While searching for more references for the new Wikipedia article (https://en.wikipedia.org/wiki/Sums_of_three_cubes) I found a horrible mistake related to this problem in Wolfram's _A New Kind of Science_, p. 789 (https://www.wolframscience.com/nks/p789--implications-for-mathematics-and-its-foundations/). Wolfram says that the smallest solution to \(x^3+y^3+z^3=2\) is the known sporadic one,

\[\begin{align}2&=1214928^3\\

&+ 3480205^3\\

&+ (-3528875)^3.

\end{align}\]

But the parametric solution has many that are smaller: \((1,1,0)\), \((7,-5,-6)\), \((49,-47,-24)\), etc.

Chris Purcell@ccppurcell@mathstodon.xyz@11011110 also z is not negative according to Wolfram there. A typo of course but still.