- Green and Sawhney proved infinitely many primes fit p² + 4q², where both p and q are prime.
- They imported Gowers norms from combinatorics — a tool never before used in prime-counting.
- The proof works by first establishing results for "rough primes," then converting to actual primes.
Mehtaab Sawhney seems to have a habit of proving things quickly. The Columbia mathematician finished his PhD in four years and has notched more than fifty proofs since starting graduate school.
But his latest result, completed with Ben Green of Oxford, surprised even the mathematicians who posed the original problem.
The pair proved that infinitely many prime numbers can be written in the form p² + 4q², where both p and q are themselves prime.
The conjecture had stood since 2018, when John Friedlander of the University of Toronto and Henryk Iwaniec of Rutgers University proposed it as a natural extension of their earlier work on primes in polynomial forms.
The Problem With Constrained Primes
Prime numbers already resist easy patterns. Requiring a prime to equal the sum of two squared primes - each multiplied by specific coefficients - narrows the field dramatically.
Traditional counting techniques struggle with such tight constraints.
Key figure
infinitely many
prime numbers can be written in the form p² + 4q², where both p and q are themselves prime
Friedlander and Iwaniec had previously shown that infinitely many primes take the form x² + 4y², without requiring x and y to be prime.
Adding that requirement seemed to push the problem beyond reach.
"There are not many results like that out there," noted Joni Teräväinen of the University of Turku.
Borrowing From Combinatorics
Green and Sawhney's approach started with a deliberate loosening. Rather than attacking primes directly, they first proved their result for "rough primes" - numbers that avoid small prime divisors but aren't necessarily prime themselves. This gave them room to maneuver.
The key move was importing Gowers norms, a tool developed by Timothy Gowers at Cambridge for problems in additive combinatorics.
What are Gowers norms?
Gowers norms are mathematical tools that measure how structured or random a set of numbers is. Originally developed for combinatorics – the study of counting and arrangement – they capture subtle patterns that standard techniques miss. Green and Sawhney used them to bridge the gap between approximate results and exact prime theorems.
These norms measure how structured or random a set of numbers appears. Nobody had applied them to this type of prime-counting problem before.
The technique worked. Gowers norms let Green and Sawhney bridge the gap between rough primes and actual primes, converting their approximate result into a precise theorem.
It's terrific. It really surprised me that they did this.
John Friedlander, Canadian Mathematician and Number Theorist
What One Proof Opens
The result does more than settle a single conjecture. It suggests that Gowers norms - developed for entirely different purposes - may crack other problems in prime number theory.
Their paper, posted to arXiv in October 2024, proves the result not just for n = 4 but for any n that equals 0 or 4 modulo 6. The generalization hints at a broader pattern waiting to be explored.
For Sawhney, the collaboration drew on Green's earlier work, which had shaped his own mathematical development. The partnership blended Green's deep expertise with a younger mathematician's willingness to import unfamiliar tools.
The proof joins a short list of results that pin down prime locations under demanding constraints.
What remains unclear is how far these cross-disciplinary methods can travel - whether Gowers norms will prove a one-time solution or a recurring key to problems that seemed locked.
Sources
- Primary Research: Green, B. & Sawhney, M. (2024). Primes of the form p² + nq². arXiv:2410.04189.
- Additional Context:
- A Math Professor Has a New Finding on Primes (Columbia News)
- Mathematicians Uncover a New Way to Count Prime Numbers (Quanta Magazine)
Fact Check: Claim-by-Claim Verification Verified
The recap closely matches the Green–Sawhney paper and the Quanta/Columbia coverage, with only mild, acceptable simplifications and no substantive misrepresentations.
Commentary
- The phrase “a tool nobody expected would work” slightly over-dramatizes the novelty of using Gowers norms, but it aligns with Quanta’s portrayal of this as a surprising new application and does not misreport what the researchers or sources claim.
Sources used for verification
Academic/Peer-reviewed:
- Primes of the form p² + nq² - arxiv.org
- Primes of the form p² + nq² (talk abstract) - stanford.edu
Other reliable sources:
- Mathematicians Uncover a New Way to Count Prime Numbers - quantamagazine.org
- A Math Professor Has a New Finding on Primes - news.columbia.edu
- Two mathematicians discover a new way to count prime numbers - earth.com
Fact-checked by Perplexity Sonar Pro on 2026-01-09
