Type: Article

On the largest prime factor of the k-fibonacci numbers

Bravo, Jhon J.; Luca, Florian;

Abstract:

Let P(m) denote the largest prime factor of an integer m >= 2, and put P(0) = P(1) = 1. For an integer k >= 2, let (F-n((k)))n >=(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 >= k+2, then P(F-n((k))) > 0.01 root log n log log n. Furthermore, we determine all the k-Fibonacci numbers F-n((k)) whose largest prime factor is less than or equal to 7.