The two-dimensional Prouhet–Tarry–Escott problem Original Research Article

In this paper we generalize the Prouhet–Tarry–Escott problem (PTE) to any dimension. The one-dimensional PTE problem is the classical PTE problem. We concentrate on the two-dimensional version which asks, given parameters image, for two different multi-sets {(x1,y1),…,(xn,yn)}, image of points from image such that image for all d,jset membership, variant{0,…,k} with jless-than-or-equals, slantd. We present parametric solutions for nset membership, variant{2,3,4,6} with optimal size, i.e., with k=n−1. We show that these solutions come from convex 2n-gons with all vertices in image such that every line parallel to a side contains an even number of vertices and prove that such convex 2n-gons do not exist for other values of n. Furthermore we show that solutions to the two-dimensional PTE problem yield solutions to the one-dimensional PTE problem. Finally, we address the PTE problem over the Gaussian integers.