Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields
Let \(S\) be a linear integer recurrent sequence of order \(k\geq 2\), and define \(P_S\) as the set of primes that divide at least one term of \(S\). We say that \(S\) is degenerate if the associated characteristic polynomial is either inseparable or has two different roots whose quotient is a root of unity. Pólya proved in 1921 that the set \(P_S\) is infinite for all non-degenerate linear recurrences \(S\) of order \(k\geq 2\). The author is interested in the question of whether \(P_S\) has a natural density (Pólya's method seems inadequate to determine this). In case \(k=2\) there are many results available, cf. \textit{C. Ballot} [Density of prime divisors of linear recurrences, Mem. Am. Math. Soc. 551 (1995; Zbl 0827.11006)]. However, essentially nothing is known for sequences of order larger than two (the main focus of this paper). The author associates a free \(R\)-algebra \(O\) to \(f\) and shows that \(p\in P_S\) iff the intersection of a certain coset \(C\) of the multiplicative subgroup \((O/pO)^*\) and \(H\) is non-empty, where \(H\) denotes the kernel of the trace map from \(O/pO\) to the finite field with \(p\) elements, and is an additive subgroup of the ring \(O/pO\). The mixture of an additive and multiplicative structure is notoriously difficult to study and this forces the author into a more conjectural approach. He uses his criterion for \(p\) to be in \(P_S\) to heuristically study the divisibility of \(S\) (in case the associated characteristic polynomial is separable). Two cases are distinguished; either \(f\) splits completely into distinct linear factors modulo \(p\), or not. Denote these sets of primes by \(T_1\) and \(T_2\) respectively. The author argues that the set of primes in \(T_1\) that divide \(S\) has a positive lower density, strictly less than \(1\) and that the primes in \(T_2\) that do not divide \(S\) have zero density. Using some results from algebraic geometry the author proves that if \(S\) is a linear recurrent sequence of order \(k\geq 3\) such that its associated characteristic polynomial satisfies a certain generalization of Artin's primitive root conjecture, and contains a \(k\)-cycle in its Galois group, then \(P_S\) contains a set of positive density. This well-written paper ends with a discussion of some numerical examples.
scientific article; zbMATH DE number 2081367
scientific article