can $4^{2n }$ be written as the sum of "three squares"?

Question

Lagrange theorem says only numbers $n \neq 4^n ( 8k+7)$ can be written as the sum of three squares. what about this one?

$$ 4= 2^2 + 0^2+ 0^2 $$

this looks acceptable to me, and yet it is excluded by Lagrange theorem.

https://en.m.wikipedia.org/wiki/Legendre%27s_three-square_theorem

Answer 1

It's not excluded; you cannot find $n$ and $k$ so that $4 = 4^n(8k + 7)$.