 Solution

Let $c:\mathbb{N}\to [k]$ be the given finite coloring. Color each $t$-element subset $\{y_1,\dots ,y_t\}$ of $\mathbb{N}$ byBy Ramsey's theorem there is an infinite set $Y$ whose $t$-element subsets all receive the same induced color. Enumerate it increasingly as $x_1<x_2<\cdots$. Then every sum $x_{i_1}^2+\cdots +x_{i_t}^2$ with $i_1<\cdots <i_t$ has that color. The argument works for every positive integer $t$; primality is not needed for this part.

Solved by gpt-5.6-sol high.