I came across the following conjecture. If you have any thoughts on how to approach it, let me know.


**Conjecture.** For any integer $n > 3$ and any Gaussian integer $z$ that is not a unit, if $z^n - z$ or $z^n + z$ is a rational integer, then $z$ is a rational integer.



Given that $n > 3$ and $z$ is a Gaussian integer, when can $z^n \pm z$ be a rational integer?
The question can be rephrased: Find the integers $n > 3$ for which there exist Gaussian integers $z$ such that
$z^n \pm z = \overline{z}^n \pm \overline{z}$. Rewriting the equation as $(z^n -\overline{z}^n)/(z-\overline{z}) = \mp 1$, one recognizes an instance of the problem of finding the terms of a Lucas sequence with no primitive divisors. The complete answer to an even more general problem (Lehmer sequences instead of Lucas sequences) was given by Bilu, Hanrot, Voutier  [J. Reine Angew. Math.
539, 75-122 (2001)](https://www.degruyter.com/document/doi/10.1515/crll.2001.080/html)  and Abouzaid  [J. Théor. Nombres Bordx.
18, No. 2, 299--313 (2006)](https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.545/) . To settle the conjecture, it suffices to look up the tables in these papers .


