Timothy Browning has discovered that
\[\begin{align}
33&=8866128975287528^3\\
&+(-8778405442862239)^3\\
&+(-2736111468807040)^3.
\end{align}\]

This settles all but one case of which two-digit numbers can be represented as a sum of three cubes. The remaining case is \(n=42\).

For more, see gilkalai.wordpress.com/2019/03 (where I found out about this) or en.wikipedia.org/wiki/Sums_of_ (new article I wrote once I found out).

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While searching for more references for the new Wikipedia article (en.wikipedia.org/wiki/Sums_of_) I found a horrible mistake related to this problem in Wolfram's _A New Kind of Science_, p. 789 (wolframscience.com/nks/p789--i). 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.

@11011110 also z is not negative according to Wolfram there. A typo of course but still.

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