A number theory problem by Nick Okita

pn+144=q2\large p^n + 144 = q^2

Let p,qp,q and nn be positive integers satisfying the equation above such that pp is a prime number. Let all the solutions of (n,p,q)(n,p,q) be denotes as (n1,p1,q1),(n2,p2,q2),,(nm,pm,qm) (n_1, p_1, q_1) , (n_2, p_2, q_2) , \ldots , (n_m, p_m, q_m) . Find k=1m(nk+pk+qk) \displaystyle \sum_{k=1}^m (n_k + p_k + q_k ) .


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