On the equivalence of certain quadratic irrationals

This paper deals with quadratic irrationals of the form $m\mathrm{/}q+\sqrt{v}$ for fixed positive integers $v$ and $q$, $v$ not a square, and varying integers $m$, $(m,q)=1$. Two numbers $m\mathrm{/}q+\sqrt{v}$, $n\mathrm{/}q+\sqrt{v}$ of this kind are equivalent (in a classical sense) if their continued fraction expansions can be written with the same period. We give a necessary and sufficient condition for the equivalence in terms of solutions of Pell’s equation. Moreover, we determine the number of equivalence classes to which these quadratic irrationals belong.