Equivalent Characterisations of Irrational Periodic Continued Fraction

Theorem
Let $x \in \R \setminus \Q$ be an irrational number.

Let $(a_n)_{n\geq 0}$ be its continued fraction.

Let $N \geq 0$ be a natural number.


 * $(1):\quad$ The sequence of partial quotients $(a_n)_{n\geq 0}$ is periodic for $n \geq N$.
 * $(2):\quad$ The sequence of complete quotients $(x_n)_{n\geq 0}$ is periodic for $n \geq N$.
 * $(3):\quad$ There exists $M > N$ such that $x_M = x_N$.