Minimal Vanishing Sums of Roots of Unity with Large Coefficients

A vanishing sum a0 + a1ζn + . . .+ an−1ζ n−1 n = 0 where ζn is a primitive n-th root of unity and the ai’s are nonnegative integers is called minimal if the coefficient vector (a0, . . . , an−1) does not properly dominate the coefficient vector of any other such nonzero sum. We show that for every c ∈ N there is a minimal vanishing sum of n-th roots of unity whose greatest coefficient is equal to c, where n is of the form 3pq for odd primes p, q. This solves an open problem posed by H.W. Lenstra Jr.