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Product of any two is perfect power

Source: Japan TST 2018 P5

January 25, 2021
number theory

Problem Statement

Prove that there are finitely many positive integers nn such that there exists a subset SS of {1,2,…,n}\{1,2,\ldots ,n\} satisfying the following conditions:
[*]SS has at least ⌊n⌋+1\lfloor \sqrt{n}\rfloor +1 elements [*] For any x,y∈Sx,y\in S, xyxy is a perfect power, i.e. xyxy is of the form aba^b for positive integers aa and b≥2b\ge 2.
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