End-symmetric continued fractions and quadratic congruences

• Autorzy: Barry R. Smith
• Języki publikacji: EN

Abstrakty

EN

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 ).

Kategorie tematyczne

• 11A55: Continued fractions
• 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 167
• Numer: 2
• Strony: 173-187

Daty

wydano

2015

Twórcy

autor

Barry R. Smith

• Department of Mathematical Sciences, Lebanon Valley College, Annville, PA 17003, U.S.A.

Identyfikatory

DOI

10.4064/aa167-2-5