On the number of residues of linear recurrences

For every nonconstant monic polynomial g∈ Z[X] , let M(g) be the set of positive integers m for which there exist an integer linear recurrence (sn)n≥0 having characteristic polynomial g and a positive integer M such that (sn)n≥0 has exactly m distinct residues modulo M. Dubickas and Novikas proved that M(X2- X- 1) = N. We study M(g) in the case in which g is divisible by a monic quadratic polynomial f∈ Z[X] with roots α, β such that αβ= ± 1 and α/ β is not a root of unity. We show that this problem is related to the existence of special primitive divisors of certain Lehmer sequences, and we deduce some consequences on M(g). In particular, for αβ= - 1 , we prove that m∈ M(g) for every integer m≥ 7 with m≠ 10 and 4 ∤ m.