4. If $x, y$ are two different real numbers, and
$$
x^{2}=2 x+1, y^{2}=2 y+1 \text {, }
$$

then $x^{6}+y^{6}=$ $\qquad$ .
4. 198 .
$$
\text { Let } S_{n}=x^{n}+y^{n} \text {. }
$$

From $x^{2}=2 x+1, y^{2}=2 y+1$, we get
$$
x^{n+2}=2 x^{n+1}+x^{n}, y^{n+2}=2 y^{n+1}+y^{n} \text {. }
$$

Thus, $S_{n+2}=2 S_{n+1}+S_{n}$.
Also, $S_{1}=2, S_{2}=6$, so
$$
S_{3}=14, S_{4}=34, S_{5}=82, S_{6}=198
$$