<s>[INST] Sum of two odd cubes plus four times a cube is zero Let $x,y$ be odd integers and let $z$ be an integer. The question is to find all solutions to the equation,
$$x^3+y^3+4z^3=0$$
Of course we have the trivial solution $(x,-x,0)$. Are there any others?
By considering the equation modulo $4$ we see that wlog $x=4k+1$ and $y=4m+3$. Numerical experimentation shows that there are no solutions with $|k|, |m| < 10000$. In fact since the cubic residues modulo $9$ are $0,\pm 1$ then we see that $z$ is a multiple of $9$ and moreover, dividing by $9^3$ if necessary, we see that $x^3$ and $y^3$ must have opposite residue modulo $9$. The same holds by considering the equation modulo $7$. I'm also aware of Broughan's theorem, but it's not clear to me whether it helps.
 [/INST] $x^3+y^3+4z^3=0$
Let $X=x/z, Y=y/z$ then we get $X^3+Y^3=-4$.
In general, $X^3+Y^3=n$ can be transformed to elliptic curve  $v^2=u^3-432n^2$.
Hence we get $v^2=u^3-6912$.
According to LMFDB, this elliptic curve has rank $0$ and has no integer solution.
Hence $x^3+y^3+4z^3=0$ has no nontrivial solution.
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