Equivalent Characterisations of Irrational Periodic Continued Fraction

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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.


The following are equivalent:

$(1):\quad$ 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 $x_M = x_N$.


Proof