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\pi(m)-\pi(n)<= (m-1)\varphi(n)/n

Source: Kosovo TST 2020 Problem 4

February 8, 2020
number theory

Problem Statement

Prove that for all positive integers mm and nn the following inequality hold: π(m)π(n)(m1)φ(n)n\pi(m)-\pi(n)\leq\frac{(m-1)\varphi(n)}{n} When does equality hold?
Proposed by Shend Zhjeqi and Dorlir Ahmeti, Kosovo