A Math Professor Has a New Finding on Primes

Prime numbers, which can only be divided by themselves and one, raise a host of interesting questions for mathematicians: At a small scale, the numbers seem to be randomly distributed, but, in aggregate, they actually show patterns that mathematicians themselves still don’t entirely understand. (If you aren’t yet convinced of the mysterious pull of primes, check out the new TV show Prime Target.)

One of prime numbers’ many mysteries was solved by Professor Mehtaab Sawhney this fall. Sawhney, who joined Columbia last year, worked with Professor Ben Green of Oxford University to prove that there are an infinite number of prime solutions to p² + 4q², where both p and q must also be prime. It’s one of more than 50 math proofs that Sawhney has solved since starting graduate school, which he completed in a very-speedy-for-a-PhD four years. The findings have been described as “phenomenally impressive” and “terrific”  by mathematicians not involved in the research.

Columbia News caught up with Sawhney to discuss his latest finding and his path to Columbia.

Can you explain your latest finding with an example?

41 is an example. 41 is a prime number, and you can write it as five squared plus four times two squared (5² + 4[2²] = 41).

149 is another example because it’s seven squared, which is 49, plus four times five squared, which is 100 (7²  + 4[5²] = 149). So here, observe that seven and five are both primes, and 149 is also prime.

There’s actually quite a lot of these at these lower levels, but they get gradually sparser as you get into higher numbers.

A big one is: 499992380034677

This is an example since 9999991² + 4 [9999907²] = 499992380034677, and both 9999907 and 9999991 are themselves prime.

What we proved is that there are an infinite number of primes that can be formed this way.

Why is this question of interest?

Mathematicians have been attempting to understanding patterns in the primes for centuries. In the 1740s, Christian Goldbach, a Prussian mathematician, mailed the preeminent mathematician of the era, Leonhard Euler, asking if every even number greater than two is the sum of two primes. More than 250 years later, although mathematicians suspect that the answer to Goldbach’s question is yes, a proof eludes us.

Mathematicians have been able, however, to prove the existence of certain patterns in the primes. For instance, in the 1800s, Johann Peter Gustav Lejeune Dirichlet proved that there are infinitely many primes ending with the digit 7. A more modern result in this direction, proved in the 1990s due to John Friedlander and Henryk Iwaniec, says that there are infinitely primes which are the sum of a square and a fourth power.

Our result falls in this long tradition of trying to understand patterns in primes by proving a small question about a very small subset of primes.

What did that week in Oxford solving this proof with Ben Green look like?

Most of the work was done in his office, bouncing ideas off one another. The process of suggesting and refining ideas at a blackboard with a collaborator is one of my favorite parts of being a mathematician.

What draws you to questions like this?

For essentially any statistical question regarding primes mathematicians have well-formed conjectures about what should be true. For more than a hundred years, we have had a precise heuristic about the number of twin primes and these heuristics also predict the number of primes of the form p² + 4q². And one can easily run numerical experiments, and the data matches these predictions extraordinarily well. Yet proving any of these predictions is a mysterious and much harder process, requiring new insight into how to probe the statistical behavior of the primes.

I really love that in math you can see a statement and know that statement is almost certainly true, but proving it is much harder.

You majored in math and minored in computer science. A lot of people go to the private sector with that background. What drew you to academia?

I think what ultimately drew me to be an academic was the freedom in what you do on a day-to-day basis, especially with your thinking time. It's that part of the job I find most freeing. I come into the office and often I have some problem, which I've chosen, that I really like and really want to think about, and I think about it. That kind of freedom is what really drew me to the profession.

What brought you to Columbia?

I grew up on Long Island. I actually came to Columbia in high school for a science honors research program that took place on Saturdays, where we took classes taught by graduate students or postdocs. I’ve always really liked the campus, and the math department has great and collegial researchers. I did my undergraduate and PhD at MIT and a master’s at Cambridge, but I knew I wanted to come back here.

People think of math as very solitary but the environment you’re in really affects what you’re working on, because people at different universities are working on different problems, and different genres of problems. I’ve found it very enriching to have connections at a few different universities. Ben Green and I proved our result when he invited me to visit him at Oxford for a week.

What are your favorite hobbies?

This is too typical for a mathematician but I love playing poker. I really enjoy the strategic aspect of it. It’s a mix of math and people reading.

I also love the Metropolitan Museum of Art. They have everything, and you can just wander and see what you discover, and find something, and you’re like, “oh my God, oh, here’s something cool. I never thought I’d see that here.”

Do you have a favorite prime number? If so, what is it and why?

I am partial to the prime two. It is so odd being the only even prime number.

Any favorite restaurants in the Morningside Heights area?

I am incredibly partial to The Hungarian Pastry Shop (having a bit of a sweet tooth).