Multiplicative Independence in k-Generalized Fibonacci Sequences

Abstract

A generalization of the Fibonacci sequence is the k-generalized Fibonacci sequence (F () )n ≥ 2 − k with some fixed integer k ≥ 2 whose first k terms are 0,…, 0, 1 and each term afterward is the sum of the preceding k terms. Carmichael’s primitive divisor theorem ensures that all members after the twelfth of the Fibonacci sequence are multiplicatively independent. Although there is no version of this theorem for k-generalized Fibonacci sequences with k > 2, here we find all the pairs of k-Fibonacci numbers that are multiplicatively dependent.