# 2012 Research Experiences for Undergraduates

Computational Algebraic Geometry, Combinatorics and
Number Theory

Research Projects

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

## Graph NIM

The undergraduate participants in this group were
Allison Nelson, Sydney Ryan and Chao Xu.
They were mentored by recent Clemson PhD graduate
Janine Janoski
They studied
the game of graphical NIM and determined families of graphs
which are first or second palyer wins.
REU Write up

## Extensions of Function Fields

The undergraduate participants in this group were
Alfeen Hasmani, Lindsey Hiltner, Angela Kraft and Daniel Scofield.
They were mentored by Clemson graduate student
Kirsti Wash.
They studied
the finite extensions of the field of formal Laurent series
over **F**_{p}.
REU Write up

Preprint published in the Rocky Mountain Journal of Mathematics

## Aliquot Cycles over Number Fields

The undergraduate participants in this group were
David Heras and Andrew Qian.
They were mentored by Clemson graduate student
Rodney Keaton.
They studied
Aliquot sequences of prime ideals in number fields for elliptic curves.
This work generalizes work of Silverman and Stange who first posed
the generalization of aliquot cycles to elliptic curves.
Preprint submitted for publication

## Frobenius distributions of elliptic curves

The undergraduate participants in this group were
Brandon Tran, Minh-Tam Trinh and Phil Wertheimer.
They were mentored by Clemson graduate student
Dania Zantout.
They studied
questions related to the distribution of primes relative to certain
properties of elliptic curves.
More precisely, they studied the distribution of the traces a_E(p) of
the Frobenius morphism of elliptic curves E. The addressed the question
of how often the trace obtains an extremal value and the question of
how often the trace is itself a prime.
REU Write up