Hurwitz's theorem (number theory)

Template:Short description 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 $$|\xi -\frac{m}{n}|<\frac{1}{\sqrt{5}{n}^2}.$$

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

See also

• Dirichlet's approximation theorem
• Lagrange number