On the Diophantine equations z2 = f(x)2 ± f(x)f(y) + f(y)2

Qiongzhi Tang
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 88—100
DOI: 10.7546/nntdm.2021.27.2.88-100
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Authors and affiliations

Qiongzhi Tang
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

Using the theory of Pell equation, we study the non-trivial positive integer solutions of the Diophantine equations z^2=f(x)^2\pm f(x)f(y)+f(y)^2 for certain polynomials f(x), which mean to construct integral triangles with two sides given by the values of polynomials f(x) and f(y) with the intersection angle 120^\circ or 60^\circ.

Keywords

  • Diophantine equation
  • Pell equation
  • Positive integer solution

2020 Mathematics Subject Classification

  • 11D25
  • 11D72

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Cite this paper

Tang, Q. (2021). On the Diophantine equations z2 = f(x)2 ± f(x)f(y) + f(y)2. Notes on Number Theory and Discrete Mathematics, 27(2), 88-100, doi: 10.7546/nntdm.2021.27.2.88-100.

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