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.

Compositional Square

This problem is inspired from a Putnam problem.