If we fixe $n\in \mathbb{N}$. I was wondring if there is an estimation of the number of the integer solutions of the equation : $$x_1^2+x_2^2+\cdots+x_n^2=n^3 $$ where $x_i>0$ for all $i=1,\cdots,n$.
In fact we can notice that if $x_i=n$ for all $i=1,\cdots,n$ is a solution and when we try to solve this equation for n given we find that:
- for $n=1$ there is $1$ solution $(1)$
- for $n=2$ there is $1$ solution $(2,2)$
- for $n=3$ there is $2$ solutions $(3,3,3)$ and $(5,1,1)$
- for $n=4$ there is $1$ solution $(4,4,4,4)$
- $\cdots$
Question: Is there any known formula or estimation for the number of solutions?