Problem 2. Determine the natural prime numbers $p$ and $q$ knowing that there exist $x, y \in \mathbb{N}^{*}$ such that $p=x^{2}+y^{2}$ and $q=x+y+1$.
Solution. From $x, y \in \mathbb{N}^{*}$ it follows that $q \geq 3$, so $q$ is an odd prime number. Thus, $x, y$ are simultaneously even or odd, which makes $p$ an even number, i.e., $p=2$. The equation $2=x^{2}+y^{2}$ has only the solution $x=y=1$, so $q=3$.