Tessellate S.T.E.M.S - Mathematics - School - Set 1 - Problem 1

Suppose $f:\mathbb{N} \rightarrow \mathbb{N}, \hspace{6pt} (\mathbb{N}=\{ 1,2,3, \cdots \})$ is a strictly increasing function such that the image of $f$ does not contain consecutive integers. Suppose, $P$ is a polynomial with coefficients as positive integers and $f(m)=P(m)$ for all perfect square integers $m$ . Under which of the following conditions on $P$ does the given data determine $f$ uniquely?

This problem is a part of Tessellate S.T.E.M.S.