On the largest prime factor of the $k-$Fibonacci numbers

TL;DR

This paper proves that for large enough indices, the largest prime factor of k-Fibonacci numbers grows faster than a constant times log log n, and it classifies those with small prime factors.

Contribution

It establishes a lower bound on the largest prime factor of k-Fibonacci numbers and classifies all with prime factors at most 7.

Findings

For n ≥ k+2, P(F_n^{(k)}) > c log log n for some constant c.

All k-Fibonacci numbers with largest prime factor ≤ 7 are explicitly identified.

Abstract

Let $P(m)$ denote the largest prime factor of an integer $m\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. Here, we show that if $n\geq k+2$, then $P(F_n^{(k)})>c\log\log n$, where $c>0$ is an effectively computable constant. Furthermore, we determine all the $k-$Fibonacci numbers $F_n^{(k)}$ whose largest prime factor is less than or equal to 7.