7. If $p$ and $q$ are both prime numbers, and $7p + q$ and $pq + 11$ are also prime numbers, find the value of $p^q + q^p$.
(1997, Jingzhou City, Hubei Province, Junior High School Mathematics Competition)
(Tip: Since $7 p+q$ is a prime number, and $7 p+q>7$, therefore, $7 p+q$ is odd. Thus, $p, q$ do not share the same parity.

If $p$ is even, then $p=2$. According to the problem, $14+$ $q, 2 q+11$ are both prime numbers. Therefore,
$$
14+q \neq \equiv 0(\bmod 3), 2 q+11 \not \equiv 0(\bmod 3),
$$

which means $q \neq 1(\bmod 3), q \neq 2(\bmod 3)$.
Hence, $q \equiv 0(\bmod 3)$. Thus, $q=3$.
At this point, $p^{q}+q^{p}=2^{3}+3^{2}=17$.
If $q$ is even, then $q=2$. According to the problem, $7 p+$ $2,2 p+11$ are both prime numbers, so,
$$
7 p+2 \neq 0(\bmod 3), 2 p+11 \neq \equiv 0(\bmod 3) \text {, }
$$

which means $p \neq 1(\bmod 3), p \neq 2(\bmod 3)$.
Hence, $p \equiv 0(\bmod 3)$. Thus, $p=3$.
At this point, $p^{q}+q^{p}=2^{3}+3^{2}=17$.)