Continued Fractions and Linear Fractional Transformations
Continued Fractions and Linear Fractional Transformations
复制标题
连分数和线性分数变换
作者：
Evan M. O’Dorney
对于任意正整数$d$，从$\inty$开始迭代变换$f(X)=(dx+k)/(x+d)$，即可得到平方根$\Sqrt{k}$的有理逼近。我们证明了当且仅当$4d^2/(k-d^2)$是一个整数时，这些近似与连分式的收敛是无限重合的，在这种情况下，连分式具有丰富的结构。它由某些可明确定义的有理数的连分式串联而成，它属于无穷多个连分式族中的一个，这些连分式族的项在两个参数中线性变化。我们还给出了轨道完全由收敛点或半收敛点组成的条件，并证明了它包含Pell方程$p^2-k q^2=\pm1$的所有解$p/q$。
Rational approximations to a square root $\sqrt{k}$ can be produced by iterating the transformation $f(x) = (dx+k)/(x+d)$ starting from $\infty$ for any positive integer $d$. We show that these approximations coincide infinitely often with continued fraction convergents if and only if $4d^2/(k-d^2)$ is an integer, in which case the continued fraction has a rich structure. It consists of the concatenation of the continued fractions of certain explicitly definable rational numbers, and it belongs to one of infinitely many families of continued fractions whose terms vary linearly in two parameters. We also give conditions under which the orbit $\{f^n(\infty)\}$ consists exclusively of convergents or semiconvergents and prove that with few exceptions it includes all solutions $p/q$ to the Pell equation $p^2 - k q^2 = \pm 1$.