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

In this paper, we fix a real quadratic field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> and take an ultimately periodic continued fraction with partial quotients in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝒪</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝒪</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> . Finally, we analyze a specific example with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ℚ</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . By the obtained results, we give a continued fraction expansion algorithm for those real quartic irrationals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> belonging to a quadratic extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> whose algebraic conjugates are all real. We prove that the expansion obtained from the algorithm is ultimately periodic and converges to the specified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> .