End-symmetric continued fractions and quadratic congruences

Barry R. Smith¹

¹ Department of Mathematical Sciences Lebanon Valley College Annville, PA 17003, U.S.A.

Acta Arithmetica, Tome 167 (2015) no. 2, pp. 173-187.

Résumé

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 $\beta ^2 \equiv \pm 1\ ({\rm mod}\ \alpha )$.

Détail

DOI : 10.4064/aa167-2-5

Keywords: fixed integer congruence equiv mod alpha has solution beta beta alpha only alpha beta has continued fraction expansion sequence quotients having finite number possible asymmetry types generalizes old theorem rational number alpha beta lowest terms has symmetric continued fraction precisely beta equiv mod alpha