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equivalent with n being even

Source: Polish MO Recond Round 1985 p5

September 9, 2024
number theoryEven

Problem Statement

Prove that for a natural number nn greater than 1, the following conditions are equivalent:
a) n n is an even number,
b) there is a permutation (a0,a1,a2,…,an−1) (a_0, a_1, a_2, \ldots, a_{n-1}) of the set {0,1,2,…,n—1} \{0,1,2,\ldots,n—1\} with the property that the sequence of residues from dividing by n n the numbers a0,a0+a1,a0+a1+a2,…,a0+a1+a2+…an−1 a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots, a_0 + a_1 + a_2 + \ldots a_{n-1} is also a permutation of this set.
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