Equal Sums of Like Powers with Minimum Number of Terms

TL;DR

This paper investigates the minimal total number of terms needed for nontrivial solutions to equal sums of like powers, providing exact values and bounds for cases where the power is less than six.

Contribution

It determines the minimal number of terms for solutions to equal sums of like powers for powers less than six, establishing exact and bounded values for these minimal sums.

Findings

(2)=4

(3)=6

7 (4) (8) (5) (10)

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,\,\ldots,\,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 $\beta(2) =4,\;\, \beta(3) = 6,\;\, 7 \leq \beta(4) \leq 8$ and $8 \leq \beta(5) \leq 10$.