On the largest prime factor of the k-Fibonacci numbers

الأصل · EN

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 (Fₙ⁽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(Fₙ⁽ᵏ⁾)>c n, where c>0 is an effectively computable constant. Furthermore, we determine all the k-Fibonacci numbers Fₙ⁽ᵏ⁾ whose largest prime factor is less than or equal to 7.