31. Find all prime numbers $p$ such that $p^{2}+14$ is also a prime number.
31. It is easy to notice that the prime number $p=3$ satisfies the condition of the problem. If $p \neq 3$, then $p^{2}$ has the form $3 \mathrm{k}+1$ (see problem 5 a). Then $p^{2}+14=3 k+15=3(k+5)$. The obtained number is composite. Therefore, the problem has a unique solution $p=3$.