# Hurwitz's theorem (number theory)

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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

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

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

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

## 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.