1. Find all primes that can be written both as a sum of two primes and as a difference of two primes.
1. Let $p$ be a prime that can be written as a sum of two primes and as a difference of two primes. Clearly, we have $p>2$. Then $p$ must be odd, so that $p=q+2=r-2$ for some odd primes $q$ and $r$.
We consider three cases.
CaSE 1. Suppose that $q \equiv 1(\bmod 3)$. Then $p$ is a multiple of 3 , implying that $p=3$. It follows that $p=3$, which means that $q=1$, a contradiction.

CASE 2. Suppose that $q \equiv 2(\bmod 3)$. Then $r \equiv 0(\bmod 3)$, which implies that $r=3$. This leads to $p=1$, which is again a contradiction.

CaSE 3. Suppose that $q \equiv 0(\bmod 3)$. Then $q=3$, and it follows that $p=5$ and $r=7$.

From the above three cases, $p=5$ is the only prime that is a sum of two primes and a difference of two primes.