Some arithmetical properties of convergents to algebraic numbers

Let xi be an irrational algebraic real number and let (p(k)/q(k))(k >= 1) denote the sequence of its convergents. Let (un)n(>= 1) be a nondegenerate linear recurrence sequence of integers, which is not a polynomial sequence. We show that if the intersection of the sequences (q(k))(k >= 1) and (un)(n >= 1) is infinite, then xi is a quadratic number. This extends an earlier work of Lenstra and Shallit (1993). We also discuss several arithmetical properties of the base -b representation of the integers q(k), k >= 1, where b >= 2 is an integer. Finally, when xi is a (possibly transcendental) non-Liouville number, we prove a result implying the existence of a large prime factor of q(k-1 qk qk+1) for large k. This is related to earlier results of Erdos and Mahler (1939), Shorey and Stewart (1983), and Shparlinskii (1987).