Hilbert 10th problem, asking for algorithm for determining whether a polynomial Diopantine equation has an integer solution, is undecidable in general, but decidable or open in some restricted families. If we classify equations by degree, the problem is trivial for linear equations, decidable for quadratic equations by the Theorem of Grunewald and Segal, but is open for cubic equations.
It is interesting to investigate what are the ``smallest'' difficult cubic equations, in the spirit of this previous question for general equations. For a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,\dots,a_k$, let $H(P)=\sum_{i=1}^k |a_i|2^{d_i}$. If we order all equations by $H$, then the smallest cubic equation I currently cannot solve is the equation$$3-y+x^2 y+y^2+x y z-2 z^2 = 0,$$of size $H=33$.
This equation after a linear substitution reduces to$$2 x^2 - y x z + 2 z^2 = y^2 - 9 y + 23,$$which is nice because it is symmetric in $x,z$. Vieta jumping seems to be difficult to apply because of coefficients $2$ near $x^2$ and $z^2$.
The question is whether this equation has any integer solutions.
