We show that for a fixed integer n ≠ ±2, the congruence x² + nx ± 1 ≡ 0 (mod α) has the solution β with 0 < β < α if and only if α/β 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 α/β > 1 in lowest terms has a symmetric continued fraction precisely when β² ≡ ±1(mod α ).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-5,
author = {Barry R. Smith},
title = {End-symmetric continued fractions and quadratic congruences},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {173-187},
zbl = {06392170},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-5}
}
Barry R. Smith. End-symmetric continued fractions and quadratic congruences. Acta Arithmetica, Tome 168 (2015) pp. 173-187. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-5/