# Hurwitz's theorem (number theory)

$\left |\xi-\frac{m}{n}\right |<\frac{1}{\sqrt{5}\, n^2}.$
The hypothesis that ξ is irrational cannot be omitted. Moreover the constant $\scriptstyle \sqrt{5}$ is the best possible; if we replace $\scriptstyle \sqrt{5}$ by any number $\scriptstyle A > \sqrt{5}$ and we let $\scriptstyle \xi=(1+\sqrt{5})/2$ (the golden ratio) then there exist only finitely many relatively prime integers m, n such that the formula above holds.