TITLE: Fermat's Last Theorem for Gaussian Integers (excluding the integers or  same pure imaginaries)
QUESTION [16 upvotes]: I am investigating solutions to Fermat's equation
$$x^n + y^n = z^n$$
with $x,y,z$ in the Gaussian integers, excluding integers and pure imaginaries.
I have found out that there are only trivial solutions for the $n=3$ and $n=4$ cases, e.g. here. 
I would be grateful if you let me know of the current status or if it is already a theorem.

REPLY [5 votes]: As mentioned above in the comments, the question has been answered on MathOverflow: it's still an open problem.
(This answer is community wiki so that it won't generate any reputation.)