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On numbers n dividing the nth term of a linear recurrence

Part of: Multiplicative number theory Sequences and sets Elementary number theory

Published online by Cambridge University Press:  23 February 2012

Juan José Alba González ,
Florian Luca ,
Carl Pomerance  and
Igor E. Shparlinski
Juan José Alba González
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autonoma de México, CP 58089, Morelia, Michoacán, México (jjalba@gmail.com; fluca@matmor.unam.mx)
Florian Luca
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autonoma de México, CP 58089, Morelia, Michoacán, México (jjalba@gmail.com; fluca@matmor.unam.mx)
Carl Pomerance
Affiliation:
Mathematics Department, Dartmouth College, Hanover, NH 03755, USA (carl.pomerance@dartmouth.edu)
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (igor.shparlinski@mq.edu.au)
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Abstract

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We give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

Keywords

linear recurrenceLucas sequencedivisibility

MSC classification

Secondary: 11B37: Recurrences 11A07: Congruences; primitive roots; residue systems 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 271 - 289
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Bilu, Yu., Hanrot, G., Voutier, P. M., Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75–122.Google Scholar
2.Canfield, E. R., Erdös, P. and Pomerance, C., On a problem of Oppenheim concerning ‘Factorisatio Numerorum’, J. Number Theory 17 (1983), 1–28.CrossRefGoogle Scholar
3.Dartyge, C., Martin, G. and Tenenbaum, G., Polynomial values free of large prime factors, Period. Math. Hungar. 43 (2001), 111–119.CrossRefGoogle Scholar
4.Erdös, P., On the normal number of prime factors of p – 1 and some other related problems concerning Euler's φ function, Q. J. Math. 6 (1935), 205–213.CrossRefGoogle Scholar
5.Erdös, P., Luca, F., and Pomerance, C., On the proportion of numbers coprime to a given integer, in Anatomy of Integers (ed. De Koninck, J.-M.), CRM Proceedings and Lecture Notes, Volume 46, pp. 47–64 (2008).CrossRefGoogle Scholar
6.Everest, G., van der Poorten, A., Shparlinski, I. E. and Ward, T., Recurrence sequences, Mathematical Surveys and Monographs, Volume 104 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
7.Gordon, D. and Pomerance, C., The distribution of Lucas and elliptic pseudoprimes, Math. Comp. 57 (1991), 825–838.CrossRefGoogle Scholar
8.Györy, K. and Smyth, C., The divisibility of an – bn by powers of n, Integers 10 (2010), 319–334.CrossRefGoogle Scholar
9.Halberstam, H. and Richert, H.-E., Sieve Methods (Academic Press, London, 1974).Google Scholar
10.Li, S., On the distribution of even pseudoprimes, Unpublished manuscript (1996).Google Scholar
11.Luca, F., On positive integers n for which Ω(n) divides Fn, Fibonacci Q. 41 (2003), 365–371.Google Scholar
12.Luca, F. and Shparlinski, I. E., Some divisibilities amongst the terms of linear recurrences, Abh. Math. Sem. Univ. Hamburg 46 (2006), 143–156.CrossRefGoogle Scholar
13.Lucas, E., Théorie des fonctions numériques simplement périodiques, Am. J. Math. 1 (1878), 184–240, 289–321.Google Scholar
14.Pomerance, C., On the distribution of pseudoprimes, Math. Comp. 37 (1981), 587–593.CrossRefGoogle Scholar
15.Schlickewei, H. P., Multiplicities of recurrence sequences, Acta Math. 176 (1996), 171–243.CrossRefGoogle Scholar
16.Schlickewei, H. P. and Schmidt, W. M., The number of solutions of polynomialexponential equations, Compositio Math. 120 (2000), 193–225.CrossRefGoogle Scholar
17.Smyth, C., The terms in Lucas sequences divisible by their indices, J. Integer Sequences 13 (2010), 10.2.4.Google Scholar
18.Somer, L., Divisibility of terms in Lucas sequences by their subscripts, in Applications of Fibonacci numbers, Volume 5, pp. 515–525 (Kluwer Academic, Dordrecht, 1993).Google Scholar