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.
by
Kartik Sharma