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arXiv · 2209.07485

Some arithmetical properties of convergents to algebraic numbers

Yann Bugeaud·Khoa D. Nguyen
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Abstract

Let $\xi$ be an irrational algebraic real number and $(p_k / q_k)_{k \ge 1}$ denote the sequence of its convergents. Let $(u_n)_{n \geq 1}$ be a non-degenerate linear recurrence sequence of integers, which is not a polynomial sequence. We show that if the intersection of the sequences $(q_k)_{k \ge 1}$ and $(u_n)_{n \geq 1}$ is infinite, then $\xi$ is a quadratic number. We also discuss several arithmetical properties of the sequence $(q_k)_{k \ge 1}$.

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BibTeXRIS

Yann Bugeaud, Khoa D. Nguyen. 2022-09-15. Some arithmetical properties of convergents to algebraic numbers. https://doi.org/10.2140/pjm.2023.326.17

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