End-symmetric continued fractions and quadratic congruences

Abstract

We show that for a fixed integer $n\ne \pm 2$, the congruence $x^2+nx\pm 1\equiv 0(\mathrm{mod}\alpha )$ 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 ${\beta }^2\equiv \pm 1(\mathrm{mod}\alpha )$.

15 pages