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

**Download 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

### References

- Bencze, M. (2012). Proposed Problem 7508, Octogon Mathematical Magazine, 13 (1B), 678.
- Chen, J. P. (2012). The squares with the form , Natural Science Journal of China West Normal University, 33 (2), 196–198, 217.
- Cohen, H. (2007). Number Theory, Vol. I: Tools and Diophantine Equations, Graduate Texts in Mathematics, Springer.
- Deza E., & Deza M. M. (2012). Figurate Numbers, Word Scientific.
- Dickson, L. E. (1934). History of the Theory of Numbers, Vol. II: Diophantine Analysis, Dover Publications.
- Eggan, L. C., Eggan, P. C., & Selfridge, J. L. (1982). Polygonal products of polygonal numbers and the Pell equation, Fibonacci Quart., 20 (1), 24–28.
- Guan, X. G. (2011). The squares with the form , Natural Science Journal of Ningxia Teachers University, 32 (3), 97–107.
- Guy, R. K. (2007). Unsolved Problems in Number Theory, Springer-Verlag.
- Hu, M. J. (2013). The positive integer solutions of the Diophantine equation ,

Journal of Zhejiang International Studies University, 4, 70–76. - Ke, S., & Sun, Q. (1980). About Indeterminate Equation, Harbin Institute of Technology Press.
- Le, M. H. (2007). The squares with the form , Natural Science Journal of Hainan University, 25 (1), 13–14.
- Peng, J. Y. (2019). The linear combination of two triangular numbers is a perfect square, Notes on Number Theory and Discrete Mathematics, 25 (3), 1–12.
- Ran, Y. X., Yan, S. J., Ran, Y. P., & Yang, X. Y. (2008). The squares with the form

, Journal of Tianshui Normal University, 28 (5), 9–15. - Sun, Z. H. (2009). On the number of representations of by , Journal of Number Theory, 129 (5), 971–989.
- Wu, H. M. (2011). The square numbers with the form , Journal of Zhanjiang Normal College, 32 (3), 20–22.

## Related papers

## 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.