Parity of Best Rational Approximations to Root 2
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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.
$\blacksquare$