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Well-known is the fact that every natural number can be represented as the sum of four integer squares — this proposition is often called Lagrange’s four-square theorem.

Perhaps less well-known is the fact that every odd natural number can be represented as the sum of four integer squares such that the sum of those integers is every odd number from $1$ to “the maximum”. For example, $$ 17 = (\pm 3)^2 + (\pm 2)^2 + (\pm 2)^2+0^2 = (\pm 4)^2 + (\pm 1)^2 + 0^2 + 0^2, $$ where $(-3)+2+2=1$ and $3+2+(-2)=3$ and $4+1=5$ and $3+2+2=7$, and the smallest set of roots summing to $9$ would have a sum of squares $\ge 19$.

This theorem was stated by Pollock, who — in the classical way — provided voluminous evidence and fascinating tables (see, for example, this paper), but no algebraic proof, as far as I can tell.

I have two questions related to this second result.

Question 1: Has a proof ever been published of Pollock’s statement?

Questions 2: Is there a formula which explicitly gives “the maximum” root-sum for any odd natural number $n$?

With regard to the second question: It is trivial, of course, to calculate the upper bound of such a “maximum”, since $n \le (\sqrt{n})^2 + a^2+b^2+c^2$. But I’m hoping there is a formula which gives the largest possible root-sum.

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    $\begingroup$ For reference, see the paragraph about Pollock toward the bottom of page 292 at books.google.com/… $\endgroup$ – Barry Cipra Sep 9 '16 at 15:31

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