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I am trying to find solutions for this diophantine equation

$$x^2+y^2+x^2y^2=4z^2$$

I am looking for advice on a procedure to find all positive integer solutions for this equations.

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Add $1$ to both sides and you get $(x^2+1)(y^2+1)=4z^2+1$, which is only possible if $x$ and $y$ are even, which reduces to another problem posted earlier today: $(4x^2+1)(4y^2+1)=4z^2+1$. math.stackexchange.com/q/162862/7933 –  Thomas Andrews Jun 25 '12 at 17:54

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