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

Abstract

Let S be a linear integer recurrent sequence of order k≥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.