Some new results on equal sums of like powers

Abstract:

The Diophantine equation $\sum \nolimits _{i = 1}^M {x_i^n = \sum \nolimits _{i = 1}^M {y_i^n} }$ is examined for $n = 3,4$ and $6$ and $M = [(n + 1)/2]$. A method for generating parametric solutions for $n = 4$ is derived and several new numerical examples for $n = 4,6$ are given. The method also applies for all other values of $M$ and possibly for values of $n$ greater than 6, too.