Topological Explorations on the Fundamental Theorem of Arithmetic

Abstract

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and investigate non-standard questions. The ideas are arranged as a series of questions for students to explore, with expositions to the instructor on how to guide them in looking for answers. The investigations culminate in a satisfying outcome: that partition topologies are the same as unique factorization topologies, those that satisfy our topological version of the Fundamental Theorem of Arithmetic.

Keywords:

• Capstone course
• seminar
• topology
• basis
• prime numbers
• Fundamental Theorem of Arithmetic

ACKNOWLEDGMENTS

Special thanks to Adalay Sexton for her role in helping us as we explored the foundations of topology and formulated the seed that would become this paper. Great appreciation to Gabriel Cantanelli for his aid in pruning our work into a more clear and practical resource. Especial thanks and appreciation to Hansel for his joyful curiosity and ability to see the wonder in things (Figure 7).

Figure 7. Hansel.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Matthew Phillips

Matthew Phillips loves reading about the history of mathematics and seeing how ideas evolve over time. He holds a BS in Pure Mathematics from the University of Texas at Arlington and continues his graduate studies in Mathematics at the University of North Texas.

Kayla Robb

Kayla Robb enjoys writing about and exploring connections between different disciplines in mathematics. She holds a BS in Mathematics with a concentration in Computer Science from the University of Texas at Arlington and plans to continue with graduate studies in Mathematics.

Barbara A. Shipman

Barbara Shipman enjoys helping people discover beautiful ideas in Mathematics in fun and creative ways. She likes to look for deep answers to simple questions, together with curious students. Dr. Shipman is Distinguished Teaching Professor at the University of Texas at Arlington.