论文标题
关于某些二次非理性的等效性
On the equivalence of certain quadratic irrationals
论文作者
论文摘要
本文处理固定正整数$ v $和$ q $，$ v $，$ v $而不是正方形的$ m/q+\ sqrt v $的二次非理性，而变化的整数$ m $，$（m，q）= 1 $。两个数字$ m/q+\ sqrt v $，$ n/q+\ sqrt v $如果可以使用同一时期编写它们的持续分数扩展，则是等效的（从经典意义上讲）。就佩尔方程的解决方案而言，我们给出了相当的必要条件。此外，我们确定了这些二次非理性属于的等效类的数量。
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.
