<s>[INST] Powers for the sum of three squares Say $n$ can be represented as a sum of three non-zero squares. (i.e. $n = a^2 +b^2+ c^2$, for some $n,a,b,c \in \mathbb{N}$)
Is it possible that every natural power of $n$ is also a sum of three non-zero
squares? (i.e. $n^k = x^2+y^2+z^2$ for $x,y,z,k \in \mathbb{N}$)
 [/INST] Yes, more generally, if,
$$N = x_1^2+x_2^2+\dots+x_m^2$$
then one can always find integer $y_i$ such that,
$$N^k = y_1^2+y_2^2+\dots+y_m^2$$

Proof: Use the expansion of
$$(a+i\sqrt{w})^k$$
To get,
$$a^2 + w = N,\\ (a^2 - w)^2 + (2 a)^2 w = (a^2 + 
        w)^2,\\ (a^3 - 3 a w)^2 + (3 a^2 - w)^2 w = (a^2 + 
        w)^3,\\ (a^4 - 6 a^2 w + w^2)^2 + (4  a^3 - 4  a w)^2 w = (a^2 + w)^4$$
and so on, then assume $w$ to be a sum of squares $w=b^2+c^2+\dots$ 
For example, for $k=3$,
$$\small\big(a^3 - 3 a (b^2+c^2+\dots)\big)^2 + \color{blue}{\big(3 a^2 - (b^2+c^2+\dots)\big)^2 \big(b^2+c^2+\dots\big)} = (a^2 + b^2+c^2+\dots)^3$$
then distribute the blue term. I trust the pattern for other $k$ is easy to discern.
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