# Hurwitz's theorem (number theory)

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.

## 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.CS1 maint: multiple names: authors list (link) - LeVeque, William Judson (1956). "Topics in number theory". Addison-Wesley Publishing Co., Inc., Reading, Mass. MR 0080682. Cite journal requires
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(help) - Ivan Niven (2013).
*Diophantine Approximations*. Courier Corporation. ISBN 0486462676.

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