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 k=n-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.
This thesis focuses on the ideal case. We extend the framework of the problem to number fields,
and prove generalizations of results from the literature. This information is used along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers.
We also extend a computation from the literature and find new lower bounds for the constant C_n associated to ideal PTE solutions. Further, we present a new algorithm that determines whether an ideal PTE solution with a particular constant exists. This algorithm improves the upper bounds for C_n and in fact, completely determines the value of C_6.
We also examine the connection between elliptic curves and ideal PTE solutions. We use quadratic twists of curves that appear in the literature to find ideal PTE solutions over number fields
