Английская Википедия:Hurwitz's theorem (number theory)
Шаблон:No footnotes Шаблон:Short description Шаблон:About 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 display="block">\left |\xi-\frac{m}{n}\right | < \frac{1}{\sqrt{5}\, n^2}.</math>
The condition that ξ is irrational cannot be omitted. Moreover the constant <math>\sqrt{5}</math> is the best possible; if we replace <math>\sqrt{5}</math> by any number <math>A > \sqrt{5}</math> and we let <math>\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>\sqrt{5}</math>.
See also
References
