Pell Number Calculator

About this calculator

Pell numbers follow the recurrence P₀=0, P₁=1, Pₙ = 2Pₙ₋₁ + Pₙ₋₂, generating 0, 1, 2, 5, 12, 29, 70, 169 … Their defining property is that the ratio Pₙ/Pₙ₋₁ gives increasingly accurate rational approximations of √2 — the same role continued fractions play, since Pell numbers are literally the denominators (with companion Pell numbers as numerators) of √2's best rational approximants.

They're named for their connection to Pell's equation x² − 2y² = ±1, a classical Diophantine equation studied since antiquity (despite the name, John Pell had little to do with it — Euler misattributed it). Number theorists, cryptography researchers exploring Diophantine-based schemes, and recreational mathematicians use Pell numbers when studying integer solutions and square-root approximation. Enter n to get any Pell number Pₙ.