In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size of the equation is substitute 2 instead of all variables, absolute values instead of all coefficients, and evaluate.

In the Mathoverlow question Can you solve the listed smallest open Diophantine equations? I list the current smallest equations for which it is open whether there exists any integer solution at all (Hilbert's 10 problem).

However, there are some famous equations, like $x^3+y^3+z^3=3$ of size $H=2^3+2^3+2^3+3=27$, for which the Hilbert's 10 problem is trivial (in this example, $x=y=z=1$ is a solution), but the equation can hardly be classified as solved, because we do not even know whether the solution set is finite.

Here, I consider more general problem: **for a given polynomial Diophantine equation, determine whether the solution set is finite, and if so, list all the solutions**. This is a much better approximation of our intuition what does it mean to solve an equation, but still avoids a subtle issue what counts as an acceptable description of the solution set if it is infinite (see What does it mean to solve an equation? for some discussion of this).

**Selected solved equations.**

The smallest equation that required a new idea turned out to be $y^2+z^2=x^3-2$ of size $H=18$, see Representing $x^3-2$ as a sum of two squares for the proof that it has infinitely many integer solutions.

Equations $ y(z^2-y)=x^3+2 $ and and $ xyz=x^3+y^2-2 $ of size $H=22$ has been listed as open and then solved by Tomita, see the answer below.

**Smallest open equations.**

The current smallest open equations are the equations $$ y^2-yz^2+x^3-2=0 $$ and $$ xyz=x^3+y^2+2 $$ of size $H=22$. These are the only remaining open equations with $H \leq 22$.

One may also study **equations of special types**. For example, the current smallest open symmetric equation (that is, invariant under cyclic shift of the variables) is
$$
x^2y+y^2z+z^2x=1
$$
of size $H=25$. The current smallest open equations in two variables are
$$
y^3+y=x^4+x
$$
and
$$
y^3-y=x^4-x
$$
of size $H=28$, the current smallest open 3-monomial equation is
$$
x^3y^2=z^3+2
$$
of size $H=42$, while the current smallest open homogeneous equation is
$$
x^4+x^3 y-y^4+y^3 z+z^4=0
$$
of size $H=80$, see Existence of rational points on some genus 3 curves.

For the listed equations, the Hilbert 10th problem is trivial, because there are some obvious small solutions. **The question**, for each of the listed equations, is whether the solution set is finite or infinite, and if finite, list the solutions.

**The plan** is to list new smallest open equations once these ones are solved. The solved equations will be moved to the "solved" section.

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