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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 https://gilkalai.wordpress.com/2019/03/09/8866128975287528%C2%B3-8778405442862239%C2%B3-2736111468807040%C2%B3/ (where I found out about this) or https://en.wikipedia.org/wiki/Sums_of_three_cubes (new article I wrote once I found out).

\[\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.

0xDE@11011110@mathstodon.xyzUpdate: Kalai now says that Browning now says that the discovery was by Andrew Booker.