$z_1 \text{ and } z_2 \text{ are the solutions of } 1-z+z^2=0$
$E=(z_1^4-z_1^3+2z_1^2-2z_1+1)^{2005}+(z_2^4-z_2^3+2z_2^2-2z_2+1)^{2005}$
Which is the value of $E$ ?
I have solved the equation:
\begin{align*}\Delta = 1-4=-3=3i^2&\Rightarrow z_{1,2}=\frac{1\pm i\sqrt{3}}{2} \end{align*}
One solution would be to write these numbers in trigonometric form. But I am sure there is an easier way if I write E differently, but I can't find it.