论文标题
有理数的平方根的单位
Units from square-roots of rational numbers
论文作者
论文摘要
令$ d，q $为自然数字，$（d，q）= 1 $，这样$ d/q> 1 $和$ d/q $不是一个正方形。令$ q $是$ q $的最小除数，以便$ q | \，q^2 $。我们表明，单位$> 1 $的环$ \ Mathbb z [\ sqrt {dq^2/q}] $与$ \ sqrt {d/q} $的某些汇合连接。在这些单元中，$ \ mathbb z [\ sqrt {dq}] $的单位扮演着一个特殊的角色，因为它们对应于$ \ sqrt {d/q} $的收敛，而这些汇聚是在每个周期结束之前发生的。我们还表明，最后提到的单元允许阅读某些二次非理性的（有限）持续扩展某些合理数量的（有限）持续扩展的（周期性的）持续分数扩展。
Let $D,Q$ be natural numbers, $(D,Q)=1$, such that $D/Q>1$ and $D/Q$ is not a square. Let $q$ be the smallest divisor of $Q$ such that $Q|\, q^2$. We show that the units $>1$ of the ring $\mathbb Z[\sqrt{Dq^2/Q}]$ are connected with certain convergents of $\sqrt{D/Q}$. Among these units, the units of $\mathbb Z[\sqrt{DQ}]$ play a special role, inasmuch as they correspond to the convergents of $\sqrt{D/Q}$ that occur just before the end of each period. We also show that the last-mentioned units allow reading the (periodic) continued fraction expansion of certain quadratic irrationals from the (finite) continued fraction expansion of certain rational numbers.
