Square Roots & Pell

Patterns that repeat forever

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Square roots have repeating continued fractions, like $\sqrt2 = [1; 2, 2, 2, \dots]$.
• Lagrange: eventually periodic ⇔ a root of a quadratic with whole-number coefficients.
• Convergents of √n solve Pell’s equation x² − ny² = ±1.
• All solutions of x² − ny² = 1 are powers of the smallest one.

Steps

1. Peeling √2
2. Periodic patterns
3. Spot the pattern
4. Pell’s equation
5. Hunting for solutions
6. The next solution