Parity of Best Rational Approximations to Root 2

Theorem

Consider the Sequence of Best Rational Approximations to Square Root of 2:

$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$

where $S_1 := \dfrac 1 1$.

The numerators of the terms of $\sequence S$ 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 $\sequence S$ is established.

Let $\dfrac {p_n} {q_n}$ be a general term of $\sequence S$.

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.

$\blacksquare$