Mathematicians Uncover a New Way to Count Prime Numbers

Mathematicians Ben Green and Mehtaab Sawhney have made a significant breakthrough in number theory by proving that there are infinitely many prime numbers of the form p² + 4q², where both p and q are prime. This result builds on a conjecture posed by Pierre de Fermat and later explored by mathematicians Henryk Iwaniec and John Friedlander. The duo's proof, which was posted online in October, utilized innovative techniques from a different area of mathematics, specifically the Gowers norm, to establish connections between various types of prime numbers. Their approach involved loosening constraints on the primes they were studying, allowing them to first prove the existence of infinitely many primes formed by "rough primes" before extending their findings to actual primes. This work not only enhances the understanding of prime distribution but also suggests that the Gowers norm could be a powerful tool for tackling other problems in number theory. The collaboration between Green and Sawhney, sparked by their meeting at a conference, exemplifies how interdisciplinary approaches can lead to unexpected advancements in mathematical research.

- Ben Green and Mehtaab Sawhney proved there are infinitely many primes of the form p² + 4q².

- Their proof utilized the Gowers norm, a technique from a different area of mathematics.

- The work builds on earlier conjectures by Fermat and Iwaniec and Friedlander.

- The findings suggest potential new applications of the Gowers norm in number theory.

- The collaboration highlights the importance of interdisciplinary approaches in mathematical research.