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Tricky problem about the difference of power of b

Source: Iranian Third Round 2020 Number Theory exam Problem4

November 21, 2020
number theorytotient function

Problem Statement

Prove that for every two positive integers a,ba,b greater than 11. there exists infinitly many nn such that the equation ϕ(an−1)=bm−bt\phi(a^n-1)=b^m-b^t can't hold for any positive integers m,tm,t.
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