A Note on Waring ’ s Number Modulo

The Waring number of the integers modulo m with respect to k-th powers, denoted by ρ(m, k), is the smallest r such that every integer is a sum of r k-th powers modulo m. This number is also the diameter of an associated Cayley graph, called the Waring graph. In this paper this number is computed when m is a power of 2. More precisely the following result is obtained: Let n, s and b be natural numbers such that b is odd, s ≥ 1 and n ≥ 4. Put k = b2. Then (i) if s ≥ n− 2, then ρ(2, k) = 2 − 1. (ii) if k ≥ 6 and s ≤ n− 3, then ρ(2, k) = 2.