Questions and important info in italics, very important ones in bold.
Here we have the system;
$V_{1}+V_{2}\cdots+V_{k}=A$ and $V_{1}^{2}+V_{2}^{2}+\cdots +V_{k}^{2}=B$ where $V_{1}$, $V_{2}$, etc. are distinct, positive integer variables.
According to my previous thread, Jykri Lahtonen assumes that the number of common solutions for the two equations as $k$ increases remains linear in the system, excluding permutations.
But how much, exactly, excluding permutations of values of the variables in the solution across them (simply divide by $k!$)?
And, how do you solve the two equations? Since, even if one of the many solutions are found, I assume the rest can be found easily applying certain equations. For example (which I later found was already stated by Thomas Andrews, IIRC), for any one set of variables $V_{1}$, $V_{2}$, etc. you can observe;
$$V_1^2+V_2^2+\cdots+V_k^2 = \left(\frac{2\left(V_{1}+V_{2}+\cdots+V_{k}\right)}{k}-V_{1}\right)^{2}+\left(\frac{2\left(V_{1}+V_{2}+\cdots+V_{k}\right)}{k}-V_{2}\right)^2+\cdots+\left(\frac{2\left(V_{1}+V_{2}\cdots+V_{k}\right)}{k}-V_{k}\right)^{2}$$
the resulting values will satisfy the system aforementioned iff twice the average of the variables is a whole number.
Assuming I could employ a computer to solve the system of equations, would it be extremely complex as the value for $k$ grows into the thousands, and so on?
Again, I'm completely lost as to what tags describe this topic perfectly. My apologies.