In my previous question Can you solve the listed smallest open Diophantine equations? I discuss the smallest equations (in some well-defined sense) for which it is not known whether they have any integer solutions. The current smallest open is the equation $y(x^3-y)=z^3+3$, and it has three variables. Because it is open for a long time, it makes sense to leave it for now, and look for the next-smallest open ones. The next-smallest open equation in at least three variables is $$ 1 + x - x^3 + x^2 y^2 + z + z^2 = 0 $$ The question is whether is has any integer solutions.
My attempt: rewrite it as $$ x(x^2-1-xy^2)=z^2+z+1=\frac{(2z+1)^2+3}{4} $$ From the original equation it is clear that $x>0$. Because $z^2+z+1$ is positive and odd, both $x$ and $x^2-1-xy^2$ are positive and odd. It is known that positive odd divisors of a ``square plus $3$'' are never $2$ modulo $3$. Hence, $x\neq 2$ mod $3$. If $x=0$ mod $3$, then $x^2-1-xy^2=2$ mod $3$, a contradiction. However, if $x=1$ mod $3$ and $y=0$ mod $3$, then $x^2-1-xy^2=0$ mod $3$, so no contradiction in this case.
On the other hand, a search with $|z|$ up to 30 millions returned no integer solutions.
