What else can the elliptic integral count?

Question

I just read this document - Jacobi's Four Square Theorem. It shows how to count the number of representations of a number as the sum of four squares.

I can follow the proof but currently it just appears like a brilliant coincidence, so I am wondering what else can you count with these?

Answer 1

"What else can you count with these?": Solutions to the Fermat equation $x^n + y^n = z^n$. (If this doesn't make immediate sense, I will add more material later, when I have more time.)