## Task A-4.1.

Determine all prime numbers $p$ and $q$ such that $p^{q}+1$ is also prime.
## Solution.

Let $p$ and $q$ be prime numbers and suppose $p^{q}+1$ is prime. Since $p, q \geqslant 2$, it follows that $p^{q}+1 \geqslant p^{2}+1 > p$. Therefore, $p^{q}+1$ is an odd prime number, which is only possible if $p=2$.

If $q=2$, then $p^{q}+1=2^{2}+1=5$ is a prime number, so this is one solution.

Otherwise, $q$ is an odd number, and then

$$
p^{q}+1=(p+1)\left(p^{q-1}-p^{q-2}+\ldots-p+1\right)
$$

is a composite number, which contradicts the assumption.