## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Monday, December 17, 2012 — 10:00 AM EST

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.

Location

MC - Mathematics & Computer Building

5046

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.