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Equation using complex numbers

$x_1, x_2$ are the solution for $x^2+x+1=0$

How much is $n \epsilon N$ so that $(x_1^2+1)^n+(x_2^2+1)^n=-1$ to be true?

The solutions to the quadratic are

$x_{1}=\frac{-1}{2}+\frac{\sqrt{3}}{2}i, \;\ x_{2}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i$

Sub these into $(x_{1}^{2}+1)^{n}+(x_{2}^{2}+1)^{n}$ and it whittles down to

$2cos(\frac{n\pi}{3})$

Now, set this equal to -1 and solve for n.

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