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Inequality on subsets of primes satisfying conditions

Source: China Mathematical Olympiad 2018 Q1

November 15, 2017
number theoryrelatively primegreatest common divisorinequalities

Problem Statement

Let nn be a positive integer. Let AnA_n denote the set of primes pp such that there exists positive integers a,ba,b satisfying a+bp and an+bnp2\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2} are both integers that are relatively prime to pp. If AnA_n is finite, let f(n)f(n) denote ∣An∣|A_n|.
a) Prove that AnA_n is finite if and only if n≠2n \not = 2.
b) Let m,km,k be odd positive integers and let dd be their gcd. Show that f(d)≤f(k)+f(m)−f(km)≤2f(d).f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).
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