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Some divisibilities amongst the terms of linear recurrences

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Abstract

Let (un)n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integersn such that either ω)(n) orΩ(n) dividesu n is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors ofn, respectively. The same also holds for the set of positive integersn such that τ(n)u n , where τ(n) is the number of the positive integer divisors of n, provided thatu n satisfies some mild technical conditions.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, México

    F. Luca

  2. Department of Computing, Macquarie University, 2109, Sydney, NSW, Australia

    I. E. Shparlinski

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  1. F. Luca
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  2. I. E. Shparlinski
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Correspondence to F. Luca or I. E. Shparlinski.

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Communicated by: C. Schweigen

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Luca, F., Shparlinski, I.E. Some divisibilities amongst the terms of linear recurrences. Abh.Math.Semin.Univ.Hambg. 76, 143–156 (2006). https://doi.org/10.1007/BF02960862

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