Definition:Periodic Continued Fraction

Periodic Continued Fraction

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 purely periodic continued fraction is a SICF whose 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.

Cycle

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

The number of elements in it is called the cycle length.

In the examples given above, the cycle length is $n$.