On the largest prime factor of the k–generalized Lucas numbers

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