A note on the Diophantine equation (x^k-1)(y^k-1)^2=z^k-1

Yangcheng Li
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 825–831
DOI: 10.7546/nntdm.2024.30.4.825-831
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Authors and affiliations

Yangcheng Li
School of Mathematical Sciences, South China Normal University
Guangzhou, People’s Republic of China

Abstract

We prove that, for k\geq10, the Diophantine equation (x^k-1)(y^k-1)^2=z^k-1 in positive integers x,y,z,k with z>1, has no solutions satisfying 1<x\leq y or 1<y<x\leq((y^k-1)^{k-2}+1)^{\frac{1}{k}}.

Keywords

  • Diophantine equation
  • Diophantine approximation

2020 Mathematics Subject Classification

  • 11D41
  • 11D61

References

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Manuscript history

  • Received: 5 July 2023
  • Revised: 7 November 2024
  • Accepted: 24 November 2024
  • Online First: 30 November 2024

Copyright information

Ⓒ 2024 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Li, Y. (2024). A note on the Diophantine equation (x^k-1)(y^k-1)^2=z^k-1. Notes on Number Theory and Discrete Mathematics, 30(4), 811-824, DOI: 10.7546/nntdm.2024.30.4.825-831.

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