End-symmetric continued fractions and quadratic congruences

TL;DR

This paper characterizes solutions to certain quadratic congruences using the symmetry properties of continued fraction expansions, extending classical results relating symmetric continued fractions to modular quadratic equations.

Contribution

It generalizes the classical theorem by linking solutions of quadratic congruences to continued fractions with specific asymmetry types.

Findings

Solutions exist iff the continued fraction has a finite set of asymmetry types.

Generalizes the classical symmetric continued fraction theorem.

Connects quadratic congruences with continued fraction asymmetry patterns.

Abstract

We show that for a fixed integer $n \neq \pm2$, the congruence $x^2 + nx \pm 1 \equiv 0 \pmod{\alpha}$ 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 $\beta^2 \equiv \pm 1 \pmod{\alpha}$.