TITLE: Complex version of the Fermat last problem
QUESTION [7 upvotes]: A complex integer is a complex number $x=m+ni$ where $m,n\in \mathbb{Z}$.

Are there complex integers $x,y,z$ with $x^3+y^3=z^3$?

REPLY [6 votes]: Lampakis 2007 provided a new proof there are no $xyz\ne 0$ solutions. It runs to several pages. Lampakis notes Feuter 1913 provided the original proof, but I couldn't find an online link to his reference, R. Feuter, Sitzungsber. Akad. Wiss. Heidelberg (Math.), 4, A, 1913 No. 25.