Rational solutions of the Diophantine equations f(x)2±f(y)2=z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)^2 \pm f(y)^2=z^2$$\end{document}

We discuss the rational solutions of the Diophantine equations f(x)2±f(y)2=z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)^2 \pm f(y)^2=z^2$$\end{document}. This problem can be solved either by the theory of elliptic curves or by elementary number theory. Inspired by the work of Ulas and Togbé (Publ Math Debrecen 76(1–2):183–201, 2010) and following the approach of Zhang and Zargar (Period Math Hung, 2018. https://doi.org/10.1007/s10998-018-0259-7) we improve the results concerning the rational solutions of these equations.