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

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

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

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.