# 2004 Research Experiences for Undergraduates

Computational Number Theory and Combinatorics

Research Projects

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

## Ramsey Theory and Saturated Graphs.

This group consisted of Kelty Allen and Jay Heumann and was
supervised by Neil Calkin and Kevin James. They implemented algorithms to
study Ramsey numbers on cyclic graphs and to study staurated graphs.

Ramsey Theory and Saturated Graphs Report

Expected Number of four-tuples in a random cyclic graph.

## Quadratic Sieve

This group consisted of Kim Bowman and Zach Cochran and was supervised
by Neil Calkin and Kevin James. They studied various probability models
which predict how many smooth numbers one must encounter before being able
to construct a square.
Quadratic Sieve Report

## Even and Odd graphs and the Congruent Number Elliptic Curves.

This group consisted of Morgan Brown, Adam King and Rob Rhoades and was
supervised by Neil Calkin and Kevin James. This group studied links between
the Selmer groups of the congruent number elliptic curves and certain types
of graphs. They calculated the probability that a graph randomly chosen from
certain types of graphs will be even.
Even and Odd Graphs Report.

Additional report on symmetric matrices.

## Modular Forms.

This group consisted of Yara Luis and Chelsea Walton. They were assisted
by two Clemson graduate students, Tim Flowers and Ethan Smith
and supervised by Neil Calkin and Kevin James.
Modular Forms Report