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).

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

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

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