End-symmetric continued fractions and quadratic congruences

Barry R. Smith. "End-symmetric continued fractions and quadratic congruences." Acta Arithmetica 167.2 (2015): 173-187. <http://eudml.org/doc/279417>.

@article{BarryR2015,
abstract = {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 ).},
author = {Barry R. Smith},
journal = {Acta Arithmetica},
keywords = {continued fraction; continuant; end-symmetric},
language = {eng},
number = {2},
pages = {173-187},
title = {End-symmetric continued fractions and quadratic congruences},
url = {http://eudml.org/doc/279417},
volume = {167},
year = {2015},
}

TY - JOUR
AU - Barry R. Smith
TI - End-symmetric continued fractions and quadratic congruences
JO - Acta Arithmetica
PY - 2015
VL - 167
IS - 2
SP - 173
EP - 187
AB - 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 ).
LA - eng
KW - continued fraction; continuant; end-symmetric
UR - http://eudml.org/doc/279417
ER -