Hurwitz's theorem (number theory)

Short description: Theorem in number theory that gives a bound on a Diophantine approximation

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{ \sqrt{5} }[/math] is the best possible; if we replace [math]\displaystyle{ \sqrt{5} }[/math] by any number [math]\displaystyle{ A \gt \sqrt{5} }[/math] and we let [math]\displaystyle{ \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].

See also

• Dirichlet's approximation theorem
• Langrange number

References

• [290} "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche"] (in German). Mathematische Annalen 39 (2): 279–284. 1891. doi:10.1007/BF01206656. https://gdz.sub.uni-goettingen.de/id/PPN235181684_0039?tify={%22pages%22:[290]}. 
• 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.. 
• Ivan Niven (2013). Diophantine Approximations. Courier Corporation. ISBN 978-0486462677.