How do I find a list (size n) of integers where the root-mean-square of the list is an integer?

Question

I already found this one, but it discusses mostly brute force. Brute force is possible of course, but are there any other ways? Is there a way to find all lists? Is there a way to find out how many combinations exist under a certain limit?

Also related: same but for first n integers

@MichielvanderBlonk So, you do want to allow repeated elements? — Peyton, Jul 30 '17 at 21:52
If $x^2+y^2=z^2$ then $(x-y)^2+(y+x)^2=2z^2$, so $(x-y,x+y)$ ives the case when $n=2$. — Thomas Andrews, Jul 30 '17 at 22:39

score 0 · Answer 1 · answered Jul 31 '17 at 00:35

If you want a list of $n$ integers whose root-mean-square equals the integer $t$, then you are looking for solutions in integers $x_1,\dots,x_n$ of $x_1^2+\cdots+x_n^2 = nt^2$. The solutions of $x_1^2+\cdots+x_n^2 = K$ for an integer $K$ are pretty well-known: for example, there are elementary exact formuals for how many there are when $n=2$ and $n=4$ (at least), an elementary characterization of when they exist for $n=3$, and an asymptotic formula (existence guaranteed) for the number of solutions when $n\ge5$. So not only are there lots of ways to generate solutions, but you can even aim for your favorite value of $t$.

a starting point: https://en.wikipedia.org/wiki/Sum_of_squares#Number_theory — Greg Martin, Jul 31 '17 at 21:46

How do I find a list (size n) of integers where the root-mean-square of the list is an integer?

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