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What I'm thinking is this: Let $a = 2n+b$, then the equation becomes $2(2n^2+2bn+b^2)=c^3$. I'm stuck here.

After this, how can I show that $a+bi$ and $a-bi$ are coprime Gaussian integers?

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As $(a,b)=1$ both can not be even together

If both are odd $(2A+1)^2+(2B+1)^2\equiv2\pmod8$

$\implies c^3$ is even $\implies c$ is even $\implies c^3\equiv0\pmod8\not\equiv2$

So, $a,b$ must be of opposite parity.

WLOG $a$ is even and $b$ is odd

$\implies a^2$ is even, $b^2$ is odd

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  • 1
    $\begingroup$ Ah, so the sum of the squares of two odd numbers can never be the cube of an integer? $\endgroup$ – turkeyhundt May 11 '17 at 18:12

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