Hurwitz's theorem is an extension of Minkowksi's Theorem and deals with rational approximations to irrational numbers. The theorem states:

For every irrational number $\alpha$, there are infinitely many coprime integers $p$ and $q$ such that:

$$\left|\alpha - \frac{p}{q} \right| < \frac{1}{\sqrt{5}q^2} $$

It turns out that the $\sqrt{5}$ term is sharp. My question is why does the $\sqrt{5}$ term appear. What properties of $\sqrt{5}$ enable this to be a sharp bound? I am looking for an intuitive understanding.

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states:

For every irrational number $\alpha$, there are infinitely many coprime integers $p$ and $q$ such that:

$$\left|\alpha - \frac{p}{q} \right| < \frac{1}{\sqrt{5}q^2} $$

It turns out that the $\sqrt{5}$ term is sharp. My question is why does the $\sqrt{5}$ term appear. What properties of $\sqrt{5}$ enable this to be a sharp bound? I am looking for an intuitive understanding.