k- Fibonacci powers as sums of powers of some fixed primes

Let S= { p1, … , pt} be a fixed finite set of prime numbers listed in increasing order. In this paper, we prove that the Diophantine equation (Fn(k))s=p1a1+⋯+ptat, in integer unknowns n≥ 1 , s≥1,k≥2 and ai≥ 0 for i= 1 , … , t such that max { ai: 1 ≤ i≤ t} = at has only finitely many effectively computable solutions. Here, Fn(k) is the nth k–generalized Fibonacci number. We compute all these solutions when S= { 2 , 3 , 5 }. This paper extends the main results of [15] where the particular case k= 2 was treated.