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## 2015 Summer Undergraduate Applied Mathematics Institute

### Projects:

Abstract: In 1950, Giussipe Giuga conjectured that an integer $n$ satisfies $\sum\limits_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$ if and only if $n$ is prime. Sixty-five years later and this problem is yet to be solved. The complexity of working in the integers has indeed proven challenging. To explore this problem further, we consider the Generalized Giuga Conjecture for ideals in number rings. We introduce the idea of correspondence between weak Giuga numbers and weak Giuga ideals. These concepts are further developed in the quadratic extensions.
• Ant Colony Optimization Applied to the Bike Sharing Problem, Cashous Bortner, Can Gürkan (download paper)
• Subrings of $\mathbb{C}$ Generated by Angles, Jackson Bahr, Arielle Roth (download paper)
Abstract: In [1], Buhler et al. considered the following scenario. Given a collection $U$ of unit magnitude complex numbers and a set $S$ of constructed points initially containing just $0$ and $1$, through each constructed point draw lines whose angles with the real axis are in $U$. The intersections of such lines are also constructed points. Upon taking the closure we form a set $R(U)$. They investigated which $U$ result in $R(U)$ being a ring.
Our main result holds for when $1 \in U$ and $\vert{U}\vert \ge 4$. We classify $R(U)$ as the set of linear combinations of elementary monomials which are the points constructed in the first step. The coefficients are taken from $Z[P] = R(U) \cap R$ which is easily calculated. We also show that when $\vert{U}\vert \ge 4$, $R(U)$ is dense in the complex plane. Furthermore, we classify $R(U)$ completely for when $1 \in U$ and $\vert{U}\vert \ge 3$, showing that $R(U)$ is a ring whenever one of the points constructed in the first step is a quadratic integer.
• Rainbow Numbers with Respect to $2$-Matchings and $3$-Matchings, Kate Borst, Jüergen Kritschgau (download paper)
Abstract: Our results focus on the rainbow numbers of the various graphs with respect to $M_2$ and $M_3$. We find the rainbow numbers for all graphs with respect to $M_2$. From then on out, the number of troublesome cases increases for rainbow numbers with respect to $M_3$. We prove that the rainbow numbers of trees with a diameter of 6 or greater have $rb(T,M_3)=\Delta +2$. We extend this result to all graphs with diameter 6 or greater. Our results suggest that $rb(G, M_3)= \Delta +2$ for unconnected graphs $G$; this is an area for further study.