Finitely Many Reduced Associated Quadratic Irrationals

Lemma
Let $n \in \Z: n > 0$.

There are finitely many reduced quadratic irrationals which are associated to $n$.

Proof
By definition, we can write an arbitrary reduced irrational as $\alpha = \dfrac{\sqrt n + P} Q$.

By definition, $\alpha > 1$ and its conjugate $\tilde{\alpha} > -1$.

So we know $\alpha + \tilde{\alpha} = \dfrac {2P} Q > 0$.

Hence with the assumption $Q > 0$ we have $P > 0$.

Since $\tilde{\alpha} < 0$ we also have $P < \sqrt n$.

Also, since $\alpha > 1$ by assumption, we have $Q < P + \sqrt n < 2 \sqrt n$.

Thus there are finitely many choices for both $P$ and $Q$, forcing finitely many reduced quadratic irrationals associated to a fixed $n$.

This amount is strictly bounded above by $2n$.