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A purely exponential Diophantine equation in three unknowns

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 xy 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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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