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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.

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    $\begingroup$ You mean at most 4 ,as $1=1^2, 2=1^2+1^2,3=1^2+1^2+1^2$ and $5=2^2+1^2$,etc. $\endgroup$ – DanielWainfleet Aug 29 '17 at 10:13
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    $\begingroup$ You might look at "Lagrange's Four-Square Theorem" in Wikipedia. $\endgroup$ – DanielWainfleet Aug 29 '17 at 10:16
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    $\begingroup$ So your Q is now exactly Lagrange's 4-Square Theorem. $\endgroup$ – DanielWainfleet Aug 29 '17 at 10:31
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    $\begingroup$ And you can find proofs on Wikipedia $\endgroup$ – P. Siehr Aug 29 '17 at 10:37
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    $\begingroup$ This link shows, though without a formal proof, that the 4 square representations of an integer N are equivalent in the sense that we can transform one into another. math.stackexchange.com/questions/2337259/… $\endgroup$ – user25406 Aug 29 '17 at 14:01

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