Iff Criterion For An Irrational Number to be Purely Periodic

# Iff Criterion For An Irrational Number to be Purely Periodic

Recall from the Periodic and Purely Periodic Continued Fractions that an irrational number $\theta = \langle a_0; a_1, a_2, ... \rangle$ is said to be purely periodic if there exists an $n \in \mathbb{N}$ such that $a_{n+r} = a_r$ for all $r$.

We are about to characterize when an irrational number has a purely periodic continued fraction expansion. Before we do that though, we need a definition.

 Definition: Let $\theta$ be a quadratic irrational with $\displaystyle{\theta = \frac{a + \sqrt{b}}{c}}$. Then the Conjugate of $\theta$ is $\displaystyle{\theta' = \frac{a - \sqrt{b}}{c}}$.

We are now ready to state the theorem.

 Theorem 1: Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$. Then $\theta$ has a purely periodic continued fraction expansion if and only $\theta$ is a quadratic irrational such that $\theta > 1$ and $-1 < \theta' < 0$.

For example, consider the irrational number:

(1)
\begin{align} \quad \theta = \frac{2 + \sqrt{5}}{3} \end{align}

It can easily be verified that $\theta > 1$ and $-1 < \theta' < 0$. So by Theorem 1, $\theta$ has a purely periodic continued fraction expansion. Indeed the continued fraction expansion for $\theta$ is:

(2)
\begin{align} \quad \theta = \langle 1; \overline{2, 2, 2, 1, 12, 1} \rangle \end{align}