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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 $y=(y_1,y_2,y_3,y_4,0,\dots,0)\in\mathbb Z^k$?

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