$2^4=16=4^2$. In fact, $\{2,4\}$ is the only pair of natural numbers with that property, i.e. if $m
This is easily seen with some analysis: For $m,n\in\mathbf{N}\backslash\{0\}$, the equation $m^n=n^m$ is equivalent to $\sqrt[m]{m}=\sqrt[n]{n}$. By calculus, we can show that the real function $t\mapsto \sqrt[t]{t}$ is strictly increasing for $t
My question: Is there an elementary proof? By elementary I mean most of all no irrational numbers, no calculus.