An improvement of the Minkowski bound for real quadratic orders using the Markoff theorem

Using the Markoff theorem on indefinite binary quadratic forms it is shown that every element of the class group of any real quadratic order of discriminant d has an ideal of norm less than or equal to 1+⌊d3⌋. This bound is best possible as there are quadratic orders of discriminant d which have ideal classes where the least norm is 1+⌊d3⌋. These discriminants are explicitly given and their connection to the Markoff conjecture is highlighted.