On Periodicity of Continued Fractions with Partial Quotients in Quadratic Number Fields

Wang, Zhaonan; Deng, Yingpu

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1053-1076

In this paper, we fix a real quadratic field $K$ and take an ultimately periodic continued fraction with partial quotients in ${𝒪}_K$. We examine the convergence of the sequence and the increase in the sizes of both the numerators and denominators of the convergent fractions. Additionally, we establish necessary and sufficient conditions for a real quartic irrational to possess an ultimately periodic continued fraction that converges to it, with partial quotients belonging to ${𝒪}_K$. Finally, we analyze a specific example with $K=\mathbb{Q}\left(\sqrt{5}\right)$. By the obtained results, we give a continued fraction expansion algorithm for those real quartic irrationals $\xi$ belonging to a quadratic extension of $K$ whose algebraic conjugates are all real. We prove that the expansion obtained from the algorithm is ultimately periodic and converges to the specified $\xi$.

Dans cet article, nous fixons un corps quadratique réel $K$ et considérons une fraction continue ultimement périodique à quotients partiels dans ${𝒪}_K$. Nous étudions le problème de convergence et examinons l’augmentation de la taille des numérateurs et dénominateurs partiels des fractions convergentes. En outre, nous établissons des conditions nécessaires et suffisantes pour qu’un nombre irrationnel quartique réel admette un développement en fraction continue ultimement périodique à quotients partiels dans ${𝒪}_K$. Enfin, nous analysons l’exemple du corps $K=\mathbb{Q}\left(\sqrt{5}\right)$. À partir des résultats obtenus, nous proposons un algorithme de développement en fraction continue des irrationnels quartiques réels $\xi$ appartenant à une extension quadratique de $K$ dont les conjugués algébriques sont tous réels. Nous démontrons que la fraction continue obtenue à partir de l’algorithme est ultimement périodique et converge vers $\xi$.

Classification : 11J70, 11Y65
Keywords: Continued fractions, quadratic fields, periodicity, Weil height

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