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

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

The hypothesis 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 ${\displaystyle \scriptstyle A>{\sqrt {5}}}$ 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

• 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 |journal= (help)
• Ivan Niven (2013). Diophantine Approximations. Courier Corporation. ISBN 0486462676.