3. Find all primes that can be written both as a sum of two primes and as a difference of two primes.
3. Let $p$ be such a prime, then $p>2$ and is therefore odd. Thus $p=q-2=r+2$ where $q, r$ are primes. If $r \equiv 1(\bmod 3)$, then $p \equiv 0(\bmod 3)$ and therefore $p=3$ and $r=1$ which is impossible. If $r \equiv 2(\bmod 3)$, then $q \equiv 0(\bmod 3)$ and thus $q=3$ and so $p=1$, again impossible. Thus $r \equiv 0$ (mod 3), which means $r=3$ and hence $p=5$ and $q=7$. Thus $p=5$ is the only such prime.