Are the following conjectures true?:
(a) For every integer $n$ there exist a number $k=2^q$ (for some positive integer $q$) and positive integers $n_1,\cdots,n_k$ such that $$ n=n_1^2+\cdots+n^2_{\frac{k}{2}}-(n^2_{\frac{k}{2}+1}+\cdots+n^2_k). $$ Now, denote by $k(n)$ the least $k$ obtained from (a) (if it is true).
(b) The set of all $k(n)$, where $n$ runs over all integers, is unbounded above.
$$1=3^2+3^2-(4^2+1^2).$$ If $n$ is a square, say, $n=k^2$, then $$n=(3k)^2+(3k)^2-((4k)^2+k^2).$$ If $n$ is not a square, and is not $2\bmod4$, then there exist $r,s$ such that $$n=r^2-s^2;$$ moreover, $s\ne0$ since $n$ is not a square. If $n=4k+2\ge6$, then $4k-1=r^2-s^2$ for some nonzero $r,s$, so $$n=r^2+2^2-(s^2+1^2).$$ Finally, $$2=3^2+1^2-(2^2+2^2).$$