Two mathematicians have cracked a centuries-old puzzle about prime numbers using an approach that no one saw coming. Ben Green of Oxford University and Mehtaab Sawhney of Columbia University proved there are infinitely many primes that can be written in a specific mathematical form, solving a problem that has stumped experts since 2018.
The breakthrough came when the duo tackled the challenge from a new angle. Rather than using traditional counting techniques, they used a mathematical tool called the Gowers norm — typically used in an entirely different area of mathematics. As reported in Quanta Magazine, this approach allowed them to prove that there are infinitely many prime numbers that can be written as p² + 4q², where both p and q are themselves prime numbers.
The mathematicians' success hinged on a clever workaround: instead of working directly with prime numbers, they first proved their result for "rough primes" — numbers that aren't divisible by small prime numbers. This intermediate step made the problem more manageable while still leading to the final solution.
The discovery's significance extends beyond just solving this single problem. By successfully applying the Gowers norm to prime number theory, Green and Sawhney have potentially opened up an entirely new toolkit for mathematicians studying prime numbers. "Because it's so new, at least in this part of number theory, there is potential to do a bunch of other things with it," said John Friedlander of the University of Toronto.
Tamar Ziegler, whose earlier work helped make this discovery possible, noted: "It's like as a parent, when you set your kid free and they grow up and do mysterious, unexpected things."
Previously:
• Kickstarting an electromechanical prime number calculator sculpture
• 'Demonic' 666 found in massive palindromic prime number
• Possession of this prime number is illegal in the United States, even if you write it on a piece of paper
• Another prime number down, infinity to go
• The world's largest prime number — visualized
