The High Schooler Who Solved a Prime Number Theorem

This problem somehow got under my skin. It seemed so obvious, yet it resisted all my attempts to prove it. It was just stubbornness on my part that I didn't want to take no for an answer. It kept going. To me, math is really about connections between things that seem totally unrelated. And it's sort of always there for you. My name is Daniel Larson. I'm a freshman at MIT. And I'm studying mathematics. When I was in high school, I got interested in Carmichael numbers. A Carmichael number is a number n such that for all integers a, a to the n minus a is divisible by n. And furthermore, this is where it gets interesting, n is not prime. So the way you construct Carmichael numbers is you build them out of prime factors, which have a special property. But the problem is it's difficult to say in advance how many prime factors these numbers will have. Without knowing how many prime factors they have, it's difficult to get an exact read on the size. Bertrand's postulate for primes says that for every large number, there exists a prime between that number and its double. And so we would like to say the same thing is true for Carmichael numbers. So what I did is I made use of this great breakthrough from a few years ago by Zhang and Maynard, which was actually about looking at small gaps between primes. And it's this incredibly exciting new technique in number theory. I had many ideas about the problem, but none of them went anywhere until I realized that I could use these results about gaps between primes. And I was able to apply that to the study of Carmichael numbers. I showed that for sufficiently large n, there exist Carmichael numbers between n and 2n. I still wasn't 100% sure that I hadn't made a mistake somewhere. So I sent the paper to Granville, who was one of the leading experts on this field. I got this manuscript from a person I didn't know called Daniel Larson. And it was a little technical, but it had ideas. It had real ideas. It's always a bit surprising when you get something really good from something you've never heard of. But then to find out the 17 in high school was mind-blowing. One thing I think is interesting about my proof is that it really relies on Fourier analysis, which is a theory from physics about decomposing functions into waves. Whenever you have a set of objects and you want to study subsets with some particular nice property, Fourier analysis lets you study the set of subsets of the objects. And so that gives you this sort of tree of possibilities expanding from this set. And now with very basic statements about the trunk of that tree, Fourier analysis lets you make very expansive claims about the branches. In my problem, the trunk was a certain set of prime numbers. And I wanted to show that you could sort of predict the location of the apples or the Carmichael numbers. There are many questions we still don't know the answer to about Carmichael numbers. And actually, some of them are very relevant for cryptography. Modern cryptography uses really large primes to decrypt messages. And so to find really large primes, the best thing to do is just test random numbers for primality. And the problem is these Carmichael numbers can register as primes, even though they're not. And that can lead to weaker messages. So we would like to understand Carmichael numbers better so that we can avoid weaker encryption. One thing I really appreciated is I have been hearing the names Maynard, Granville, Pomerantz for so many years. These are like legends. But a random person emailed them. And they responded. And they looked at the work. And I'm just so incredibly grateful. He was doing work at the level of a top-quality PhD student. This super surprised me. I mean, this is definitely out of the ordinary, even for, you know, the US has a couple of prodigies every year in mathematics, right? But he's exceptional, even amongst them. I think this is really a great thing about the number theory community, that people are so welcoming and supportive of new people trying to get into it. So, you know, I encourage everybody to try their hand at various problems. And, you know, don't be afraid to write these big names. They will write back.