We will be featuring different members of the Brilliant community, so that you can get to know them better. For the sixth issue, we are featuring Patrick Corn, who used to be a research mathematician in number theory. He provides us with numerous insights into different approaches for hard number theory problems.
One of his solutions that I like, is to Solve \( p^x - y^3 = 1\). Even though the problem looks simple, most of the other solutions required quite a lot of work. Patrick got to the crux of the problem, which allowed for a nice clean solution
An intriguing problem that he would like you to solve, is to provide a simple proof that there exists no knight's tour on any \( 4 \times n \) chessboard. This seems hard to generalize, but he has provided hints for a coloring approach.
1. Tell us more about yourself.
I used to be a research mathematician. All of my colleagues were smarter and older than me. I was in number theory / arithmetic geometry. My specific area of research was in exceptions to the Hasse principle. There is a lot of complicated machinery, which can be used to attack appealingly down-to-earth Diophantine equations. For example, one of the results of my PhD thesis showed that there are no nontrivial integral solutions to \(w^2 = 34(x^4+y^4+z^4) \).
I liked research and teaching, but I wanted more stability and I wasn't sure if I'd be able to sustain a research program for an entire career. Now I work in finance at an electronic market making firm, and all of my colleagues are smarter and younger than me. The math that is required is surprisingly simple, although we have several math and physics PhDs on the trading floor. It's not the career path I expected, but I'm very happy where I am.
No regrets about leaving research mathematics. Sometimes I miss teaching a little bit. In particular, I taught a couple classes in cryptography and number theory that I really enjoyed. My impression was that there's a lot of room left for clever and relatively elementary new ideas in cryptography research, especially given how recently it became a topic of intense study.
2. What developed your interest in research mathematics?
I got my first taste of "real" mathematics at the Ross Program, at Ohio State. I can't recommend it enough. It's exactly the kind of program that young Brilliant users should aspire to enter. There are offshoots that are also great (e.g. PROMYS, Texas State), but the Ross Program especially has been a fantastic incubator for world-class mathematicians for decades.
I took several great classes as an undergrad at Harvard. Two standouts are the somewhat infamous Math 55 my freshman year, and a wonderful class taught by Elkies on group theory and combinatorics. The professors who taught my math courses at Harvard were uniformly brilliant lecturers as well, which in retrospect was pretty lucky.
3. What is one fun fact about yourself that the Brilliant community doesn’t know about.
My Erdos number is 3. (Can you find the path to Erdos?)
4. What do you want to accomplish in the next few years?
Keep my wife and kids happy.
5. What do you wish for Brilliant?
Contest math problems are fun to work on, and require quite a bit of cleverness and intelligence to solve. But they are generally not "real" mathematics. So many amazing young prospective mathematicians expend so much effort solving and creating these problems. I'd like to see Brilliant help steer them toward explorations of more fundamental and deep mathematics, to show them what doing mathematics professionally is really like. And I want to stress that the Brilliant staff who comment on and pose problems generally do a really good job of this as it is, from what I've seen. Keep up the good work.