The Prouhet–Tarry–Escott problem, indecomposability of polynomials and Diophantine equations - Hajdu, L.; Papp, Á.; Tijdeman, R. (The Ramanujan journal, 2022)
In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of A ⊆ { 1 , ⋯ , n } in k -sets such that the first k - 1 symmetric polynomials of the elements of the k -sets coincide. Then we apply this result to derive a decomposability result for the polynomial f A ( x ) : = ∏ x ∈ A ( x - a ) . Finally we prove two theorems on the structure of the solutions ( x , y ) of the Diophantine equation f A ( x ) = P ( y ) where P ( y ) ∈ Q [ y ] and on shifted power values of f A ( x ) .