Further to this question, A hard Cubic Diophantine equation I considered the more general equation $N=x^3+y^3+z^3-2xyz$.
As $x^3+y^3+z^3-2xyz$ is homogenous, any solution $(N_0,x_0,y_0,z_0)$ to $x^3+y^3+z^3-2xyz=N$ gives a set of solutions $(k^3N_0,kx_0,ky_0,kz_0)$.
In particular, any solution with positive $N$ gives a solution for $-N$ using $k=-1$, so it’s not necessary to further consider negative $N$
For example, $(N,x,y,z)=(13,-3,2,2)$ gives $(-13,3,-2,-2)$
Clearly, $(x,y,z)$ are interchangeable.
Using a small search up to $N=152$ I’ve found, with $a,b,k$ integer, just
$$(N,x,y,z)=(0,-a,0,a)$$
$$(N,x,y,z)=(19k^3,(-b-1)k,3k,(b-1)k)$$
Examples as $(N,x,y,z)$
$$(0,-45,0,45)$$ $$(19,-6,3,4)$$ $$(152,-12,6,8)$$
I’ve noticed other $N$ that seem possible candidates, but haven’t spotted the patterns. For example,
$$(9,-1575,583,1163)$$ $$(9,-944,522,545)$$ $$(9,-703,-198,838)$$ $$(9,-323,-187,457)$$ $$(9,-167,80,108)$$ $$(9,-162,86,97)$$ $$(9,-47,-34,72)$$ $$(9,-7,4,4)$$ $$(9,-2,1,2)$$ $$(9,0,1,2)$$ $$(9,1,2,2)$$
Other $N$ values that look interesting are $6,17,33,37,48,51,72,93,96,107,114,117,136$
My question:
Apart from $0$ and numbers of the form $19k^3$, for what $N$ does $N=x^3+y^3+z^3-2xyz$ have infinite integer solutions?
Update 5th March 2018
I’m also interested in:
values of $N$ where all solutions are known
values of $N$ where it can be shown that there are a finite number of solutions
values of $N$ where it can be shown that there no solutions.