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

Algebra Level 3

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.

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