Squares with three nonzero digits

Michael A. Bennett, Adrian Maria Scheerer

Research output: Chapter in Book/Report/Conference proceedingChapterResearchpeer-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 languageEnglish
Title of host publicationNumber Theory - Diophantine Problems, Uniform Distribution and Applications
Subtitle of host publicationFestschrift in Honour of Robert F. Tichy's 60th Birthday
PublisherSpringer International Publishing AG
Pages83-108
Number of pages26
ISBN (Electronic)9783319553573
ISBN (Print)9783319553566
DOIs
Publication statusPublished - 1 Jun 2017

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Q-function
Digit
Odd
Polynomial
Integer
Family

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Bennett, M. A., & Scheerer, A. M. (2017). Squares with three nonzero digits. In Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday (pp. 83-108). Springer International Publishing AG . https://doi.org/10.1007/978-3-319-55357-3_4

Squares with three nonzero digits. / Bennett, Michael A.; Scheerer, Adrian Maria.

Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday. Springer International Publishing AG , 2017. p. 83-108.

Research output: Chapter in Book/Report/Conference proceedingChapterResearchpeer-review

Bennett, MA & Scheerer, AM 2017, Squares with three nonzero digits. in Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday. Springer International Publishing AG , pp. 83-108. https://doi.org/10.1007/978-3-319-55357-3_4
Bennett MA, Scheerer AM. Squares with three nonzero digits. In Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday. Springer International Publishing AG . 2017. p. 83-108 https://doi.org/10.1007/978-3-319-55357-3_4
Bennett, Michael A. ; Scheerer, Adrian Maria. / Squares with three nonzero digits. Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday. Springer International Publishing AG , 2017. pp. 83-108
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