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 $a^3 + b^3$

Question

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 $a^3 + b^3$ .

Answer 1

Let $x=a+bi$, where $\{a,b\}\subset\mathbb R$.

Thus, $$(a+bi)^3=9+46i$$ or $$a^3-3ab^2=9$$ and $$3a^2b-b^3=46,$$ which gives $$46(a^3-3ab^2)=9(3a^2b-b^3)$$ or $$(2a+3b)(23a^2-48ab+3b^2)=0,$$ which gives $$2a+3b=0.$$

Thus, $$3\cdot\frac{9}{4}b^3-b^3=46,$$ which gives $b=2$, $a=-3$ and $a^3+b^3=-19.$

Answer 2

$\mathbb{Z}[i]$ 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.