Last week, my computer broke. No writing, no programming, no email or book promotion. It was a perfect opportunity for me to rediscover how much I love math.
I did a lot of it, and good timing. This summer I have been granted a directed research opportunity at the university. The problem of interest: one that has fascinated me since my adviser introduced it to me in 2011, the congruent number problem. Prior to this, I had been working on generating a lot of data for the sake of understanding some of the patterns behind the scenes. Doing algebra and work by hand gave me a fresh angle which has now given me direction for exploration.
No, it's not the Reimann Hypothesis, but if anything my studies have taught me that exploring such a topic requires lots of careful development and time; I am simply not ready for it yet. Fortunately, a lot of the work I did - premature and amateur though it was at the time - has useful carry-over to this problem, itself a special case of the Birch-Swinnerton-Dyer Conjecture (which, it turns out, is deeply intertwined with the Riemann Hypothesis).
My interest in number theory and, in particular, the patterns and properties of the primes, is leading me into a fascinating little universe, and I am delighted - and excited - to have this opportunity to work full time as an apprentice researcher. The congruent number problem is a great framework for learning about elliptic curves and the properties of group operations they exhibit when analyzed over finite fields. And this is a great starting point for asking the question, "When will an elliptic curve have rational solutions?"
So what is the congruent number problem, exactly?
You could google it, but most of what you will find might be quite technical. For those of you who are not mathematicians, but are interested, it's not complicated:
You have learned about a right angle triangle as a child. Many people know the Pythagorean Theorem: the sum of the squares of the perpendicular sides of a triangle is equal to the square of the hypotenuse. You also might remember the formula for the area of a triangle from high school: area equals one half base times height.
What if I said to you: find me a right angle triangle that has area 1, 2, 3, ... and so on? In other words, what if I gave you a whole number and said to find me a right angle triangle whose area equals this number - call it "n".
It would be easy to do if you used your calculator. You can just pick any number for one of the perpendicular sides, say 1. Then the other side will be 2n, when you cross multiply, and the hypoteneuse will be the square root of 4n squared plus one.
That's easy; for a mathematician, easy is not interesting. See, depending on what you choose for n, your calculator might put out something like 1.71415444... for the hypotenuse - in other words, a number that doesn't round nicely. But it's still an answer, and if you were an architect you'd have the measurement you need to make sure the pieces of your triangular frame fit together.
So we look for rational numbers - a number that is a fraction. Like 1/3 or 4/7 or 34567 / 356733562984498 - you name it. Now let's ask the question again: what triangles of whole number area exist so that we have all rational sides?
This is the congruent number problem. It turns out these numbers are 5,6,7, 13, 14, 15, 20, 21, 22, ... this list goes on and on. 6 is the simplest example - that's the area of a 3,4,5 triangle (3 is a fraction: 3/1, as it 4 and 5, and you can quickly check that the area of this triangle is indeed 6).
You can try to find congruent numbers with some algebra, but it's messy. When I started with this problem, I used a spreadsheet and captured some triangles by brute force, but there were others, such as the triangle with area 4, that had no answer. In mathematics, no answer is not good enough - one must prove there is none.
Fast-forwarding a little, we can use some algebra tricks to construct a special curve called an elliptic curve (it has nothing to do with an ellipse), where the area is built into a curve that varies with x and y - x is related to the sides of the triangle (specifically, x = (c/2)^2 ).
Why go and make it messy like that? Mathematics is a bit like a complicated game of minesweeper. Sometimes you must go around to other corners and work out things you know, before that will open up what you need for the part where you are stuck. And so it is with elliptic curves. After all, it was these wonderful little creatures that led to Andrew Wiles' proof of Fermat's Last Theorem.
And because of the mad rush to solve Fermat's Last Theorem, we know a lot about elliptic curves. Particularly, we know that if you analyze them over finite fields, you will find certain properties that give a clue to whether they have rational solutions or not. In other words, the elliptic curves we construct with that integer area of interest, should they have rational points (since x is related to our sides) will thus give us a triangle with rational sides and integer area.
Now, the biggest question: why do we care? Who ever needs to understand such things about triangles? After all, the architects are not going to be concerned with 0.00000001 of a centimeter of accuracy.
Well, the answer is that it's not really about triangles at all. The congruent number problem is about numbers, particularly prime numbers. A triangle is a nice picture for us to relate a deep number theory problem about special ways that collections of products of primes can be formed; it's just one of many unique questions that represent the universe of numbers and their properties.
The devil's in the details, and I have my hands full of devils right now. Though angels would be more appropriate. This project is a marriage of computer programming and the skills I am developing, and it is great fun.
I write this tonight, at the end of my day, after getting in my hour-long fix of writing, and I'm already looking forward to getting up early and doing it all over again tomorrow.