MathDB

6

Part of 2019 Turkey MO (2nd round)

Problems(1)

n-separating (shouting LTE)

Source: Turkey National Mathematical Olympiad 2019, Problem 6

12/24/2019
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.
number theorynumber theory unsolvednumber theory proposed