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Permutation of 1,2,...,n and complete residue system

Source: Serbian National Olympiad 2012, Problem 4

April 4, 2012
modular arithmeticnumber theory proposednumber theory

Problem Statement

Find all natural numbers nn for which there is a permutation (p1,p2,...,pn)(p_1,p_2,...,p_n) of numbers (1,2,...,n)(1,2,...,n) such that sets {p1+1,p2+2,...,pn+n}\{p_1 +1, p_2 + 2,..., p_n +n\} and {p1−1,p2−2,...,pn−n}\{p_1-1, p_2-2,...,p_n -n\} are complete residue systems mod  n\mod n.
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