The Diophantine equation
$$x^3 + y^3 + z^3 = 42$$
was recently solved by
Booker and Sutherland:
[Sum of three cubes for 42 finally solved](https://www.eurekalert.org/pub_releases/2019-09/uob-sot090619.php).


Is there a clean partition of the form of those
polynomial equations all of which *do* have integer solutions,
and those that are known to be undecidable
(following the negative solution of Hilbert's 10th)?
Or—in the absence of a clean partition—can at least
the equations be partitioned into: 
$$\{ \textrm{solvable, undecidable, unknown} \}$$


Am I correct that the status
of $x^3 + y^3 + z^3 = c$ is unknown
except for certain values of $c$?
I ask this naively; not my expertise. 




---


Related MO question:
[What are the solutions of this Diophantine equation?](https://mathoverflow.net/a/339954/6094).



Which Hilbert's 10th polynomials are known to have solutions?
The negative solution of [Hilbert's tenth problem](https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem) means that there is no algorithm that would recognize the solvable polynomial equations. Hence the "clean partition" you are looking for does not exist.


Of course, one can partition the polynomials any way one likes. The partition solvable/non-solvable/undecidable is a legitimate one, except that one "does not know" which polynomials belong to the individual parts.


