Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$?
$$p^2 q^2 \geq 3 p^2 q + 3p^2 + 3pq^2 + 3pq + 3p + 3q^2 + 3q + 3$$
I tried to use Wolfram Alpha, and it says that there are $11$ possible solutions, all of which have a negative $q$.
Can anybody here validate Wolfram Alpha's computation? I am thinking that this may be another one of those instances where its existing numerical algorithm limits its capability to search the solution space.
(Postscript: I was unable to see a diophantine-inequality tag, so feel free to remove the diophantine-equations tag if it is inappropriate.)
The inequality as it currently stands has infinitely many solutions. For look for solutions with $p=q\ge 1$. Then each term on the right is $\le 3p^3$. There are $8$ of them, so the right-hand side is $\le 24p^3$.
The left-hand side is $p^4$, which is $\ge 24p^3$ if $p\ge 24$.