On Vanishing Sums of Roots of Unity

An unsolved problem in number theory asked the following: For a given natural number m, what are the possible integers n for which there exist mth roots of unity α1,…,αn ∈ C such that α1 + ··· + αn = 0? We show in this paper that the set of all possible n's is exactly the collection of N-combinations of the prime divisors of m, where N denotes the set of all non-negative integers. The proof is long and involves a subtle analysis of minimal vanishing sums of mth roots of unity, couched in the setting of integral group rings of finite cyclic groups. Our techniques also recovered with ease some of the classical results on vanishing sums of roots of unity, such as those of Redei, de Bruijn, and Schoenberg.