On the Monoid Generated by a Lucas Sequence



A Lucas sequence is a sequence of the general form \(v_{n} = (\phi ^{n} -\overline{\phi }^{n})/(\phi -\overline{\phi })\), where ϕ and \(\overline{\phi }\) are real algebraic integers such that \(\phi +\overline{\phi }\) and \(\phi \overline{\phi }\) are both rational. Famous examples include the Fibonacci numbers, the Pell numbers, and the Mersenne numbers. We study the monoid that is generated by such a sequence; as it turns out, it is almost freely generated. We provide an asymptotic formula for the number of positive integers ≤ x in this monoid, and also prove Erdős–Kac type theorems for the distribution of the number of factors, with and without multiplicity. While the limiting distribution is Gaussian if only distinct factors are counted, this is no longer the case when multiplicities are taken into account.

2010 Mathematics Subject Classification

11N37 11B39 



C. Heuberger is supported by the Austrian Science Fund (FWF): P 24644-N26. Parts of this paper have been written while he was a visitor at Stellenbosch University.

S. Wagner is supported by the National Research Foundation of South Africa, grant number 96236.


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Authors and Affiliations

  1. 1.Institut für Mathematik, Alpen-Adria-Universität KlagenfurtKlagenfurtAustria
  2. 2.Department of Mathematical SciencesStellenbosch UniversityStellenboschSouth Africa

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