# Hurwitz's theorem (number theory)

**Short description**Theorem in number theory that gives a bound on a Diophantine approximation

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

- [math]\displaystyle{ \left |\xi-\frac{m}{n}\right |\lt \frac{1}{\sqrt{5}\, n^2}. }[/math]

The condition that *ξ* is irrational cannot be omitted. Moreover the constant [math]\displaystyle{ \scriptstyle \sqrt{5} }[/math] is the best possible; if we replace [math]\displaystyle{ \scriptstyle \sqrt{5} }[/math] by any number [math]\displaystyle{ \scriptstyle A \gt \sqrt{5} }[/math] and we let [math]\displaystyle{ \scriptstyle \xi=(1+\sqrt{5})/2 }[/math] (the golden ratio) then there exist only *finitely* many relatively prime integers *m*, *n* such that the formula above holds.

The theorem is equivalent to the claim that the Markov constant of every number is larger than [math]\displaystyle{ \sqrt{5} }[/math].

## References

- "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximate representation of irrational numbers by rational fractions)" (in German).
*Mathematische Annalen***39**(2): 279–284. 1891. doi:10.1007/BF01206656.(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. - Ivan Niven (2013).
*Diophantine Approximations*. Courier Corporation. ISBN 0486462676.

Original source: https://en.wikipedia.org/wiki/ Hurwitz's theorem (number theory).
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