Mei Jiang and Yangcheng Li

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 2, Pages 105–115

DOI: 10.7546/nntdm.2020.26.2.105-115

**Full paper (PDF, 200 Kb)**

## Details

### Authors and affiliations

Mei Jiang

*School of Mathematics and Statistics, Changsha University of Science and Technology,
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering,
Changsha, 410114, China
*

Yangcheng Li

*School of Mathematics and Statistics, Changsha University of Science and Technology,
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering,
Changsha, 410114, China
*

### Abstract

By the theory of Pell equation and congruence, we study the problem about the linear combination of two polygonal numbers is a perfect square. Let denote the -th -gonal number. We show that if , is not a perfect square, and there is a positive integer solution of satisfying

then the Diophantine equation has infinitely many positive integer solutions . Moreover, we give conditions about such that the Diophantine equation has infinitely many positive integer solutions .

### Keywords

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

### 2010 Mathematics Subject Classification

- 11D09
- 11D72

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## Cite this paper

Jiang, M., & Li, Y. (2020). The linear combination of two polygonal numbers is a perfect square. *Notes on Number Theory and Discrete Mathematics*, 26 (2), 105-115, DOI: 10.7546/nntdm.2020.26.2.105-115.