MMN-3664

On the Diophantine equations z^2=f(x)^2 ± f(y)^2 involving Laurent polynomials II.

  • Yong Zhang, School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, 410114 Changsha, People's Republic of China, zhangyongzju@163.com
  • Qiongzhi Tang, School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, 410114 Changsha, People's Republic of China, tangqiongzhi628@163.com
  • Yuna Zhang, School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, 410114 Changsha, People's Republic of China, zhangyunacsust@163.com

Abstract

By the theory of Pell's equation, we give conditions for $f(x)=b+\frac{c}{x}$ with $b,c\in \mathbb{Z}\backslash \{0\}$ such that the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ have infinitely many solutions $x,y\in \mathbb{Z}$ and $z\in \mathbb{Q}$, which gives a positive answer to Question 3.2 of Zhang and Shamsi Zargar \cite{Zhang-Zargar3}. By the theory of elliptic curve, we study the non-trivial rational solutions of the above Diophantine equations for Laurent polynomials $f(x)=\frac{\prod_{t=0}^{n}(x+k^t)}{x}$, $\frac{\prod_{t=0}^{n}(x-k^t)(x+k^t)}{x}$, $n\geq1$, $k\in \mathbb{Z}\backslash \{0,\pm1\}$, and give a positive answer to Question 3.1 of Zhang and Shamsi Zargar \cite{Zhang-Zargar3}.


Vol. 23 (2022), No. 2, pp. 1023-1036
DOI: 10.18514/MMN.2022.3664


Download: MMN-3664