On the equivalence of certain quadratic irrationals

TL;DR

This paper characterizes when quadratic irrationals of the form m/q + sqrt(v) are equivalent based on their continued fractions and Pell's equation, and counts the number of equivalence classes among them.

Contribution

It provides a necessary and sufficient condition for equivalence of these irrationals and determines the number of their equivalence classes.

Findings

Equivalence characterized by solutions to Pell's equation.

Derived a criterion for when two irrationals are equivalent.

Counted the total number of equivalence classes.

Abstract

This paper deals with quadratic irrationals of the form $m/q+\sqrt v$ for fixed positive integers $v$ and $q$, $v$ not a square, and varying integers $m$, $(m,q)=1$. Two numbers $m/q+\sqrt v$, $n/q+\sqrt v$ 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.