Hurwitz's theorem (number theory)

This article is about a theorem in number theory. For other uses, see Hurwitz's theorem.

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

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

The hypothesis that ξ is irrational cannot be omitted. Moreover the constant Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \scriptstyle \sqrt{5}

is the best possible; if we replace Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \scriptstyle \sqrt{5}
by any number Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \scriptstyle A > \sqrt{5}
and we let Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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.