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 \( (n_1, p_1, q_1) , (n_1, p_1, q_1) , \ldots , (n_m, p_m, q_m) \). Find \( \displaystyle \sum_{k=1}^m (n_k + p_k + q_k ) \).