The Dirichlet density of set of primes related to the Lucas sequences

For a Lucas sequence of the first kind, denoted by [Formula: see text], defined by [Formula: see text], where a and b are nonzero rational numbers and D is a square free integer, let [Formula: see text] be the least positive integer [Formula: see text] such that [Formula: see text], whenever it exists. We derive a formula for the density of primes p satisfying [Formula: see text], where [Formula: see text] is a given positive integer.