, at least in this part of number theory, there’s potential to do a lot of other things,” Friedlander said. Mathematicians now hope to expand the scope of the Gowers norm even more—to try to use it to solve other problems in number theory beyond the prime count. , “said Ziegler. “It’s like being a parent, when you release your child and they grow up and do mysterious and unexpected things.” Original story reprinted with permission from Quanta Magazine, an independent publication of the Simons Foundation whose mission is to increase public understanding about science by covering research developments and trends in mathematics and the physical and life sciences.
