I thought of this problem after thinking of how $n=3$ seems to have no solution to the below problem.
What numbers $n$ exist such that there are $n$ integers $a_1,a_2,a_3,a_4,...,a_n$ such that their standard deviation is a natural number? (Take the numbers as a population)
Overall progress: $n=1$ and $n=2k$ for $k$ is an integer both work.
Progress for the current case I'm working on: $n=3$ is very difficult to find, but thinking a lot, the current problem is to find $x^2+xy+y^2=6p^2$ for an integer $p.$ From there, I could use complex factoring, but the sum of squares is just as unwieldy.
I think that this can be proved/disproved using NT, but I'm stuck on how to go from there.
What do I do now?