On repdigits as product of $k$-Fibonacci and $k$-Lucas numbers
Safia Seffah, Salah Eddine Rihane, Alain Togbé
Received March 25, 2024. Published online January 27, 2025.
Abstract: For an integer $k\geq2$, let $(F_n^{(k)})_{n\geq-(k-2)}$, $(L_n^{(k)})_{n \geq-(k-2)}$ be $k$-Fibonacci and $k$-Lucas sequences, respectively. 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. In this paper, we determine all possibilities such that $F_n^{(k)} L_m^{(k)}$ can represent a repdigit.
Keywords: $k$-Fibonacci numbers; $k$-Lucas numbers; repdigits; linear form in logarithms; reduction method
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Affiliations: Safia Seffah (corresponding author), University of Science and Technology Houari Boumediene, Faculty of Mathematics, Department of Algebra and Number Theory, Arithmetic, Coding, Combinatorics and Formal Calculus Laboratory, BP 32, El Alia, 16111 Bab Ezzouar, Algiers, Algeria, e-mail: safiaseffah58@gmail.com, safia.seffah@usthb.dz; Salah Eddine Rihane, National Higher School of Mathematics, P.O.Box 75, Mahelma 16093, Sidi Abdellah, Algiers, Algeria, e-mail: salahrihane@hotmail.fr, salaheddine.rihane@nhsm.edu.dz; Alain Togbé, Department of Mathematics and Statistics, Purdue University Northwest, 2200 169th Street Hammond, IN 46323, USA and Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany, e-mail: atogbe@pnw.edu
