Definition:Periodic Continued Fraction

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Let $\left[{a_1, a_2, a_3, \ldots}\right]$ be a simple infinite continued fraction.

Let the partial quotients be of the form:

$\left[{r_1, r_2, \ldots, r_m, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, \ldots}\right]$

that is, ending in a block of partial quotients which repeats itself indefinitely.

Such a SICF is known as a periodic continued fraction.

The notation used for this is $\left[{r_1, r_2, \ldots, r_m, \left \langle{s_1, s_2, \ldots, s_n}\right \rangle}\right]$, where the repeating block is placed in angle brackets.

Purely Periodic Continued Fraction

A periodic continued fraction is a purely periodic continued fraction if its partial quotients are of the form:

$\left[{\left \langle{s_1, s_2, \ldots, s_n}\right \rangle}\right]$

That is, all of its partial quotients form a block which repeats itself indefinitely.


The repeating block in a periodic (or purely periodic) continued fraction $F$ is called the cycle of $F$.