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 approximation of irrational numbers by rational numbers)". Mathematische Annalen (in German) 39 (2): 279–284. doi:10.1007/BF01206656. JFM 23.0222.02. 
• 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.