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

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

See also

• Dirichlet's approximation theorem
• Lagrange 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.