Pell's Equation

Find whole numbers x, y with x² − 2y² = 1. The smallest is (3, 2): 9 − 8 = 1. The next is (17, 12): 289 − 288 = 1. Then (99, 70), then (577, 408)… each solution is roughly 1 + √2 times the last, exploding in size. This is Pell's equation, and it has infinitely many solutions for every non-square D — a fact that connects square roots, continued fractions, and the structure of number rings.