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

Let $$S=\{p_{1},\ldots ,p_{t}\}$$ be a fixed finite set of prime numbers listed in increasing order. In this paper, we prove that the Diophantine equation $$(F_n^{(k)})^s=p_{1}^{a_{1}}+\cdots +p_{t}^{a_{t}}$$ , in integer unknowns $$n\ge 1$$ , $$s\ge 1,~k\ge 2$$ and $$a_i\ge 0$$ for $$i=1,\ldots ,t$$ such that $$\max \left\{ a_{i}: 1\le i\le t\right\} =a_t$$ has only finitely many effectively computable solutions. Here, $$F_n^{(k)}$$ 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.