Parity of Best Rational Approximations to Root 2

Theorem
Consider the Sequence of Best Rational Approximations to Square Root of 2:
 * $\left\langle{S}\right\rangle := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$

where $S_1 := \dfrac 1 1$.

The numerators of the terms of $\left\langle{S}\right\rangle$ are all odd.

For all $n$, the parity of the denominator of term $S_n$ is the same as the parity of $n$.

Proof
First the parity of the numerators of the terms of $\left\langle{S}\right\rangle$ is established.

Let $\dfrac {p_n} {q_n}$ be a general term of $\left\langle{S}\right\rangle$.

By Relation between Adjacent Best Rational Approximations to Root 2:
 * $p_{n + 1} = p_n + 2 q_n$

Thus if $p_n$ is odd then so is $p_{n + 1}$.

But $p_1 = 1$ is odd.

So $p_n$ is odd for all $n$, by Principle of Mathematical Induction.

The denominators are the Pell numbers.

The result follows from Parity of Pell Numbers.