How many twin prime pairs are there whose sum is a prime power? (We call two prime numbers twin primes if their difference is 2.)
Let $p$ and $p+2$ be twin primes such that their sum is a power of some prime $q$:

$$
p+(p+2)=q^{k} \quad(k \geq 1 \text { integer). Then }
$$

$2(p+1)=q^{k}$, so $q$ must be divisible by 2, hence $q=2$: $2(p+1)=2^{k}$, from which $p=2^{k-1}-1$ and $p+2=2^{k-1}+1$ are twin primes. Since $2^{k-1}$ is not divisible by 3, but among three consecutive natural numbers, one is certainly divisible by 3, either $3 \mid p$ or $3 \mid p+2$, and since both are prime, only $p=3$ is possible. In this case, $p+2=5$ is also prime, and their sum is $8=2^{3}$.

Therefore, there is only one solution, the pair $(3,5)$.