The purpose of this paper is to generalize some seminal results in the literature concerning the interrelationships between Legendre symbols and continued fractions. We introduce the power of ideal theory into the arena. This allows significant improvements over the existing results via the infrastructure of real quadratic fields.
Let d ≥ 2 be a square-free integer and for all n ≥ 0, let be the length of the continued fraction expansion of . If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].
À partir d’une courbe elliptique définie sur le corps des classes de Hilbert d’un corps quadratique imaginaire et à multiplicité complexe par l’anneau des entiers de , on construit des fonctions elliptiques. Nous établissons des formules produits relatives à ces fonctions. De ce fait, nous obtenons une formulation analytique du lemme de Gauss généralisé ainsi qu’une expression explicite pour le symbole quadratique de Legendre défini sur l’anneau des entiers du corps quadratique imaginaire. Comme...
In the previous paper [15], we determined the structure of the Galois groups of the maximal unramified extensions of imaginary quadratic number fields of conductors under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors ) and give a table of . We update the table (under GRH). For 19 exceptional fields of them, we determine . In particular, for , we obtain , the fourth Hilbert class field of . This is the first example of a number field whose...