The Case of the Impossible Triangles

The Congruent Number Problem and Elliptic Curves

Chapter 1: The Congruent Number Problem

Choose an integer N. Can you find a right triangle with area N whose sides are all rational numbers?

If so, N is called a "congruent number."

A right triangle with area 30 whose sides are NOT all rational numbers

(\sqrt{136} isn't rational)

A right triangle with area 30 whose sides are all rational numbers

This shows 30 is a congruent number.

Your Turn!

Is 24 a congruent number?

Can you find a right triangle with area 24 whose sides are all rational numbers?

Scaling Rational Right Triangles Gives Rational Right Triangles

6 is a congruent number

Your Turn!

Is 7 a congruent number?

Can you find a right triangle with area 7 whose sides are all rational numbers?

Your Turn!

Is 16 a congruent number?

Can you find a right triangle with area 16 whose sides are all rational numbers?

Can you find a right triangle with area 30 whose side lengths are all rational numbers?
This problem is called the "congruent number problem". Sounds simple, right? But as it turns out, solving this problem leads to some very interesting areas of math. In particular, the congruent number problem has tons of interesting connections to number theory and these things called elliptic curves, which are one of applied math's biggest success stories.

One of the most common questions people ask about math is "Does this have any real-world applications?". For elliptic curves, the answer is "absolutely". Every time you connect to the internet, your computer uses elliptic curves to encrypt your connection so hackers can't steal your credit card numbers when you shop online.
Non-applied mathematicians also care a lot about elliptic curves because they connect many different subfields of math, such as number theory, geometry, and abstract algebra. In fact, they're so important that there's a million-dollar prize offered by the Clay Mathematics Institute for solving an unsolved problem about elliptic curves called the BSD conjecture. To mathematicians, elliptic curves are serious business.

This presentation is an introduction to elliptic curves and how they solve the congruent number problem. First we'll introduce the congruent number problem, then reframe the problem in a beautifully geometric way that's easier to solve, and finally discover how elliptic curves let us generate new triangles from old ones. Let's dive right in!

First, let's take a look at the congruent number problem. Here's the problem: for a given number N, can you find a right triangle with area N whose side lengths are all rational numbers?
Let's start with N=30, which is a particularly easy example. Our goal: find a right triangle with area 30 whose side lengths are all rational numbers.

For example, here's a random triangle with area 30, where we've drawn the lengths of the triangle's three sides. We're looking for triangles where all the sides are rational numbers, so even though this triangle has area 30, it doesn't mean that 30 is a congruent number: one side has length \sqrt{136}, and \sqrt{136} isn't a rational number.

In this other triangle, however, all three sides' lengths are rational numbers. We did it! Because all three sides of this triangle are rational, and it has area 30, this triangle proves 30 is a congruent number.

What about other numbers? For example, is 24 a congruent number? Drag the red circle to explore triangles, and see if you can find a right triangle with area 24 whose sides are all rational numbers.

This triangle has area 24! Therefore, 24 is a congruent number.

Alright, so we know 30 and 24 are congruent numbers. What about other numbers?

Once we know 24 is a congruent number, we can prove that 6 is a congruent number for free.
If we start with the 6-8-10 triangle with area 24 we already found, and multiply each side's length by \frac{1}{2}, we get a 3-4-5 triangle. Since we scale the both the width and height by a factor of 1/2, the area goes down by a factor of (\frac{1}{2})^2 = \frac{1}{4}, giving us a triangle with area 24 \cdot \frac{1}{4} = 6. Since our original triangle had rational sides, and we multiplied by the rational number \frac{1}{2}, our new triangle also has all-rational sides. Therefore, this triangle shows 6 is a congruent number.

What about the number after six, 7? Can you find a right triangle with area 7 and rational sides?

Turns out 7 is a congruent number, but even for really small numbers it can become really hard to find right triangles. The simplest rational right triangle with area 7 has sides \frac{35}{12}, \frac{24}{5}, and \frac{337}{60}. N=7 hasn't even reached double-digits, and these triangles are already involving fractions with numbers over a hundred!

Things get even more hairy for the next congruent number N=13 - it's not that much bigger than 7, but the simplest rational triangle with area 13 has sides \frac{780}{323}, \frac{323}{30}, and - wait for it - \frac{106921}{9690}. Those may be huge numbers in the numerator and denominator, but they're still rational.

And what about 16? Is 16 is a congruent number? Try for yourself - can you find a triangle with area 16 and rational sides?

Turns out: For n=16, it's impossible! There are no right triangles whatsoever with rational sides with area 16.
But... how would you prove that it's impossible? Right triangles, rational sides, and area 16 are all straightforward conditions. Don't take my word for it - how do you know there isn't some triangle out there with area 16 whose sides are ridiculously complicated fractions? Hunting for triangles manually clearly won't be enough. Is there a systematic way to figure out whether or not N is a congruent number?
To find out, we'll need to reframe the question and take a look at some geometry.

Next Chapter