Computational investigations of the Prouhet-Tarry-Escott Problem

We describe a method for searching for ideal symmetric solutions to the Prouhet-Tarry-Escott Problem. We report results of extensive searches for solutions of sizes up to 12. We found two solutions of size 10 that are smaller by two orders of magnitude than the solution found by A. Letac in the 1940s, which was the smallest size 10 solution known before our search. 1. The Prouhet-Tarry-Escott Problem The Prouhet-Tarry-Escott Problem (PTE Problem) is an old unsolved problem in Diophantine number theory. In its most general setting the PTE Problem asks for two distinct multisets of integers X = {x1, . . . , xn} and Y = {y1, . . . , yn} such that