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

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.

Jhon J J Bravo¹ Florian Luca^{2 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

Published

May 2014

Manuscript received

29 January 2013

Manuscript revised

6 May 2013