Minimal vanishing sums of roots of unity with large coefficients

A vanishing sum a 0 + a 1 ζ n + … + a a − 1ζ n n − 1 = 0 , where ζ n is a primitive n th root of unity and the a i s are non‐negative integers is called minimal if the coefficient vector ( a 0 , … , a n −1 ) does not properly dominate the coefficient vector of any other such non‐zero sum. We show that for every c ∈ℕ there is a minimal vanishing sum of n th roots of unity with its greatest coefficient equal to c , where n is of the form 3 pq for odd primes p , q . This solves an open problem posed by Lenstra Jr.