2010/11/04 by Timothy Caley, Caley, Timothy
Mathematics · #11D72 #11Y50 (Primary) 11P05 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D72 #msc:11P05 #msc:11Y50
paper · pdf · doi:10.48550/arxiv.1011.1262
14 pages
arxiv created 2011/02/13 · arxiv updated 2011/02/15
Given natural numbers n and k, with n>k, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of \Z, say X=\x1,...,xn\ and Y=\y1,...,yn\, such that x1i+...+xni=y1i+...+yni 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-1Y 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.