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Math professor proves new pattern in prime numbers
Mehtaab Sawhney has proven that there are infinitely many prime numbers of a specific kind.
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Malvika Choudhary - Tue, 22, Apr 2025
- 16:57 EDT
- 76
Mehtaab Sawhney / Columbia University
A mathematician at Columbia University co-authored a major breakthrough in the study of prime numbers — a field full of ancient puzzles and modern mysteries.
Mehtaab Sawhney, who joined Columbia's department of Mathematics last year, and Oxford professor Ben Green have proven that there are infinitely many prime numbers of a specific kind: ones that can be written as p² + 4q², where both p and q are also primes.
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In simpler terms, they found that if you take two prime numbers, square one of them, and add four times the square of the other, the result can also be a prime — and this isn’t just a one-off case, but a pattern that continues infinitely.
Though the formula may sound technical, the wider significance lies in understanding how prime numbers — which at first glance appear random — can follow predictable patterns.
“Our result falls in the long tradition of trying to understand patterns in primes by proving a small question about a very small subset,” Sawhney said in an interview with Columbia News.
The result draws from long-standing mathematical conjectures and numerical observations. “For essentially any statistical question regarding primes, mathematicians have well-formed conjectures about what should be true,” Sawhney said. “Yet proving any of these predictions is a mysterious and much harder process.”
The proof was finalized during a week-long visit to Oxford, where the two mathematicians worked closely together. “Most of the work was done in his office, bouncing ideas off one another,” Sawhney recalled. “That process of suggesting and refining ideas at a blackboard with a collaborator is one of my favorite parts of being a mathematician.”
At just 26, Sawhney has authored or co-authored over 50 mathematical papers and earned several of the most prestigious honors in his field, including the 2024 Clay Research Fellowship, the 2021 Frank and Brennie Morgan Prize, and Best Student Paper Awards at STOC and ITCS.
He grew up in Long Island and graduated as valedictorian from Commack High School. In a time when many with a math and computer science background gravitate toward tech or finance, Sawhney chose academic research.
“What drew me was the freedom to think,” he said. “Often I come into the office with a problem I’ve chosen myself, and I get to spend my time working on it. That’s what I love most about this profession.”
Sawhney earned both his bachelor’s and doctorate degrees in mathematics from MIT, completing his doctorate in just four years. He was also named a Churchill Scholar in 2020 and briefly studied at the University of Cambridge.
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