New results on an anti-Waring problem

Abstract

The number $N\left(k,r\right)$ is defined to be the first integer such that it and every subsequent integer can be written as the sum of the $k$-th powers of $r$ or more distinct positive integers. For example, it is known that $N\left(2,1\right)=129$, and thus the last number that cannot be written as the sum of one or more distinct squares is 128. We give a proof of a theorem that states if certain conditions are met, a number can be verified to be $N\left(k,r\right)$. We then use that theorem to find $N\left(2,r\right)$ for $1\le r\le 50$ and $N\left(3,r\right)$ for $1\le r\le 30$.