The domain of a function $f$ is the set of natural numbers. The function is defined as follows: $f(n) = n + \lfloor{\sqrt{n}}\rfloor$. Prove that for every natural number $m$ the following sequence contains at least one perfect square $m, f(m), f^{2}(m), f^{3}(m),...$
Asked
Active
Viewed 52 times
3
-
This is trivial, but if every such sequence contains at least one square then it must contain infinitely many squares, since we can start the sequence from $f^{(i)}(m)$ for all $i$. – idok Jan 06 '18 at 19:33