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