The Prouhet-Tarry-Escott problem for Gaussian integers

TL;DR

This paper extends the Prouhet-Tarry-Escott problem to Gaussian integers, providing new theoretical results and computationally discovering ideal solutions in this complex integer setting.

Contribution

It generalizes existing results to Gaussian integers and finds new ideal solutions using both theoretical and computational methods.

Findings

Generalized PTE problem to Gaussian integers

Proved new theoretical results in this setting

Discovered new ideal solutions computationally

Abstract

Given natural numbers $n$ and $k$, with $n>k$, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of $\Z$, say $X=\{x_1,...,x_n\}$ and $Y=\{y_1,...,y_n\}$, such that \[x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for $i=1,...,k$. Many partial solutions to this problem were found in the late 19th century and early 20th century. When $n=k-1$, we call a solution $X=_{n-1}Y$ ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. In 2007, Alpers and Tijdeman gave examples of solutions to the PTE problem over the Gaussian integers. This paper extends the framework of the problem to this setting. We prove generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers.