The number of zero sums modulo m in a sequence of length n

We prove a result related to the Erdős‐Ginzburg‐Ziv theorem: Let p and q be primes, α a positive integer, and m ∈{ p α , p α q }. Then for any sequence of integers c = { c 1 , c 2 ,…, c n } there are at least(⌊ 1 2 n ⌋m) + (⌈ 1 2 n ⌉m)subsequences of length m , whose terms add up to 0 modulo m (Theorem 8). We also show why it is unlikely that the result is true for any m not of the form p α or p α q (Theorem 9).