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Two weird numbers are coprime

Source: Kazakhstan National Olympiad (10-11).5

March 22, 2023
number theory

Problem Statement

Given are positive integers a,b,m,ka, b, m, k with k≥2k \geq 2. Prove that there exist infinitely many nn, such that gcd⁡(φm(n),⌊an+bk⌋)=1\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1, where φm(n)\varphi_m(n) is the mm-th iteration of φ(n)\varphi(n).
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