Squares with three digits

TL;DR

This paper surveys known results and provides new insights into integers whose squares have exactly three decimal digits, focusing on elementary proofs and extending some results to general bases.

Contribution

It consolidates scattered knowledge on three-digit squares and offers new elementary proofs and conjectures, with some results generalized beyond base 10.

Findings

Characterization of integers with three-digit squares

Elementary proofs of key properties

Extension of results to arbitrary bases

Abstract

We consider integers whose squares have just three decimal digits. Examples are e.g. given by $2108436491907081488939581538^2 = 4445504440405440505004450045555054500055550554550445444$ and $10100000000010401000000000101^2 = 102010000000210100200000110221001000002101002000000010201$. The aim of this paper is to summarize the current knowledge on squares with three digits, scattered around webpages and newsgroup postings, and to add a few new insights. While we will mostly focus on the base $B=10$, several results are presented for general values of $B$. The used mathematical tools are completely elementary. However, we give complete proofs of all statements or explicitly state them as conjectures.