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

In this paper, we fix a real quadratic field K and take an ultimately periodic continued fraction with partial quotients in (9K. 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 (9K. Finally, we analyze a specific example with K = Q(v/5). By the obtained results, we give a continued fraction expansion algorithm for those real quartic irrationals 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 .