The question and answer from this post seem to imply that the polynomials $z^n - 1$ and $(z+1)^n - 1$ (over $\Bbb C[x])$ will be relatively prime if and only if $6 \nmid n$.
Is this true? If so, is there a quick, direct justification that this is the case?
The geometric view. Not sure about an algebraic proof.
If they have a common factor, they have a common complex root.
If they have a common root, then there is some $n$th root of unity $\alpha$ such that $\alpha-1$ is also an $n$th root of unity.
Viewed on the unit circle, there are only two points where a horizontal chord on the unit circle has length equal to $1$, specifically the cases $y=\pm \sqrt{3}/2.$ These in turn yield $z$ to be one of the primitive sixth roots of unity.
Alternatively, the two unit circles $|z|=1$ and $|z+1|=1$ have two points in common. Those two points are the primitive cubed roots of $1.$ In those cases $z+1$ is a primitive sixth root of $1$ so we'd need $6\mid n.$
