WARING’S NUMBER MOD M

Abstract. Let p be an odd prime and γ(k, pn) be the smallest positive integer s such that every integer is a sum of s k-th powers (mod pn). We establish γ(k, pn) ≤ [k/2] + 2 and γ(k, pn) √k provided that k is not divisible by (p − 1)/2. Next, let t = (p − 1)/(p − 1, k), and q be any positive integer. We show that if φ(t) ≥ q then γ(k, pn) ≤ c(q)k1/q for some constant c(q). These results generalize results known for the case of prime moduli. 1