# Hurwitz's theorem (number theory)

In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m, n such that

${\displaystyle \left|\xi -{\frac {m}{n}}\right|<{\frac {1}{{\sqrt {5}}\,n^{2}}}.}$

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