# Super fun complex number question

Algebra Level 5

Suppose $$z$$ is a complex number such that $$z^5 + 1 = 0.$$

If $$z^4 + z^2 + 1 = z^m + z^n$$, then find $$\left|m - n\right|$$ - 1, where $$m$$ and $$n$$ are the smallest possible positive integers for which the listed equation is true.

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