The linear combination of two triangular numbers is a perfect square

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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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


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 2n is not a perfect square, the Diophantine equation

    \[1+n\binom y 2=z^2\]

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

    \[m\binom x 2+n\binom y 2=z^2\]


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

2010 Mathematics Subject Classification

  • 11D09
  • 11D72


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Cite this paper

Peng, 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.

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