Squares with three nonzero digits

TL;DR

This paper classifies integers whose squares have at most three digits in various bases and explores related equations, revealing solutions mostly from known families or with small exponents, using Padé approximants in a novel way.

Contribution

It provides a complete classification of squares with limited base-q digits and analyzes related exponential equations using p-adic Padé approximants.

Findings

All such squares are characterized for specified bases.

Solutions to the equations are either from known polynomial families or have small exponents.

The method introduces a novel p-adic approach using Padé approximants.

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.