IMO 2011 Shortlist N8

Let k be a positive integer and set n = 2k + 1. Prove that n is a prime number if and only if the following holds: there...

imomathematicsolympiadshortlistnumber-theory

IMO 2011 Shortlist N8

Category: Number Theory

Problem

Let k be a positive integer and set n = 2k

    1. Prove that n is a prime number if and only if the following holds: there is a permutation a1,...,an−1 of the numbers 1,2,...,n − 1 and a sequence of integers g1,g2,...,gn−1 such that n divides gai i −ai+1 for every i ∈ {1,2,...,n−1}, where we set an = a1.