On Prouhet's solution to the equal powers problem

Eugène Prouhet is considered to be one of the fathers of combinatorics on words. In his pioneering work, he constructed a solution to the equal powers problem in number theory (later known as the Prouhet-Tarry-Escott problem) based on positions of symbols in words obtained by iteration of a uniform endomorphism of the free monoid over a finite alphabet. We show that composition of uniform morphisms of free monoids is a fairly powerful tool to obtain solutions to a more general problem, which covers the major generalizations of Prouhet's theorem appearing in literature.