top
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 ).
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 -