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 condition that ξ is irrational cannot be omitted. Moreover the constant ${\sqrt {5}}$ is the best possible; if we replace ${\sqrt {5}}$ by any number $A>{\sqrt {5}}$ and we let $\xi =(1+{\sqrt {5}})/2$ (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 ${\sqrt {5}}$.

See also

• Dirichlet's approximation theorem
• Langrange number

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. S2CID 119535189.
• 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 978-0-19-921986-5.{{cite book}}: CS1 maint: multiple names: authors list (link)
• LeVeque, William Judson (1956). Topics in number theory. Addison-Wesley Publishing Co., Inc., Reading, Mass. MR 0080682.
• Ivan Niven (2013). Diophantine Approximations. Courier Corporation. ISBN 978-0486462677.