### Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields

Let $S$ be a linear integer recurrent sequence of order $k\ge 3$, and define ${P}_{S}$ as the set of primes that divide at least one term of $S$. We give a heuristic approach to the problem whether ${P}_{S}$ has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that ${P}_{S}$ has positive lower density for “generic” sequences $S$. Some numerical examples are included.