IT-CESENATICO 2001 Problem 3

Consider the equation $$x^{2001}=y^x .$$

• Find all pairs $(x,y)$ of solutions where $x$ is a prime number and $y$ is a positive integer.
• Find all pairs $(x,y)$ of solutions where $x$ and $y$ are positive integers.

(Remember that $2001=3 \cdot 23 \cdot 29$ .)