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prime power criterion with a @ b = \frac{a - b}{gcd(a, b)}

Source: Dutch IMO TST2 2012 p1

January 10, 2020
number theoryPerfect powerprimeGCDgreatest common divisor

Problem Statement

For all positive integers aa and bb, we de ne a@b=a−bgcd(a,b)a @ b = \frac{a - b}{gcd(a, b)} . Show that for every integer n>1n > 1, the following holds: nn is a prime power if and only if for all positive integers mm such that m<nm < n, it holds that gcd(n,n@m)=1gcd(n, n @m) = 1.
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