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

TL;DR

This paper investigates the largest prime factors of k-generalized Lucas numbers, establishing a lower bound related to the logarithm of the logarithm of n, and classifies those with small prime factors.

Contribution

It provides a new lower bound for the largest prime factor of k-generalized Lucas numbers and completely characterizes those with prime factors at most 7.

Findings

For n ≥ k+1, P(L_n^{(k)}) > (1/86) log log n.

All k-generalized Lucas numbers with largest prime factor ≤ 7 are identified.

The results extend understanding of prime factors in linear recurrence sequences.

Abstract

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ 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 $P(0)=P(\pm 1)=1$. We show that if $n \ge k + 1$, 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$.