Multiplicative dependence between k-Fibonacci and k-Lucas numbers

Abstract

A generalization of the well-known Fibonacci and Lucas sequences are the k-Fibonacci and k-Lucas sequences with some fixed integer $$k\ge 2$$ . 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. Here we find all pairs of k-Fibonacci and k-Lucas numbers multiplicatively dependent.