Multiplicative independence in the sequence of k-generalized Lucas numbers

Abstract

Let ${\left(L_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the sequence of $k$-generalized Lucas numbers for some fixed integer $k\ge 2$, whose first $k$ terms are $0,\dots ,0,2,1$ and each term afterward is the sum of the preceding $k$ terms. In this paper, we find all pairs of the $k$-generalized Lucas numbers that are multiplicatively dependent.