5. Find all triples $(m, n, p)$ such that $p^{n}+144=m^{2}$ (where $m, n \in \mathbf{Z}_{+}$, and $p$ is a prime).
[4]
(22nd Italian Mathematical Olympiad)
Hint: Convert the original equation to
$$
p^{n}=(m+12)(m-12) \text {. }
$$

The solutions are
$$
(m, n, p)=(13,2,5),(20,8,2),(15,4,3) .
$$