Recurrence with prescribed number of residues

Dubickas, Artūras; Novikas, Aivaras

doi:10.1016/j.jnt.2020.01.004

Cited by 3 publications

(3 citation statements)

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“…Now suppose that (ii) holds. Then Lemma 3.2(p6) and (7) give that p | Res(f, Φ m ). Consequently, by Lemma 3.1, we get that (i) holds.…”

Section: Proof Of Theorem 12mentioning

confidence: 95%

“…, s r−1 ∈ Z and a positive integer M for which the linear recurrence s has exactly m distinct residues modulo M . Dubickas and Novikas [7] proved that M(X 2 − X − 1) = N and stated that the problem of determining M(g) "may be very difficult in general". The first step of their proof is a lemma regarding roots of X 2 − X − 1 modulo p that have a prescribed multiplicative order [7, Lemma 3].…”

Section: Introductionmentioning

confidence: 99%

“…As consequences of Theorem 1.2, Lemma 1.1, and results on the existence of primitive divisors of terms of Lehmer sequences (Lemma 3.4 and 3.7 below), we obtain the following results on M(g). Given two specific polynomials f, g ∈ Z[X] satisfying the hypothesis of Theorem 1.3 and Theorem 1.4, one can try to determine M(g) by using the aforementioned theorems and by employing [7,Lemma 6]. However, this requires a meticulous inspection of the numerical values of certain linear recurrences of characteristic polynomial g, and a detailed case-by-case analysis, as the one done by Dubickas and Novikas for M(X 2 − X − 1) [7,.…”

Section: Introductionmentioning

confidence: 99%

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On the number of residues of linear recurrences

Sanna¹

2021

Preprint

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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(X 2 − 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.

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“…Now suppose that (ii) holds. Then Lemma 3.2(p6) and (7) give that p | Res(f, Φ m ). Consequently, by Lemma 3.1, we get that (i) holds.…”

Section: Proof Of Theorem 12mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%