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

Abstract

Let $P(m)$ denote the largest prime factor of an integer $m\ge 2$, and put $P(0)=P(1)=1$. For an integer $k\ge 2$, let $(F_n^{(k)}{)}_{n\ge 2-k}$ 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 $n\ge k+2$, 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.

15 pages