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

Algebra Level 3

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

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

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