On the divisibility of the rank of appearance of a Lucas sequence

Abstract

Let [Formula: see text] be a Lucas sequence and, for every prime number [Formula: see text], let [Formula: see text] be the rank of appearance of [Formula: see text] in [Formula: see text], that is, the smallest positive integer [Formula: see text] such that [Formula: see text] divides [Formula: see text], whenever it exists. Furthermore, let [Formula: see text] be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes [Formula: see text] such that [Formula: see text] divides [Formula: see text], as [Formula: see text].