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Permutation of the set gives all the residues modulo n

Source: 1st TASIMO Day2, Problem6

May 19, 2024
abstract algebranumber theorycomplete residue system

Problem Statement

We call a positive integer n≥4n\ge 4 beautiful if there exists some permutation {x1,x2,…,xn−1}\{x_1,x_2,\dots ,x_{n-1}\} of {1,2,…,n−1}\{1,2,\dots ,n-1\} such that {x11, x22, …,xn−1n−1}\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\} gives all the residues {1,2,…,n−1}\{1,2,\dots, n-1\} modulo nn. Prove that if nn is beautiful then n=2p,n=2p, for some prime number p.p.
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