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By the theory of elliptic curves, we study the nontrivial rational parametricsolutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pmg(y)^2$ for some simple Laurent polynomials $f$ and $g$.

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