Equal Sums of Like Powers with Minimum Number of Terms

Abstract

This paper is concerned with the diophantine system, ${\sum }_{i=1}^{s_1}{x}_i^r={\sum }_{i=1}^{s_2}{y}_i^r,r=1,2,\dots ,k,$ where $s_1$ and $s_2$ are integers such that the total number of terms on both sides, that is, $s_1+s_2,$ is as small as possible. We define $\beta (k)$ to be the minimum value of $s_1+s_2$ for which there exists a nontrivial solution of this diophantine system. We find nontrivial integer solutions of this diophantine system when $k<6$, and thereby show that $\beta (2)=4,\beta (3)=6,7\le \beta (4)\le 8$ and $8\le \beta (5)\le 10$.

14 pages