Equivalent Characterisations of Irrational Periodic Continued Fraction
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Theorem
Let $x \in \R \setminus \Q$ be an irrational number.
Let $\sequence {a_n}_{n \mathop \ge 0}$ be its continued fraction.
Let $N \ge 0$ be a natural number.
The following statements are equivalent:
- $(1):\quad$ The sequence of partial denominators $\sequence {a_n}_{n \mathop \ge 0}$ is periodic for $n \ge N$.
- $(2):\quad$ The sequence of complete quotients $\sequence {x_n}_{n \mathop \ge 0}$ is periodic for $n \ge N$.
- $(3):\quad$ There exists $M > N$ such that $x_M = x_N$.
Proof
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