Solve in integers the equation
$x^{2} + y^{2} = (x - y)^{3} .$

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Solution

If we denote a $=$ x $-$ y and b $=$ x $+$ y, the equation rewrites as $a^{2} + b^{2}$ = $2 a^{3}$ , $2 a - 1$ = $\frac{b^{2}}{a^{2}}$ .
We see that 2a-1 is the square of a rational number, hence it is the square of an (odd) integer. Let 2a-1 $=$ $(2 n + 1)^{2}$ .
It follows that $a = 2 n^{2} + 2 n + 1$ and $b = a (2 n + 1)$ = $(2 n + 1) (2 n^{2} + 2 n + 1)$ .
Finally, we obtain $x$ = $2 n^{3} + 4 n^{2} + 3 n + 1$ , $y$ = $2 n^{3} + 2 n^{2} + n$ , where n is an arbitrary integer number.

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