Given $a^{3} + b^{3} + c^{3} - abc$, determine all the possible values of the expression where $A$, $B$, and $C$ are non-negative integers.
Note that this has been solved here
But I need help understanding why the proof generalizes when we let $f(A,A, A+1)$, $f(A, A, A-1)$ and $f(A, A+1, A-1)$.
The proof doesn't say anything about the following $f(A, A+j, A+k)$ where $k,j \in \mathbb{N}$.
But obviously it probably does as the proof is correct, I just can't seem to understand where. Any help understanding this would be great, thanks!
When it considers $f(a,a,a\pm 1)$ and $f(a+1,a,a-1)$ it wants to show that you can get all numbers not divisible by $3$ or divisible by $9$. These examples are sufficient to get all those numbers. There is no need in this phase to consider any more values of the arguments.
The second phase is to show that numbers divisible by $3$ but not divisible by $9$ cannot be expressed. Here you have to consider other values for the arguments because (so far) it could be that there is some set that will express $3$, for example. It proves this by considering the numbers $\bmod 3$.