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n-separating (shouting LTE)

Source: Turkey National Mathematical Olympiad 2019, Problem 6

December 24, 2019
number theorynumber theory unsolvednumber theory proposed

Problem Statement

Given an integer n>2n>2 and an integer aa, if there exists an integer dd such that n∣ad−1n\mid a^d-1 and n∤ad−1+⋯+1n\nmid a^{d-1}+\cdots+1, we say aa is n−n-separating. Given any n>2, let the defect of nn be defined as the number of integers aa such that 0<a<n0<a<n, (a,n)=1(a,n)=1, and aa is not n−n-separating. Determine all integers n>2n>2 whose defect is equal to the smallest possible value.
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