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In a …
23. December 2024
Ben Green, a mathematician at the University of Oxford, has been fascinated by the mysterious patterns that characterize prime numbers. His interest led him to meet Mehtaab Sawhney, an exceptional mathematician from graduate school, with whom he collaborated on a problem posed by Friedlander and Iwaniec.
The two decided to tackle a problem involving counting primes made by squaring two other primes and adding them together. However, they soon realized that directly counting the number of such primes was not possible. Instead, they loosened their constraint slightly and considered a weaker version of the problem, where the numbers getting squared only had to be “roughly” prime.
Rough primes are much easier to find than primes and are significantly less randomly distributed. By using rough primes, Green and Sawhney were able to solve the problem. The Gowers norm played a crucial role in their solution, allowing them to make a surprising connection to another area of math.
The breakthrough demonstrates that the Gowers norm can be used as a powerful tool in number theory, even in new domains. Mathematicians are now eager to explore the scope of the Gowers norm further, hoping to broaden its applications and solve other problems in number theory beyond counting primes.
Friedlander notes that “it’s a lot of fun for me to see things I thought about some time ago have unexpected new applications.” The discovery is seen as a significant development, offering new possibilities for solving long-standing problems. This breakthrough highlights the importance of interdisciplinary approaches in advancing mathematical knowledge.
By combining their expertise in number theory with that of other mathematicians, Green and Sawhney were able to overcome seemingly insurmountable problems. Ziegler notes that “it’s a lot of fun for me to see this new application.” The potential for the Gowers norm to be used in other areas of math has sparked excitement among researchers, showcasing the power of collaboration and innovation in mathematics.