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I have done many attempts of computations to get such integers $x, y, z$ for which $390=x^3+y^3+z^3$, but I can't however $390 \neq 4\bmod 9$ or $-4 \bmod 9$ which means there are solutions? Any help?

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  • $\begingroup$ Wolfram Alpha gives $x=13, y=4, z=-11$; presumably permutations of those work too $\endgroup$ – J. W. Tanner Apr 20 '20 at 3:25
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    $\begingroup$ Sorry I meant 390 not 930, just a wrong typo $\endgroup$ – zeraoulia rafik Apr 20 '20 at 3:33
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    $\begingroup$ see wiki entry of sum of 3 squares. As of September 2019, $390$ is the second smallest positive $n$ which we don't know whether the equation $x^3 + y^3 + z^3 = n$ has a solution or not. $\endgroup$ – achille hui Apr 20 '20 at 4:02
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It's actually an open problem. We still don't know whether numbers such as $114, 390, 579, 627, 633, 732, 906, 921$ and $975$ are the sum of $3$ cubes.

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