Junyao Peng

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 3, Pages 1—12

DOI: 10.7546/nntdm.2019.25.3.1-12

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## Details

### Authors and affiliations

Junyao Peng

*Chongqing Fuling No.15 Middle School
Chongqing, 400000, China*

*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 basic properties of Pell equation and the theory of congruence, we investigate the problem about the linear combination of two triangular numbers is a perfect square. First, we show that if 2*n* is not a perfect square, the Diophantine equation

has infinitely many positive integer solutions (*y,z*). Second, we prove that if *m,n* are some special values, the Diophantine equation

### Keywords

- Triangular 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.
- 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 Quarterly, 20(1), 24–28.
- Guan, X. G. (2011). The squares with the form , Natural Science Journal of Ningxia Teachers University, 32(3), 97–107.
- Hu, M. J. (2013). The positive integer solutions of the Diophantine equation , Journal of Zhejiang International Studies University, 4, 70–76.
- Ke, Z., & Sun, Q. (1980). Talk about the 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.
- 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.

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## Cite this paper

APAPeng, J. (2019). The linear combination of two triangular numbers is a perfect square. Notes on Number Theory and Discrete Mathematics, 25(3), 1-12, doi: 10.7546/nntdm.2019.25.3.1-12.

ChicagoPeng, Junyao. “The Linear Combination of Two Triangular Numbers Is a Perfect Square.” Notes on Number Theory and Discrete Mathematics 25, no. 3 (2019): 1-12, doi: 10.7546/nntdm.2019.25.3.1-12.

MLAPeng, Junyao. “The Linear Combination of Two Triangular Numbers Is a Perfect Square.” Notes on Number Theory and Discrete Mathematics 25.3 (2019): 1-12. Print, doi: 10.7546/nntdm.2019.25.3.1-12.