Fermat's Last Theorem for Gaussian Integers (excluding the integers or same pure imaginaries)
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.
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.)