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.

令\（S = \ {p_ {1}，\ ldots，p_ {t} \} \）为固定的有限质数集，以递增顺序列出。在本文中，我们证明了Diophantine方程\（（F_n ^ {（k）}）^ s = p_ {1} ^ {a_ {1}} + \ cdots + p_ {t} ^ {a_ {t}} \），以整数未知数\（n \ ge 1 \），\（s \ ge 1，〜k \ ge 2 \）和\（a_i \ ge 0 \）表示\（i = 1，\ ldots，t \ ），使得\（\ max \ left \ {a_ {i}：1 \ le i \ le t \ right \} = a_t \）仅具有有限的许多有效可计算解。这里，\（F_N ^ {（K）} \）是Ñ个ķ -generalized斐波那契数。当\（S = \ {2,3,5 \} \）时，我们计算所有这些解决方案。本文扩展了[15]的主要结果，其中处理了特定情况\（k = 2 \）。