Indication of Gaussian integers solutions for $x^2+y^2=z^3$.

Question

I know the existence of infinitely many integer solutions for $x^2+y^2=z^3$. But my concern is the existence of Gaussian Integers solution of this diophantine equation. I try to search for relevant papers but most discuss in terms of integer solutions only. Thank you.

Answer 1

$$z^3=uv,\qquad z,u,v \in \Bbb{Z}[i]$$ then $$z^3=x^2+y^2, \qquad x = \frac{u+v}{2}, y = \frac{u-v}{2i}\in \frac12 \Bbb{Z}[i]$$

$x,y$ are in $\Bbb{Z}[i]$ iff $u-v\in 2\Bbb{Z}[i]$ which implies that $u=v=z\bmod 2 \Bbb{Z}[i]$.

Every solution is of this form.

In particular there is no solution with $z=1+i$

and there is one iff $z\in 2\Bbb{Z}[i]$ or $1+2\Bbb{Z}[i]$.