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Repdigits as Product of Terms of k -Bonacci Sequences

Author

Listed:
  • Petr Coufal

    (Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic)

  • Pavel Trojovský

    (Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic)

Abstract

For any integer k ≥ 2 , the sequence of the k -generalized Fibonacci numbers (or k -bonacci numbers) is defined by the k initial values F − ( k − 2 ) ( k ) = ⋯ = F 0 ( k ) = 0 and F 1 ( k ) = 1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form a a … a , with a ∈ [ 1 , 9 ] ) in the sequence ( F n ( k ) F n ( k + m ) ) n , for m ∈ [ 1 , 9 ] . This result generalizes a recent work of Bednařík and Trojovská (the case in which ( k , m ) = ( 2 , 1 ) ). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method).

Suggested Citation

  • Petr Coufal & Pavel Trojovský, 2021. "Repdigits as Product of Terms of k -Bonacci Sequences," Mathematics, MDPI, vol. 9(6), pages 1-10, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:682-:d:522040
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    References listed on IDEAS

    as
    1. Pavel Trojovský, 2019. "On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes," Mathematics, MDPI, vol. 7(8), pages 1-10, August.
    2. Pavel Trojovský, 2020. "Fibonacci Numbers with a Prescribed Block of Digits," Mathematics, MDPI, vol. 8(4), pages 1-7, April.
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