On the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ involving Laurent polynomials

Abstract

Using the theory of elliptic curves, we study the nontrivial rational (parametric) solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f(x)$.