1. If two prime numbers $p, q$ satisfy $3 p^{2}+5 q=517$, then $p+q=$ $\qquad$
II. 1.15 or 103.
If $p$ and $q$ are both odd, then $3p^2 + 5q$ is even, which contradicts the given condition. Therefore, at least one of $p$ or $q$ must be even.
Since $p$ and $q$ are both prime numbers, $p=2$ or $q=2$.
When $p=2$,
$3 \times 2^2 + 5q = 517 \Rightarrow q = 101$ (prime).
In this case, $p + q = 2 + 101 = 103$.
When $q=2$,
$3p^2 + 5 \times 2 = 517 \Rightarrow p = 13$ (prime).
In this case, $p + q = 13 + 2 = 15$.