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Squares with three digits

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arxiv 2112.00444 v2 pith:NXVTZA5I submitted 2021-12-01 math.NT math.CO

Michael Gei{ss}er , Theresa Körner , Sascha Kurz , Anne Zahn This is my paper · ORCID
classification math.NTmath.CO
keywords digitssquaresthreearoundbasecompletecompletelyconjectures
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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.

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