### Abstract

We determine all integers n such that n^{2} 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 t^{2}, 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 |
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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 | |

Publication status | Published - 1 Jun 2017 |

### ASJC Scopus subject areas

- Mathematics(all)

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## 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