Timothy Browning has discovered that

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 (where I found out about this) or (new article I wrote once I found out).

Update: Kalai now says that Browning now says that the discovery was by Andrew Booker.

While searching for more references for the new Wikipedia article ( I found a horrible mistake related to this problem in Wolfram's _A New Kind of Science_, p. 789 ( Wolfram says that the smallest solution to \(x^3+y^3+z^3=2\) is the known sporadic one,
&+ 3480205^3\\
&+ (-3528875)^3.
But the parametric solution has many that are smaller: \((1,1,0)\), \((7,-5,-6)\), \((49,-47,-24)\), etc.

@11011110 I wonder if every such number can be written as the sum of three cubes in *infinitely many ways*.

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

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