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Find all integer solutions of the equation $x^2+y^2=(x-y)^3$.

MATHPRITOM wrote:Find all integer solutions of the equation $x^2+y^2=(x-y)^3$.

Since $x^2+y^2\ge 0$, $x\ge y$. Say $x=y+k$.
$\ \ \ k^3=2y^2+2yk+k^2$
$\Longrightarrow 2k^3=4y^2+4yk+k^2+k^2$
$\Longrightarrow k^2(2k-1)=(2y+k)^2$
Since, $\gcd(k^2,2k-1)=1$, $2k-1$ must be a perfect square, say $2k-1=(2t+1)^2$
Then $k=2t^2+2t+1$
Set this. Then you will get the general solution.