Some new results on equal sums of like powers

Abstract

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