Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

Streszczenie

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