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Repdigits as sums of two \(k\)-Fibonacci numbers

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Abstract

For an integer \(k\ge 2\), let \((F_{n}^{(k)})_{n}\) be the \(k\)-Fibonacci sequence which starts with \(0,\ldots ,0,1\) (\(k\) terms) and each term afterwards is the sum of the \(k\) preceding terms. In this paper, we find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion) which are sums of two \(k\)-Fibonacci numbers. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker–Davenport reduction method. This paper is an extended work related to our previous work (Bravo and Luca Publ Math Debr 82:623–639, 2013).

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Acknowledgments

Part of this work was done when the first author was visiting Santiago (Chile) in July–October of 2013; he thanks the Center for Mathematical Modeling (CMM) of the University of Chile for its hospitality during his visit. J. J. B. was partially supported by CONACyT from Mexico and University of Cauca, Colciencias from Colombia. F. L. was supported in part by Projects PAPIIT IN104512, CONACyT 163787, CONACyT 193539 and a Marcos Moshinsky Fellowship.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del Cauca, Calle 5 No 4-70, Popayán, Cauca, Colombia

    Jhon J. Bravo

  2. Mathematical Institute, UNAM Juriquilla, 76230 , Santiago de Querétaro, Querétaro de Arteaga, Mexico

    Florian Luca

  3. School of Mathematics, University of the Witwatersrand, P. O. Box Wits 2050, Johannesburg, South Africa

    Florian Luca

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  1. Jhon J. Bravo
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  2. Florian Luca
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Correspondence to Florian Luca.

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Communicated by J. Schoißengeier.

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Bravo, J.J., Luca, F. Repdigits as sums of two \(k\)-Fibonacci numbers. Monatsh Math 176, 31–51 (2015). https://doi.org/10.1007/s00605-014-0622-6

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  • DOI: https://doi.org/10.1007/s00605-014-0622-6

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