Yann Bugeaud (Université de Strasbourg): On the continued fraction expansion of algebraic numbers

Online Number Theory Seminar

Abstract: Let $\xi = [a_0; a_1, a_2, \ldots ]$ be an irrational algebraic real number and $(p_k / q_k)_{k \ge 1}$ denote the sequence of its convergents. We survey various (mostly arithmetical) properties of the sequences $(a_j)_{j \ge 1}$ and $(q_k)_{k \ge 1}$. Let $(u_n)_{n \ge 1}$ 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 $(u_n)_{n \ge 1}$ is infinite, then $\xi$ is a quadratic number, a recent result obtained jointly with Khoa Nguyen.

For access please contact the organizers (ntrg[at]science.unideb.hu).