Squares with three nonzero digits

We determine all integers $n$ such that $n^2$ has at most three base-$q$ digits for $q\in \{2,3,4,5,8,16\}$. More generally, we show that all solutions to equations of the shape $$Y^2=t^2+M\cdot q^m+N\cdot q^n,$$ where $q$ is an odd prime, $n>m>0$ and $t^2,\mathrm{\mid }M\mathrm{\mid },N<q$, either arise from “obvious” polynomial families or satisfy $m\le 3$. Our arguments rely upon Pad'e approximants to the binomial function, considered $q$-adically.