Hurwitz's theorem (number theory)

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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

The condition that ξ is irrational cannot be omitted. Moreover the constant is the best possible; if we replace by any number and we let (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 .


  • 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. {{cite journal}}: External link in |volume= (help)(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.{{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. {{cite journal}}: Cite journal requires |journal= (help)
  • Ivan Niven (2013). Diophantine Approximations. Courier Corporation. ISBN 0486462676.