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Three variable, third degree Diophantine equation
Has anyone seen the proof that $x^{3} +y^{3}=3z^{3}$ has no solutions in Eisenstein integers? I have tried to mimic the proof for $x^{3} +y^{3}=z^{3}$ having no solutions in Eisenstein integers and am stuck on the "hard case" of when $\lambda |z$ where $\lambda=1-\omega$ and $\omega$ is a primitive cubic root of unity. Thanks for any help you can offer.