On the largest prime factor of the ratio of two generalized Fibonacci numbers

A generalization of the well-known Fibonacci sequence is the k  -generalized Fibonacci sequence (Fn(k))n≥2−k for some integer k≥2k≥2, whose first k   terms are 0,…,0,10,…,0,1 and each term afterwards is the sum of the preceding k   terms. In this paper, we look at the prime factors of the reduced rational number Fn(k)/Fm(ℓ) as max⁡{m,n,k,ℓ}max⁡{m,n,k,ℓ} tends to infinity.