JBMO-SL 2001 Problem 2

Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$ , where $k,l,m$ are positive integers. Find $n$ such that:

i) In the set $P_n$ there are infinitely many squares.

ii) In the set $P_n$ there are no squares.