<s>[INST] Solving a Diaphontine Equation Find all integral solutions to the equation
$(x^2 + 1)(y^2 + 1) + 2(x − y)(1 − xy) = 4(1 + xy)$.
I've gotten this far:
$x^2y^2+x^2+y^2+1+2x-2y-2x^2y-2xy^2=4+4xy$
$(x-y)^2+x^2y^2-2x^2y-2xy^2+2x-2y-2xy=3$
What do I do from here?
 [/INST] Let $A=(x^2+1)(y^2+1)$, $B=2(y-x)(1-xy)$, $C=4(1+xy)$. If $|x|,|y| \geq 4$, then we have $A > 4C$ and if $|x|,|y| \geq 6$, then we have $3A>4B$.
Thus for $|x|,|y| \geq 6$, there's no solution. Now the remaining cases can be solved by calculation.
Now I prove just one of the claims above, i.e. $A>4C$ for $|x|,|y| \geq 4$: Let $a=|x|,b=|y|$, then $4C \leq 16|C| \leq 16(1+ab) < (a^2+1)(b^2+1)$ because $a^2b^2 \geq 16ab$ and $a^2 \geq 16$.
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