# For which $x,y, z$ we have $390=x^3+y^3+z^3$ with $x, y, z$ integers?

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?

• Wolfram Alpha gives $x=13, y=4, z=-11$; presumably permutations of those work too – J. W. Tanner Apr 20 '20 at 3:25
• Sorry I meant 390 not 930, just a wrong typo – zeraoulia rafik Apr 20 '20 at 3:33
• 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. – achille hui Apr 20 '20 at 4:02

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.