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 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).
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,
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*.
@gnivasch According to https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf this is conjectured to be true
@11011110 also z is not negative according to Wolfram there. A typo of course but still.
A Mastodon instance for maths people. The kind of people who make \(\pi z^2 \times a\) jokes.
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