We discuss the rational solutions of the Diophantine equations $$f(x)^2 \pm f(y)^2=z^2$$f(x)2±f(y)2=z2. 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.

中文翻译：

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丢番图方程的有理解 $$f(x)^2 \pm f(y)^2=z^2$$ f ( x ) 2 ± f ( y ) 2 = z 2

我们讨论丢番图方程 $$f(x)^2 \pm f(y)^2=z^2$$f(x)2±f(y)2=z2 的有理解。这个问题可以通过椭圆曲线理论或初等数论来解决。受到 Ulas 和 Togbé 工作的启发（Publ Math Debrecen 76(1-2):183-201, 2010）并遵循 Zhang 和 Zargar 的方法（Period Math Hung，2018. https://doi.org/10.1007/ s10998-018-0259-7) 我们改进了关于这些方程的有理解的结果。