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Multiplicative dependence between k-Fibonacci and k-Lucas numbers

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Abstract

A generalization of the well-known Fibonacci and Lucas sequences are the k-Fibonacci and k-Lucas sequences with some fixed integer \(k\ge 2\). For these sequences the first k terms are \(0,\ldots ,0,1\) and \(0,\ldots ,0,2,1\), respectively, and each term afterwards is the sum of the preceding k terms. Here we find all pairs of k-Fibonacci and k-Lucas numbers multiplicatively dependent.

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Acknowledgements

The first author was supported in part by Project 71228 (Universidad del Valle). The second author worked on this project during a visit to the FLAME in Morelia, Mexico, in August 2019. This author also thanks the Universidad del Valle for support during his Ph.D. studies. The third author was supported in part by Grant CPRR160325161141 from the NRF of South Africa and the Focus Area Number Theory Grant RTNUM19 from CoEMaSS Wits. Part of this work was done when the third author visited the Max Planck Institute for Mathematics in Bonn, Germany in Fall 2019. He thanks this institution for hospitality and support.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100-00, Cali, 25360, Colombia

    Carlos A. Gómez & Jhonny C. Gómez

  2. School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa

    Florian Luca

  3. Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia

    Florian Luca

  4. Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico

    Florian Luca

Authors
  1. Carlos A. Gómez
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  2. Jhonny C. Gómez
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  3. Florian Luca
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Correspondence to Carlos A. Gómez.

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Gómez, C.A., Gómez, J.C. & Luca, F. Multiplicative dependence between k-Fibonacci and k-Lucas numbers. Period Math Hung 81, 217–233 (2020). https://doi.org/10.1007/s10998-020-00336-z

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  • DOI: https://doi.org/10.1007/s10998-020-00336-z

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