Squares with three nonzero digits

Michael A. Bennett; Adrian Maria Scheerer

doi:10.1007/978-3-319-55357-3_4

Squares with three nonzero digits

Michael A. Bennett^*, Adrian Maria Scheerer

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

We determine all integers n such that n² has at most three base-q digits for q ε (2, 3, 4, 5, 8, 16). More generally, we show that all solutions to equations of the shape where q is an odd prime, n > m > 0 and t², πMπ,N < q, either arise from "obvious" polynomial families or satisfy m ≤ 3. Our arguments rely upon Padé approximants to the binomial function, considered q-adically.

Original language	English
Title of host publication	Number Theory - Diophantine Problems, Uniform Distribution and Applications
Subtitle of host publication	Festschrift in Honour of Robert F. Tichy's 60th Birthday
Publisher	Springer International Publishing AG
Pages	83-108
Number of pages	26
ISBN (Electronic)	9783319553573
ISBN (Print)	9783319553566
DOIs	https://doi.org/10.1007/978-3-319-55357-3_4
Publication status	Published - 1 Jun 2017

ASJC Scopus subject areas

Mathematics(all)

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10.1007/978-3-319-55357-3_4

Cite this

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AB - We determine all integers n such that n2 has at most three base-q digits for q ε (2, 3, 4, 5, 8, 16). More generally, we show that all solutions to equations of the shape where q is an odd prime, n > m > 0 and t2, πMπ,N < q, either arise from "obvious" polynomial families or satisfy m ≤ 3. Our arguments rely upon Padé approximants to the binomial function, considered q-adically.

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Squares with three nonzero digits

Abstract

ASJC Scopus subject areas

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