jmc

First notice that $5^n\equiv 2^n(\mathrm{mod}3)$, which should make our computations easier. For $n=1,2,3,4$, we get $2^n\equiv 2,1,2,1(\mathrm{mod}3)$ respectively and $n^5\equiv 1,2,0,1(\mathrm{mod}3)$ respectively. Since we have a congruent pair at $n=\boxed{4}$, we don't need to look any further.