Multiplicative dependence of k -Fibonacci numbers with the Fibonacci, Lucas, and Pell sequences

Journal article

The k–generalized Fibonacci sequence (Fm(k))m≥2-k is the linear recurrent sequence of order k whose first k terms are 0, …, 0, 1 and each term afterwards is the sum of the preceding k terms. The case k=2 corresponds to the well known Fibonacci sequence. In Gómez and Luca (Lith. Math. J. 56(4):503–517, 2016), the multiplicative independence between terms of the same k-generalized Fibonacci sequence was studied. In this paper, we find all the multiplicative dependent pairs (Fm(k), un) where un is a Fibonacci, a Lucas or a Pell number.

Authors: Gómez, C; C. Gómez, J; Luca, F

• Alternative title: Multiplicative dependence of k -Fibonacci numbers..
• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Ramanujan Journal
• Volume: 69
• Issue: 2
• Article number: 48
• Publication date: 2026-01-31
• Acceptance date: 2025-12-18
• DOI: 10.1007/s11139-025-01306-0
• EISSN: 1572-9303
• ISSN: 1382-4090
• Language: English
• Keywords: 11D61 11J86; Lower bounds for nonzero linear forms in logarithms of algebraic numbers; Primary 11B39; k-generalized Fibonacci numbers; Diophantine approximation; Effective solution for exponential Diophantine equation