Periodic continued fraction

Definition

An infinite simple continued fraction $[a_0; a_1, a_2, \dots]$ is periodic if there are integers $N \ge 0$ and $k \ge 1$ with $a_{n+k} = a_n$ for all $n \ge N$. It is purely periodic if this holds with $N = 0$.

Remarks

One writes the repeating block with a bar, for example $\sqrt{2} = [1; \overline{2}]$ and $\sqrt{3} = [1; \overline{1, 2}]$. By theorems of Euler and Lagrange, a real number has a periodic continued fraction if and only if it is a quadratic irrational. That is, it is an irrational root of a quadratic equation with integer coefficients. The period of $\sqrt{d}$ gives the fundamental solution of Pell's equation.

Explanations

L1 human

A periodic continued fraction repeats the same block of terms forever. The golden ratio $[1; 1, 1, 1, \dots]$ is the simplest example.