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I am investigating solutions to Fermat's equation $$x^n+y^n=z^n$$ with $x,y,z$ in the Gaussian integers, excluding solutions in excluding $\mathbb{Z}$ or $i\mathbb{Z}$ .

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.

P.S.: This same question was asked on Math.SE but it has now drawn drowned under the fold and I thought I will have better chances of getting answers here.
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Fermat's Last Theorem for Gaussian Integers ( excluding $\mathbb{Z}$ or $i\mathbb{Z}$ )

I am investigating solutions to Fermat's equation $$x^n+y^n=z^n$$ with $x,y,z$ in the Gaussian integers, excluding solutions in excluding $\mathbb{Z}$ or $i\mathbb{Z}$ .

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.

P.S.: This same question was asked on Math.SE but it has now drawn under the fold and I thought I will have better chances of getting answers here.