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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/124/02/0141-0154

    • Keywords

       

      Generalized Fibonacci and Lucas numbers; lower bounds for nonzero linear forms in logarithms of algebraic numbers; repdigits.

    • Abstract

       

      For an integer $k\geq 2$, let $(L_n^{(k)})_n$ be the 𝑘-Lucas sequence which starts with $0,\ldots,0,2,1$ (𝑘 terms) and each term afterwards is the sum of the 𝑘 preceding terms. In 2000, Luca (Port. Math. 57(2) 2000 243-254) proved that 11 is the largest number with only one distinct digit (the so-called repdigit) in the sequence $(L_n^{(2)})_n$. In this paper, we address a similar problem in the family of 𝑘-Lucas sequences. We also show that the 𝑘-Lucas sequences have similar properties to those of 𝑘-Fibonacci sequences and occur in formulae simultaneously with the latter.

    • Author Affiliations

       

      Jhon J J Bravo1 Florian Luca2 3

      1. Departamento de Matemáticas Universidad del Cauca, Calle 5 No 4–70, Popayán, Colombia
      2. Mathematical Institute, UNAM Juriquilla Juriquilla, 76230, Santiago de Querétaro, Querétaro de Arteaga, México
      3. School of Mathematics, University of the Witwatersrand, P. O. Box Wits 2050, Johannesburg, South Africa

    • Dates

       
      Published
      May 2014
      Manuscript received
      29 January 2013
      Manuscript revised
      6 May 2013

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