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Related to sum of three squares and this question.

Let $l_1,l_2,l_3 \in \mathbb{Z}[x,y]$ and $n \in \mathbb{Z}$.

Assume that $n$ is not a cube and not twice cube.

Let $f=l_1(x,y)^3+l_2(x,y)^3+l_3(x,y)^3-n$.

Is it possible $f$ to factors as $f(x,y)=q(x,y)g(x,y)$ where $\deg q=2$ and $q=0$ is absolutely irreducible and have infinitely many integral points, giving infinitely many representations of $n$ as sum of three cubes?

For $n= -1$ this is possible:

$l_1=2*x - 1;l_2=-x + y;l_3=-2*x - y;q=x^2 + 9*x*y + 9*y^2 + 12*x - 6$

Possible approach is to equate coefficients symbolically.

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  • $\begingroup$ Isn't $-1$ a cube ? $\endgroup$
    – Henri Cohen
    Commented Oct 22, 2019 at 15:21
  • $\begingroup$ @HenriCohen Yes, it is cube. Just to show it is possible in general. $\endgroup$
    – joro
    Commented Oct 22, 2019 at 15:48

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