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

Abstract

Let $U=(U_n{)}_{n\ge 0}$ be a Lucas sequence and, for every prime number $p$, let ${\rho }_U(p)$ be the rank of appearance of $p$ in $U$, that is, the smallest positive integer $k$ such that $p$ divides $U_k$, whenever it exists. Furthermore, let $d$ be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes $p\le x$ such that $d$ divides ${\rho }_U(p)$, as $x\to +\infty$.