Infinitely Many Primes of the Form p² + 4q²: The Green and Sawhney Breakthrough
How Mathematicians Found a New Way to Count Primes
Today we’re talking about something fundamental: prime numbers. Primes are the basic building blocks of mathematics, yet their distribution along the number line remains one of the greatest unsolved mysteries. Everyone knows there are infinitely many primes, but predicting exactly where they appear is impossible.
Recently, two mathematicians — Ben Green (Oxford) and Mehtaab Sawhney (Columbia University) — proved that infinitely many primes of a very special form exist. To do it, they drew on methods originally developed in a completely different branch of mathematics.
The Breakthrough: Primes Represented by Quadratic Forms
It has long been known that prime numbers can follow recognizable patterns. For instance, some primes can be written as the sum of squares of two other primes: 13 = 2² + 3². Another family takes the form p² + 4q², where both p and q are themselves prime — for example, 41 = 5² + 4 × 2².
In 2018, mathematicians Friedlander and Iwaniec posed a precise question about this second family: do infinitely many primes of the form p² + 4q² exist? Green and Sawhney proved that the answer is yes.
The Proof Strategy: From Rough Primes to True Primes
Rather than working with ordinary primes directly, Green and Sawhney started with a simplified stand-in: so-called “rough” primes — numbers not divisible by the smallest primes 2, 3, 5, and 7. This relaxed setting made the first step of the argument tractable.
The key was then to show that infinitely many rough numbers of the required quadratic form exist, before lifting the result to genuine primes.
An Unexpected Connection: Gowers Norms in Number Theory
The bridge between the two steps came from an unexpected source — Gowers norms, a tool originally designed to measure the “randomness” of functions. Informally, the Gowers norm of order k measures how structured or chaotic a function on a finite group is: a small norm means the behavior is essentially random; a large norm means hidden structure is present.
By applying this theory, Green and Sawhney were able to transfer the result from rough primes to actual primes — a move that had not been seen in this context before.
Why This Result Matters for Number Theory
This breakthrough carries significance well beyond the specific statement it proves:
A new tool for studying primes. Gowers norms can now be applied in analytic number theory, opening a new avenue of attack on longstanding problems.
Generalizability. The method is not tailored to p² + 4q² alone — it could extend to other families of primes defined by polynomial expressions.
A rare advance in a hard field. Progress on questions about primes represented by quadratic forms is genuinely uncommon, making this result a landmark in the area.
Ultimately, the work confirms that primes are not as chaotic as they appear. There is order hidden within their structure — and mathematicians are only beginning to uncover it.
