# Hurwitz's theorem (number theory)

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Short description
Theorem in number theory that gives a bound on a Diophantine approximation

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

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

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

The theorem is equivalent to the claim that the Markov constant of every number is larger than $\displaystyle{ \sqrt{5} }$.