Abstract
A generalization of the Fibonacci sequence is the k-generalized Fibonacci sequence (F (k) n )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.
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Ruiz, C.A.G., Luca, F. Multiplicative Independence in k-Generalized Fibonacci Sequences. Lith Math J 56, 503–517 (2016). https://doi.org/10.1007/s10986-016-9332-1
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DOI: https://doi.org/10.1007/s10986-016-9332-1