MathDB

B4

Part of 1983 Putnam

Problems(1)

Sequence contains a perfect square

Source: Putnam 1983

6/14/2017
Problem. Let f:R0+→R0+f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+ be a function defined as f(n)=n+⌊n⌋ ∀ n∈R0+.f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+. Prove that for any positive integer m,m, the sequence m,f(m),f(f(m)),f(f(f(m))),…m,f(m),f(f(m)),f(f(f(m))),\ldots contains a perfect square.
number theoryPutnamfunctionSequence