Let $k$ and $s$ be odd positive integers such that \[\sqrt{3k-2}-1 \le s \le \sqrt{4k}.\] Show that there are nonnegative integers $t$, $u$, $v$, and $w$ such that \[k=t^{2}+u^{2}+v^{2}+w^{2}, \;\; \text{and}\;\; s=t+u+v+w.\]
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Tags: Additive Number Theory
Let $k$ and $s$ be odd positive integers such that \[\sqrt{3k-2}-1 \le s \le \sqrt{4k}.\] Show that there are nonnegative integers $t$, $u$, $v$, and $w$ such that \[k=t^{2}+u^{2}+v^{2}+w^{2}, \;\; \text{and}\;\; s=t+u+v+w.\]