On the Largest Prime factor of the k-generalized Lucas numbers

2023/11/21 by Batte, Herbert, Luca, Florian · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.13047

Abstract

Let (L_n^(k))_{n≥ 2-k} be the sequence of k--generalized Lucas numbers for some fixed integer k≥ 2 whose first k terms are 0,…,0,2,1 and each term afterwards is the sum of the preceding k terms. For an integer m, let P(m) denote the largest prime factor of m, with P(0)=P(± 1)=1. We show that if n ≥ k + 1, then P (L_n^(k) ) > (1/86) log log n. Furthermore, we determine all the k--generalized Lucas numbers L_n^(k) whose largest prime factor is at most 7.

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