# 2005 Research Experiences for Undergraduates

Computational Number Theory and Combinatorics

Research Projects

** Please note that some of these works are still in progress! **

## Counting Kings.

This group consisted of Shaina Race, Keith Schneider and Matthew Yancey and
was
supervised by Neil Calkin, Kevin James and Shannon Lockard.
They studied the number of placements of nonattacking kings on chessboards
of various shapes and dimensions.

Counting Kings: Explicit Formulas, Recurrence Relations,
and Generating Functions! Oh My!

Counting Kings: As easy as lambda_1, lambda_2, lambda_3,
...

## Lang-Trotter Conjecture.

This group consisted of George Schaefer, Cole South and Julie Wu and
was supervised by Tim Flowers, Neil Calkin and Kevin James.
They extended work of a 2002 REU group to prove that the Lang-Trotter
conjecture is true in an average sense when one averages over elliptic
curves with a rational point of a given order.
REU Report

The Lang-Trotter Conjecture
on average for elliptic curves with torsion.

## Quadratic Sieve

This group consisted of
Katie Field, Tina Little and Ryan Witko and was
supervised by Shannon Purvis, Tim Flowers, Neil Calkin and Kevin James.
They attempted to extend work of our 2004 REU quadratic sieve project and
to better understand the running time for the quadratic sieve factoring
algorithm.
REU Report