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Improved Bounds on the Anti-Waring Number

The Anti-Waring number, N(k, r), is defined to be the least integer such that it and every larger integer can be written as the sum of the kth powers of r or more distinct positive integers. Several authors have examined this variation of the classical Waring problem. We provide improved bounds for N(k, r) in general and when k = 2. We then connect this problem to the theory of partitions. We use traditional counting arguments, as well as a generating function methodology that has not yet been applied to finding the Anti-Waring number.