Hurwitz's theorem (number theory)

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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 \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.

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