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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.

Fix an element $\alpha$ in a quadratic field $K$. Define $S$ as the set of rational primes $p$, for which $\alpha$ has maximal order modulo $p$. Under the assumption of the generalized Riemann hypothesis, we show that $S$ has a density. Moreover, we give necessary and sufficient conditions for the density of $S$ to be positive.