1. If prime numbers $p$ and $q$ satisfy the equation $3 p+5 q=31$, then the value of $\log _{2} \frac{p}{3 q+1}$ is $\qquad$.
1. -3 or 0

Solution: From the condition, we know that one of $3 p$ and $5 q$ must be even, and the other must be odd.
If $3 p$ is even, then $p=2$, and thus $q=5$. In this case,
$$
\log _{2} \frac{p}{3 q+1}=\log _{2} \frac{2}{3 \times 5+1}=\log _{2} \frac{1}{8}=-3,
$$

If $5 q$ is even, then $q=2$, and thus $p=7$. In this case,
$$
\log _{2} \frac{p}{3 q+1}=\log _{2} \frac{7}{3 \times 2+1}=\log _{2} 1=0 .
$$

Therefore, the value of $\log _{2} \frac{p}{3 q+1}$ is -3 or 0.