On the largest prime factor of the k-Fibonacci numbers

2012/10/15 by Bravo, Jhon J., Luca, Florian
#11B39 #11J86 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1210.4101

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))>cloglog 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.

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