# Sum of four squares from sum of $k\geq5$ squares - scaled version $\mathsf I$

Every natural number is sum of $$4$$ squares and every natural number bigger than $$169$$ is sum of five squares.

Suppose $$n>169$$ and $$x'x=n$$ at a vector $$x\in\mathbb Z^5$$ then can we find a matrix $$M\in\mathbb Z^{5\times5}$$ with

a. Each entry of $$M$$ are bound by in absolute value by $$r'_{n,5}$$ at some $$r'_{n,5}\in\mathbb Z$$

b. $$(M')^{-1}(M)^{-1}=r_{n,5}I_{5\times 5}$$ at some $$r_{n,5}\in\mathbb Z$$

c. $$Mx=y\in\mathbb Z^5$$ with $$y=(y_1,y_2,y_3,y_4,0)\in\mathbb Z^5$$ form

such that $$y'y=r_{n,5}n$$ holds?

1. If so what is the minimum $$r_{n,5}$$ and $$r'_{n,5}$$ achievable at given $$n$$?

2. If we replace $$5$$ by $$k$$ then what is the minimum $$r_{n,k}$$ where $$y$$ has form $$y=(y_1,y_2,y_3,y_4,0,\dots,0)\in\mathbb Z^k$$?