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A purely exponential Diophantine equation in three unknowns

Periodica Mathematica Hungarica volume 84, pages 287–298 (2022)Cite this article

Abstract

For any fixed integers a and b greater than 1, we study the Diophantine equation \(a^x+(ab+1)^y=b^z\). First, we describe a heuristic list of the positive integer solutions x, y and z of the equation. Finally, we solve the equation in some particular cases, which supports the validity of our list of solutions.

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Acknowledgements

We would like to thank the referee for the helpful remarks. The first author is supported by JSPS KAKENHI (No. 20K03553). The third author is supported by JSPS KAKENHI (No. 18K03247).

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Authors and Affiliations

  1. Division of Pure and Applied Science, Faculty of Science and Technology, Gunma University, Tenjin-cho 1-5-1, Kiryu, 376-8515, Japan

    Takafumi Miyazaki

  2. Professor Emeritus, Seikei University, Kichijhoji Kitamachi 3-3-1, Musashino-shi, 180-8633, Japan

    Masaki Sudo

  3. Division of Mathematical Sciences, Department of Integrated Science and Technology, Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 870–1192, Japan

    Nobuhiro Terai

Authors
  1. Takafumi Miyazaki
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  2. Masaki Sudo
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  3. Nobuhiro Terai
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Correspondence to Takafumi Miyazaki.

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Miyazaki, T., Sudo, M. & Terai, N. A purely exponential Diophantine equation in three unknowns. Period Math Hung 84, 287–298 (2022). https://doi.org/10.1007/s10998-021-00405-x

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  • DOI: https://doi.org/10.1007/s10998-021-00405-x

Keywords

  • S-unit equation
  • Purely exponential equation
  • Baker’s method
  • Simultaneous non-Archimedean valuations

Mathematics Subject Classification

  • 11D61
  • 11J86