Question 5
Let $k$ be a positive integer and let $n_1, n_2, \dots, n_k \gt 1$ be $k$ distinct integers such that
$$
n_1! + \dots + n_k! = a^{n_1}
$$
for some positive integer $a$. Prove that at least one of $n_1, n_2, \dots, n_k$ is prime.
$$
n_1! + \dots + n_k! = a^{n_1}
$$
for some positive integer $a$. Prove that at least one of $n_1, n_2, \dots, n_k$ is prime.