3. Find all integers $n>2$ such that the number $n^{n-2}$ is the $n$-th power of an integer.

(Patrik Bak)
SOLUTION. We will show that only $n=4$ satisfies the condition. For brevity, we will use "n-th power" instead of "n-th power of an integer" throughout the solution.

A positive integer is an $n$-th power if and only if each prime number in its prime factorization appears with an exponent that is a multiple of $n$. Therefore, if two integers are both $n$-th powers and their quotient is an integer, then this quotient must also be an $n$-th power. Applying this statement to the $n$-th powers $n^{n}$ and $n^{n-2}$ (as given in the problem), we get that the number $n^{n} / n^{n-2}=n^{2}$ is also an $n$-th power.

Below, we will prove that for every integer $n \geqq 5$, the inequalities $1^{n} < 2^{n-1}$ and $2^{n-1} < 3^{n-2}$ hold (the first inequality is obvious for $n \geqq 2$). We will prove the second inequality by mathematical induction.