The Prouhet-Tarry-Escott problem for Gaussian integers
Timothy Caley · Mathematics of Computation · 2012
Given natural numbers n n and k k , with n > k n>k , the Prouhet-Tarry-Escott ( pte ) problem asks for distinct subsets of Z \mathbb {Z} , say X = { x 1 , … , x n } X=\{x_1,\ldots ,x_n\} and Y = { y 1 , … , y n } Y=\{y_1,\ldots ,y_n\} , such that \[ x 1 i + … + x n i = y 1 i + … + y n i x_1^i+\ldots +x_n^i=y_1^i+\ldots +y_n^i \] for i = 1 , … , k i=1,\ldots ,k . Many partial solutions to this problem were found in the late 19th century and early 20th century. When n = k − 1 n=k-1 , we call a solution X = n