Ta có : \(\left\{{}\begin{matrix}\left(x+\sqrt{2017+x^2}\right)\left(\sqrt{2017+x^2}-x\right)=2017\\\left(x+\sqrt{2017+x^2}\right)\left(y+\sqrt{2017+y^2}\right)=2017\end{matrix}\right.\)
\(\Rightarrow\sqrt{2017+x^2}-x=y+\sqrt{2017+y^2}\)
\(\Leftrightarrow x+y=\sqrt{2017+x^2}-\sqrt{2017+y^2}\left(1\right)\)
\(\left\{{}\begin{matrix}\left(y+\sqrt{2017+y^2}\right)\left(\sqrt{2017+y^2}-y\right)=2017\\\left(y+\sqrt{2017+y^2}\right)\left(x+\sqrt{2017+x^2}\right)=2017\end{matrix}\right.\)
\(\Rightarrow\sqrt{2017+y^2}-y=x+\sqrt{2017+x^2}\)
\(\Leftrightarrow x+y=\sqrt{2017+y^2}-\sqrt{2017+x^2}\left(2\right)\)
Lấy (1) + (2) \(\Leftrightarrow2\left(x+y\right)=0\Leftrightarrow x+y=0\Leftrightarrow x=-y\)
\(T=x^{2017}+y^{2017}=-y^{2017}+y^{2017}=0\)
