Can anyone help me find the number of solutions to the equation:
$$ x^2 + y^2 + xy = (xy)^2 $$
Let me give a brief account of what I've tried to proceed with:
Case 1: One of $x$ and $y$ is odd. This results into a contradiction where the parity of LHS and RHS differs.
Case 2: Both of $x$ and $y$ are even. I have shown that no solution apart from $x=y=0$ exists and I've proved that using Infinite Descent.
Now I'm stuck in case $3$ where both $x$ and $y$ are odd.
Can anyone help me get out of this?
Thanks a lot!!