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

The condition that ξ is irrational cannot be omitted. Moreover, the constant is the best possible; if we replace by any number and we let (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 .

See also

• Dirichlet's approximation theorem
• Lagrange number

References