PhD thesis defense seminar

Timothy Caley, Pure Mathematics, University of Waterloo

"The Prouhet-Tarry-Escott problem"

Given natural numbers $n$ and $k$, with $n>k$, the Prouhet-Tarry-Escott (\textsc{pte}) problem asks for distinct subsets of $\mathbb{Z}$, say $X=\{x_1,\ldots,x_n\}$ and $Y=\{y_1,\ldots,y_n\}$, such that
\[x_1^i+\ldots+x_n^i=y_1^i+\ldots+y_n^i\] for $i=1,\ldots,k$. 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 $X=_{n-1}Y$ {\it 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, proving generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the \textsc{pte} problem in the Gaussian integers.

We extend some computations finding new lower bounds for the constant
$C_n$ associated to ideal {\sc pte} solutions. Further, we present a new
algorithm that determines whether an ideal {\sc 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 {\sc pte} solutions. We use quadratic twists of curves that appear in the literature to find ideal {\sc pte} solutions over number fields.