Prove any positive integer $n$ can be the summed squares of four special nonnegative integers $a_1, a_2,a_3, a_4$ satisfying both $a_1$ and $a_2+24\cdot a_3$ are perfect squares, such that: $$n=\sum_{k=1}^4a_k^2$$
The problem Lagrange's four-square theorem can be easily verified for most of the small enough positive integers, but the representation might not be unique. For example:
$$316224^2+1504^2+288^2+192={316224^2+1376^2+672^2+192^2}=10^{11}$$
$${316227^2+693^2+64^2+15^2={316227^2+ 693^2+ 55^2+ 36}^2=10^{11}+99}$$
$${9^2+4^2+1^2+1^2}=8^2+5^2+3^2+1^2=99$$
In order to make it unique, we add one more constraint to the four integers: one of the four is a perfect square, and the $a_2+24\cdot a_3$ sum of other two is also a perfect suqare.
Will this continue to hold for :
Any positive integers that can be represented as the summed square of four nonnegative integers ; and the representation is not unique.