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.

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## Comments

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TopNewest@Patrick Corn: You said:

Can you elaborate on this part? I agree that most of these Olympiad problems boils down to spotting patterns or following a certain algorithm, so they are not "real" mathematics as you calls it. It's almost comparable to how Bobby Fischer describes chess:

you know how to play, they have so many examples of what to do from this position... and that is why I don’t like chess any more... It is all just memorization and prearrangementCan you tell us what doing mathematics professionally is really like in further detail? I'm curious to know. Thanks!

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Well, I'm by no means an expert. But generally: there is a lot of collaboration, a lot of reading papers, a lot of learning. Good research questions have been thought about by many great minds already. The temptation is to learn just enough about the problem to understand the statement, and then to attack it using the (fairly meager) tools you know already. But usually the right approach is to learn as much as you can about related topics, until you become an expert on the problem's environment, and then you have a better idea what is going on and how best to attack the problem. You carry this out primarily by talking to other mathematicians, reading their papers, going to their lectures, etc.

The platonic ideal of this approach is probably Alexander Grothendieck, who is one of the giants of 20th-century mathematics. Here is a nice article about him, with some really wonderful analogies in the first couple of pages:

http://www.landsburg.com/grothendieck/mclarty1.pdf

A good way to start, the way most people start in their PhD program, is to find an advisor who is willing to give you an idea of a subject to study and a good research problem to attack. In my case this was a special case of a fairly general conjecture. It would be nice if a system of polynomial equations, with some suitable assumptions, would have solutions over the rational numbers if and only if it has solutions mod \( N \) for every integer \( N \) as well as solutions over the real numbers. This is called the "Hasse Principle." But this turns out to be false, and nicely enough the counterexamples all turn out to fail for roughly the same reason. (One such counterexample is the above polynomial equation--you can check that it has solutions mod \( N \) for every \( N \).) This was first noticed and described in the early 1970s; it is called the "Brauer-Manin obstruction" to rational points.

https://en.wikipedia.org/wiki/Manin_obstruction

So the modified conjecture is that every counterexample will fail for that specific reason and no other, and remarkably this seems to be true under some mild assumptions about the system of equations (although the "general conjecture," whatever that is, is still unproven). My thesis consisted roughly of developing some computational tools to verify this conjecture in certain interesting cases.

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My Erdos number is 2; I wrote with Martin Gardner who wrote with Erdos!

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What is a piece of advice that you would like to give an aspiring mathematics undergraduate?

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Congratulations 🎊 Patrick Corn. What are you thinking 💭 to do now?

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Me too i want to research and publish paper .but dont know where to start from?! Help

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Congrats Patrick Sir for being featured, what's your dream job other than what you are doing now ?

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As far as the 200 followers problem goes, it took me a little while playing with it before I hit on the substitution to turn it into a quadratic. Once you get there, then you can do the usual number theory tricks, factoring, looking mod \( p \), etc. I liked that one a lot.

Haven't looked at the other one yet; it's on my list! As a number theorist my first instinct would be to use the theory of elliptic curves, which might not be the intended approach. :)

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@Patrick Corn Congratulations for being featured. Seeing your erdos number i feel like why did you quit research. Anyways that's your decision. Can you tell me how is Erdos number calculated?

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Erdős number is basically the

collaborative distancebetween Paul Erdős and another person which is measured by the authorship of scientific research papers.Paul Erdős himself has an Erdős number of \(0\). Those who directly co-authored research papers with him has Erdős number of \(1\). Those who directly didn't co-author a paper with Erdős but co-authored a paper with someone having an Erdős number of \(1\), they have an Erdős number of \(2\), and so on.

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I know what erdos number is, but didnt know how it was calculated.But Thanks for informing.

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I don't want to invent new theories or formulas but I just want to keep learning maths my entire lifetime like u.

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I agree with the contest math stuff.

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