A number theory problem by Nick Okita

Number Theory Level 5

$\large p^n + 144 = q^2$

Let $p,q$ and $n$ be positive integers satisfying the equation above such that $p$ is a prime number. Let all the solutions of $(n,p,q)$ be denotes as $(n_1, p_1, q_1) , (n_2, p_2, q_2) , \ldots , (n_m, p_m, q_m)$ . Find $\displaystyle \sum_{k=1}^m (n_k + p_k + q_k )$ .

A number theory problem by Nick Okita

This question was taken from IME (Instituto Militar de Engenharia) 2009-2010 entrance exam.