Advanced Diophantine Equations

Take y² = x³ − x. Instead of asking 'which numbers solve this?', plot the solutions as points on a curve — and a graceful loop appears. This is an elliptic curve, and its rational points (solutions in fractions) form a secret group: you can 'add' two points to get a third. Mordell proved this group is finitely generated, and the deep question of its size — the rank — is the Birch–Swinnerton-Dyer conjecture, a $1,000,000 Clay problem.