Compositional Square

Number Theory Level 5

Define a function \(\displaystyle f(n) = n + \left \lfloor \sqrt{n} \right \rfloor\).

Find the smallest value of \(k\) such that the composite function,
\(f^{k} (2017) = \underbrace{f\circ f \circ f \circ \cdots \circ f}_{k \text{ times}} (2017) \) can be expressed as \(m^2\) for some positive integer \(m\).

Submit your answer as \(m-k\).

\[\] Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

This problem is inspired from a Putnam problem.

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