First suppose that $a$ is not a square hence $p^{2s+1}||a$ where $p$ is prime. If $p^{2s+2}|b$ then $p^{2s+1}||a+bc$ for all $c\in N^+$ and $a+bc$ is never a square. Now suppose that $a=k^2$ and take any $b\in N^+$. Then $a+(2k+b)b=(k+b)^2$ hence we can take $c:=2k+b$.