On the discriminator of Lucas sequences

Annales mathématiques du Québec, Feb 2018

We consider the family of Lucas sequences uniquely determined by \(U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),\) with initial values \(U_0(k)=0\) and \(U_1(k)=1\) and \(k\ge 1\) an arbitrary integer. For any integer \(n\ge 1\) the discriminator function \(\mathcal {D}_k(n)\) of \(U_n(k)\) is defined as the smallest integer m such that \(U_0(k),U_1(k),\ldots ,U_{n-1}(k)\) are pairwise incongruent modulo m. Numerical work of Shallit on \(\mathcal {D}_k(n)\) suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every \(k\ge 1\) there is a constant \(n_k\) such that \({\mathcal D}_{k}(n)\) has a simple characterization for every \(n\ge n_k\). The case \(k=1\) turns out to be fundamentally different from the case \(k>1\).

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On the discriminator of Lucas sequences

On the discriminator of Lucas sequences Bernadette Faye 0 1 2 3 4 5 Florian Luca 0 1 3 4 5 Pieter Moree 0 1 3 4 5 B Pieter Moree 0 1 3 4 5 Florian Luca 0 1 3 4 5 0 School of Mathematics, University of the Witwatersrand , P. O. Box Wits, Wits 2050 , South Africa 1 AIMS-Senegal , km 2 Route de Joal, BP 1418 Mbour , Senegal 2 , Czech Republic 3 École Doctorale de Mathématiques et d'Informatique, Université Cheikh Anta Diop de Dakar , BP 5005, Dakar Fann , Senegal 4 Department of Mathematics, Faculty of Sciences, University of Ostrava , 30 Dubna 22, 701 03 Ostrava 5 Max Planck Institute for Mathematics , Vivatsgasse 7, 53111 Bonn , Germany We consider the family of Lucas sequences uniquely determined by Un+2(k) = (4k +2)Un+1(k)−Un (k), with initial values U0(k) = 0 and U1(k) = 1 and k ≥ 1 an arbitrary integer. For any integer n ≥ 1 the discriminator function Dk (n) of Un (k) is defined as the smallest integer m such that U0(k), U1(k), . . . , Un−1(k) are pairwise incongruent modulo m. Numerical work of Shallit on Dk (n) suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every k ≥ 1 there is a constant nk such that Dk (n) has a simple characterization for every n ≥ nk . The case k = 1 turns out to be fundamentally different from the case k > 1. FRENCH ABSTRACT Pour un entier arbitraire k ≥ 1, on considère la famille de suites de Lucas déterminée de manière unique par la relation de récurrence Un+2(k) = (4k + 2)Un+1(k) − Un (k), et les valeurs initiales U0(k) = 0 et U1(k) = 1. Pour tout entier n ≥ 1, la fonction discriminante Dk (n) de Un (k) est définie comme le plus petit m tel que U0(k), U1(k), . . . , Un−1(k) soient deux à deux non congruents modulo m. Des travaux numériques de Shallit sur Dk (n) suggère qu'il en existe une caractérisation relativement simple. Dans cet article, on démontre que c'est en effet le cas en établissant que pour tout - k ≥ 1, il existe une constante nk telle que Dk (n) possède une caractérisation simple pour tout n ≥ nk . Le cas k = 1 se révèle être de nature totalement différente du cas k > 1. Mathematics Subject Classification 11B39 · 11B50 1 Introduction The discriminator of a sequence a = {an }n≥1 of distinct integers is the sequence given by Da(n) = min{m : a0, . . . , an−1 are pairwise distinct modulo m}. In other words, Da(n) is the smallest integer m that allows one to discriminate (tell apart) the integers a0, . . . , an−1 on reducing modulo m. Note that since a0, . . . , an−1 are n distinct residue classes modulo Da(n) it follows that Da(n) ≥ n. On the other hand obviously Da(n) ≤ max{a0, . . . , an−1} − min{a0, . . . , an−1}. Put Da = {Da(n) : n ≥ 1}. The main problem is to give an easy description or characterization of the discriminator (in many cases such a characterization does not seem to exist). The discriminator was named and introduced by Arnold, Benkoski and McCabe in [1]. They considered the sequence u with terms u j = j 2. Meanwhile the case where u j = f ( j ) with f a polynomial has been well-studied, see, for example, [3,9,10,16]. The most general result in this direction is due to Zieve [16], who improved on an earlier result by Moree [9]. In this paper we study the discriminator problem for Lucas sequences (for a basic account of Lucas sequences see, for example, Ribenboim [13, 2.IV]). Our main results are Theorem 1 (k = 1) and Theorem 3 (k > 2). Taken together with Theorem 2 (k = 2) they evaluate the discriminator Dk (n) for the infinite family of second-order recurrences ( 1 ), with for each k at most finitely many n that are not covered. All members in the family ( 1 ) have a characteristic equation that is irreducible over the rationals. Very recently, Ciolan and Moree [6] determined the discriminator for another infinite family, this time with all members having a reducible characteristic equation. For every prime q ≥ 7 they computed the discriminator of the sequence uq ( j ) = 3 j − q(−1) j+(q−1)/2 4 , j = 1, 2, 3, . . . that was first considered in this context by Jerzy Browkin. The case q = 5 was earlier dealt with by Moree and Zumalacárregui [11], who showed that, for this value of q, the smallest positive integer m discriminating uq ( 1 ), . . . , uq (n) modulo m equals min{2e, 5 f }, where e is the smallest integer such that 2e ≥ n and f is the smallest integer such that 5 f ≥ 5n/4. Despite structural similarities between the present paper and [6] (for example the index of appearance z in the present paper plays the same role as the period ρ in [6]), there are also many differences. For example, Ciolan and Moree have to work much harder to exclude small prime numbers as discriminator values. This is related to the sequence of good discriminator candidate values in that case being much sparser, namely being O(log x ) for the values ≤ x , versus log2 x . In our case one has to work with elements and ideals in quadratic number fields, whe (...truncated)


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Bernadette Faye, Florian Luca, Pieter Moree. On the discriminator of Lucas sequences, Annales mathématiques du Québec, 2018, pp. 1-21, DOI: 10.1007/s40316-017-0097-7