End-symmetric continued fractions and quadratic congruences
Volume 167 / 2015
Acta Arithmetica 167 (2015), 173-187
MSC: Primary 11A55; Secondary 11A05.
DOI: 10.4064/aa167-2-5
Abstract
We show that for a fixed integer $n \not =\pm 2$, the congruence $x^2 + nx \pm 1 \equiv 0 \ ({\rm mod}\ \alpha )$ has the solution $\beta $ with $0 < \beta < \alpha $ if and only if $\alpha /\beta $ has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number $\alpha /\beta > 1$ in lowest terms has a symmetric continued fraction precisely when $\beta ^2 \equiv \pm 1\ ({\rm mod}\ \alpha )$.
