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2018 PAMO Shortlist: Sequence eventually contains a square

Source: 2018 Pan-African Shortlsit - A7

May 6, 2019
algebranumber theoryIterationInteger sequence

Problem Statement

Let f(n)=n+⌊n⌋f(n) = n + \lfloor \sqrt{n} \rfloor. Prove that for every positive integer mm, the integer sequence m,f(m),f(f(m)),…m, f(m), f(f(m)), \dots contains at least one square of an integer.
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