<s>[INST] The complex equation $x^3 = 9 + 46i$ has a solution of the form $a + bi$ where $a,b\in \mathbb Z$. Find the value of $a^3 + b^3$ The complex equation $x^3 = 9 + 46i$ has a solution of the form $a + bi$ where $a,b\in \mathbb Z$. Find the value of $a^3 + b^3$ .
 [/INST] $\mathbb{Z}[i]$ is a Euclidean domain, hence a UFD. Since the norm of $9+46i$ is $9^2+46^2=13^3$, $(a+bi)^3=9+46i$ implies $a^2+b^2=13$. It follows that $a,b\in\{-3,-2,2,3\}$ and to check that the only solution is $a=-3$ and $b=2$ is straightforward.
</s>