Forty Minutes With Fields Medallist Manjul Bhargava

Arjun Bhagoji, Nithyanand Rao and Raghavi Rao Kodati in The Fifth Estate:

What are you working on at the moment?

ScreenHunter_1280 Jul. 30 19.49The truth is I haven’t done much math or music or poetry or anything in the past year. But I hope to. I hope to get back to it soon.

I can tell you what I’ve been working on. If you pick a random whole number…Do you know what a square-free number is? Square-free number means that when you factor it, no prime occurs more than once. So 6 is square-free: it’s 2 × 3. But 12 is not square-free: it’s 2 × 2 × 3. So 2 is repeated.

So suppose you pick a random whole number, what’s the probability that it’s square-free? The answer has been known for a long time. The answer is 6/π2. It’s unexpected, right? The π — there’s no circles here or anything, right? You’re asking for the probability of a whole number being square-free. And the answer is 6/π2. Here, π appears in this magical way in this number theory problem, not a geometry problem. So this is something that fascinated me.

So one thing that I’ve been thinking about lately is: Often in number theory, you need to know about square-free numbers. If you have a polynomial with whole number coefficients and you look at its values when you plug in whole numbers, what’s the probability that the value of the polynomial is square-free? It depends on the polynomial, of course. For even a simple polynomial, x4 + 1, the answer is not known. What’s the probability that a random value of x4 + 1 is square-free? That’s one question that I work on.

More here. [Thanks to Ali Minai.]