Three Diophantine equations concerning the polygonal numbers

Yong Zhang, Mei Jiang and Qiongzhi Tang
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 736–746
DOI: 10.7546/nntdm.2025.31.4.736-746
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Authors and affiliations

Yong Zhang
School of Mathematics and Statistics, Changsha University of Science and Technology
Hunan, Changsha, 410114, People’s Republic of China

Mei Jiang
School of Mathematics and Statistics, Changsha University of Science and Technology
Hunan, Changsha, 410114, People’s Republic of China

Qiongzhi Tang
School of Mathematics and Statistics, Changsha University of Science and Technology
Hunan, Changsha, 410114, People’s Republic of China

Abstract

Many authors investigated the problem about the linear combination of two polygonal numbers being a perfect square, i.e., the Diophantine equation

    \[mP_k(x)+nP_k(y)=z^2,\]

where P_k(x) denotes the x-th k-polygonal number and m,n are positive integers. In this note, we continue the study of this problem in another direction and consider three Diophantine equations

    \[mP_k(x)-1=z^2,\quad mP_k(x)-nP_k(y)=z^2,\quad mP_k(x)-nP_k(y)=1.\]

By the theory of Pell equations and congruences, we obtain some conditions such that the above three Diophantine equations have infinitely many positive integer solutions.

Keywords

  • Polygonal number
  • Diophantine equation
  • Pell equation
  • Positive integer solution

2020 Mathematics Subject Classification

  • 11D09
  • 11D72

References

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Manuscript history

  • Received: 24 April 2025
  • Revised: 6 October 2025
  • Accepted: 26 October 2025
  • Online First: 28 October 2025

Copyright information

Ⓒ 2025 by the Authors.
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

Zhang, Y., Jiang, M., & Tang, Q. (2025). Three Diophantine equations concerning the polygonal numbers. Notes on Number Theory and Discrete Mathematics, 31(4), 736-746, DOI: 10.7546/nntdm.2025.31.4.736-746.

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