Ideal solutions in the Prouhet-Tarry-Escott problem

Abstract

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