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

The hypothesis that ξ is irrational cannot be omitted. Moreover the constant is the best possible; if we replace by any number and we let (the golden ratio) then there exist only finitely many relatively prime integers m, n such that the formula above holds.

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