Let $a, b$, and $c$ be integers. Suppose that $a$ and $b$ are coprime and that $a^2 +b^2 = c^3$. Show that $a\not\equiv b \pmod 2$ and that $c$ is odd

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?

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

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