Vol. 7, No. 2, 2014

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New results on an anti-Waring problem

Chris Fuller, David R. Prier and Karissa A. Vasconi

Vol. 7 (2014), No. 2, 239–244
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 $1\le r\le 30$.

Keywords
number theory, Waring, anti-Waring, series
Primary: 11A67
Milestones
Received: 24 April 2013
Revised: 10 July 2013
Accepted: 24 July 2013
Published: 16 November 2013

Communicated by Nigel Boston
Authors
 Chris Fuller Department of Mathematics Cumberland University Lebanon, TN 37087 United States David R. Prier Department of Mathematics Gannon University Erie, PA 16541-0001 United States Karissa A. Vasconi 1440 Heinz Avenue Sharon, PA 16146 United States