End-symmetric continued fractions and quadratic congruences
Acta Arithmetica, Tome 167 (2015) no. 2, pp. 173-187
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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 $\beta ^2 \equiv \pm 1\ ({\rm mod}\ \alpha )$.
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
Affiliations des auteurs :
Barry R. Smith 1
@article{10_4064_aa167_2_5,
author = {Barry R. Smith},
title = {End-symmetric continued fractions and quadratic congruences},
journal = {Acta Arithmetica},
pages = {173--187},
publisher = {mathdoc},
volume = {167},
number = {2},
year = {2015},
doi = {10.4064/aa167-2-5},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa167-2-5/}
}
Barry R. Smith. End-symmetric continued fractions and quadratic congruences. Acta Arithmetica, Tome 167 (2015) no. 2, pp. 173-187. doi: 10.4064/aa167-2-5
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