#### Thesis Title

Prime And Irreducible Elements In Z_X , Hamiltonian Cycles In Concentric Circle Graphs, And Fibonacci Sequences In Z_X

#### Date of Graduation

Summer 2000

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Kishor Shah

#### Subject Categories

Mathematics

#### Abstract

This thesis is composed of three distinct works. In the first part, we characterize the set of all prime elements and the set of all irreducible elements in Zx. Our characterization shows that the set of irreducible elements is contained in the set of prime elements; in particular, our results imply that 3, 9 and 27 are prime elements, but not irreducible elements in Z₆₀. In the second part, we define a new type of graph: a concentric circle graph. We develop the theory of concentric circle graphs and present polynomial time algorithms for determining Hamiltonian cycles in certain types of concentric circle graphs. In the third part, we discuss the Fibonacci sequence in Zx. All of the theorems stated in our thesis are new to us.

#### Copyright

© Christopher Mueller

#### Recommended Citation

Mueller, Christopher, "Prime And Irreducible Elements In Z_X , Hamiltonian Cycles In Concentric Circle Graphs, And Fibonacci Sequences In Z_X" (2000). *MSU Graduate Theses*. 1004.

https://bearworks.missouristate.edu/theses/1004

**Dissertation/Thesis**