On Periodicity of Continued Fractions with Partial Quotients in Quadratic Number Fields
Year: 2025
Type: article
Abstract: 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 = ℚ ( 5 ) . By the obtained results, we give a continued fraction expansion algorithm for those real quartic irrationals ξ 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 ξ .
DOI: 10.5802/jtnb.1307
Authors: Zhaonan Wang, Yingpu Deng
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4409003594