MathDB

The domain of a function

f

\mathbb{N}

(The set of natural numbers). The function is defined as follows :

f(n)=n+\lfloor\sqrt{n}\rfloor

where

\lfloor k\rfloor

denotes the nearest integer smaller than or equal to

k

. 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),\cdots

The notation

f^k

denotes the function obtained by composing

f

with itself

k

times.

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