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}
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