Sums of Two Squares

Fermat’s Christmas theorem

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• A prime is a sum of two squares exactly when it is 2 or of the form $4k + 1$.
• Squares leave remainder 0 or 1 mod 4, so 4k + 3 numbers can never work.
• The number of grid points inside a circle of radius $r$ is about $\pi r^2$.

Steps

1. Points on a circle
2. Which primes?
3. Why 4k + 3 fails
4. Fermat’s Christmas theorem
5. Multiplying sums
6. Counting inside the circle
7. Spot the sum