On vanishing sums for roots of unity

Abstract

Consider the $m$-th roots of unity in {\bf C}, where $m>0$ is an integer. We address the following question: For what values of $n$ can one find $n$ such $m$-th roots of unity (with repetitions allowed) adding up to zero? We prove that the answer is exactly the set of linear combinations with non-negative integer coefficients of the prime factors of $m$.