Hurwitz's theorem (number theory)

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 {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 {displaystyle {sqrt {5}}} is the best possible; if we replace {displaystyle {sqrt {5}}} by any number {displaystyle A>{sqrt {5}}} and we let {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 {displaystyle {sqrt {5}}} .

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. 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. Categories: Diophantine approximationTheorems in number theory

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