Squares with three nonzero digits

Abstract

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 $t^2, |M|, N < q$, either arise from "obvious" polynomial families or satisfy $m \leq 3$. Our arguments rely upon Pad\'e approximants to the binomial function, considered $q$-adically.