The linear combination of two polygonal numbers is a perfect square

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
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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 P_k(x) denote the x-th k-gonal number. We show that if k\geq5, 2(k-2)n is not a perfect square, and there is a positive integer solution (Y',Z') of Y^2-2(k-2)nZ^2=(k-4)^2n^2-8(k-2)n satisfying

    \[Y'+(k-4)n\equiv0\pmod{2(k-2)n}, Z'\equiv0\pmod{2},\]

then the Diophantine equation 1+nP_k(y)=z^2 has infinitely many positive integer solutions (y,z). Moreover, we give conditions about m,n such that the Diophantine equation mP_k(x)+nP_k(y)=z^2 has infinitely many positive integer solutions (x,y,z).

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.

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