Linear recurrence sequences without zeros

On the number of residues of linear recurrences

“…By Lemma 3.1, we have that p | R + m and p ≡ 1 (mod m). Hence, from (6) and the fact that n | m, we get that p | Φ n (γ, δ) and p ≡ 1 (mod n). Therefore, Lemma 3.2(p6) implies that p is a primitive divisor of u n (γ, δ), and (ii) follows.…”

“…Thus, from Lemma 3.2(p6), it follows that p | Φ n (γ, δ). Consequently, by (6), we get that either p | R + m or p | R − m . In the first case, (i) follows immediately from Lemma 3.1.…”

“…It is easily seen that s is ultimately periodic modulo M , for every positive integer M , and purely periodic if (c r , M ) = 1. Indeed, properties of linear recurrences modulo M have been studied intensively, including: which residues modulo M appear in the s and how frequently [4,6,9,12,15,17], and for which positive integers M the linear recurrence s contains a complete system of residues modulo M [2,5,16,18]. Let M(g) denote the set of positive integers m such that there exist initial conditions s 0 , .…”

“…It is clear that 1 ∈ M(g). By Zsigmondy's theorem [14, p. 1], for every integer a / ∈ {−1, 0, +1} and for every positive integer m with (a,m) / ∈ {(±2 v − 1, 2) : v ≥ 1} ∪ {(−2, 3),(2,6)}, there exists a prime number p such that ord p (a) = m. Hence, by Lemma 1.1 with f (X) = X − a, we get that m ∈ M(g).…”