7. Let $p$ be a prime number, and $q=4^{p}+p^{4}+4$ is also a prime number. Find the value of $p+q$.

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(Tip: Mod 3 we get $q=4^{p}+p^{4}+4 \equiv 1^{p}+p^{4}+$ $1 \equiv p^{4}+2(\bmod 3)$. Also, $p^{2} \equiv 0,1(\bmod 3)$, so $p^{4}$ $\equiv 0,1(\bmod 3)$.

If $p^{4} \equiv 1(\bmod 3)$, then $q \equiv p^{4}+2 \equiv 1+2 \equiv$ $0(\bmod 3)$. But $q=4^{p}+p^{4}+4>3$, which contradicts the fact that $q$ is a prime number.

If $p^{4} \equiv 0(\bmod 3)$, then $p=3$. In this case, $q=4^{p}+$ $p^{4}+4=149$ is a prime number, which meets the requirement. Therefore, $p+q=$ 152.)