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 $\scriptstyle {\sqrt {5}}$ 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.

References

Hurwitz, A. (1891). "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximate representation of irrational numbers by rational fractions)". Mathematische Annalen (in German). 39 (2): 279–284. doi:10.1007/BF01206656. JFM 23.0222.02. (note: a PDF version of the paper is available from the given weblink for the volume 39 of the journal, provided by Göttinger Digitalisierungszentrum)
G. H. Hardy, Edward M. Wright, Roger Heath-Brown, Joseph Silverman, Andrew Wiles (2008). "Theorem 193". An introduction to the Theory of Numbers (6th ed.). Oxford science publications. p. 209. ISBN 0-19-921986-9.
LeVeque, William Judson (1956). "Topics in number theory". Addison-Wesley Publishing Co., Inc., Reading, Mass. MR 0080682.
Ivan Niven (2013). Diophantine Approximations. Courier Corporation. ISBN 0486462676.

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