Ideal Solutions in the Prouhet-Tarry-Escott problem

For given positive integers m m and n n with m > n m>n , the Prouhet–Tarry–Escott problem asks if there exist two disjoint multisets of integers of size n n having identical k k th moments for 1 ≤ k ≤ m 1\leq k\leq m ; in the ideal case one requires m = n − 1 m=n-1 , which is maximal. We describe some searches for ideal solutions to the Prouhet–Tarry–Escott problem, especially solutions possessing a particular symmetry, both over Z \mathbb {Z} and over the ring of integers of several imaginary quadratic number fields. Over Z \mathbb {Z} , we significantly extend searches for symmetric ideal solutions at sizes 9 9 , 10 10 , 11 11 , and 12 12 , and we conduct extensive searches for the first time at larger sizes up to 16 16 . For the quadratic number field case, we find new ideal solutions of sizes 10 10 and 12 12 in the Gaussian integers, of size 9 9 in Z [ i 2 ] \mathbb {Z}[i\sqrt {2}] , and of sizes 9 9 and 12 12 in the Eisenstein integers.