Novel Introductions to Number Theory

Thursday, July 27, 2017, 3 pm – 4:35 PM, Salon C-6

This session invites presenters to share interesting ways in which to introduce undergraduate students to topics in number theory. These “tastes” of number theory may be demonstrations, in-class activities, projects, proofs, or ways in which to guide undergraduates to explore and learn about areas of number theory while improving their ability to write proofs. Those discussing demonstrations or in-class activities are encouraged to share key portions. Presenters are welcome to share their first experiences teaching topics in number theory or how they have modified their approaches over time. Presentations related to teaching topics with which students experience difficulty and student reaction as well as information about successes and failures are encouraged. Abstracts should provide a glimpse of the demonstration, in-class activity, project, or proof to be discussed and information about the related topics in number theory in addition to the software or application, if any, used. Those whose presentations are dependent upon software or tablet explorations must provide their own laptop or tablet.

Organizer: Sarah L. Mabrouk, Framingham State University

3 PM – 3:15 PM	A Group Activities Approach to Number Theory Stefan Erickson, Colorado College Presentation File
	Number theory has long served as our “introduction to proofs” course at Colorado College. Our innovative block schedule provides extended periods of classroom time and intensive focus on one subject at a time. This flexibility has encouraged our professors to create an active learning environment. Over the years, I have developed handouts that guide students to find patterns in numbers. Topics include Pythagorean triples, linear Diophantine equations, Euler’s Theorem and totient function, and quadratic reciprocity. Through working in small groups in class, students discover the natural beauty of number theory and stoke their interest in theoretical mathematics. I will begin by briefly presenting my general philosophy about teaching number theory under Colorado College’s Block Plan. I will also share a sample of my in-class worksheets. My hope is that the audience will take away some new ideas of how to make number theory fun and engaging to their students.
3:20 PM – 3:35 PM	Presenting MAA Articles on Number Theory Susan H. Marshall, Monmouth University
	In this presentation, we’ll talk about an assignment given to my junior Number Theory class last year. The course carries a general education requirement entitled “reasoned oral discourse.” Students must “orally present mathematical ideas and information in a reasoned and effective manner with attention to elements of vocal and nonverbal quality.” This requirement gave rise to our final project, where students were assigned an article to read and then present the results to the rest of the class. Articles in MAA publications such as College Mathematics Journal and Math Horizons (especially those that had won writing awards) were chosen to ensure the students were able to understand the content. A happy side effect was the range of topics all students were exposed to during the presentations, including typical topics such as the digits of pi and innovative topics such as applying number theory to video games. Equally satisfying was their increased enthusiasm for the subject. We’ll discuss the particulars of the assignment, how it went, and how it might be improved in future semesters. with attention to elements of vocal and nonverbal quality.
3:40 PM – 3:55 PM	The Wehmueller Conjecture Everette L. May, Salisbury University
	This talk is the story of a student’s attempt to build on the ideas of Fermat’s Last Theorem. In the spring of 2010, after watching The Proof , the NOVA/BBC video of Andrew Wiles’s successful quest to prove the theorem, Kara Wehmueller, one of my students in discrete mathematics, stated that, instead of trying to use three integers a, b, and c such that a3 + b3 = c3, one should look for four integers a, b, c, and d (a “Wehmueller Quadruple” ) such that a3 + b3 + c3 = d3. In general, she said, for any integer n > 1, one should search for n + 1 integers x(1), x(2),... x(n), x(n + 1) such that x(1)n + x(2)n + ... + x(n)n = x(n + 1)n. Additionally, she exhibited the Wehmueller Quadruple (3,4,5,6). Finally, she stated the following: The Wehmueller Conjecture. For each integer n > 1 there is a “Wehmueller n-tuple.” Thus began a new “Fermatian” quest. The talk will detail the (unfinished) history of that quest, suggest some directions in which it might proceed, and seek the help of the audience in resolving the Wehmueller Conjecture.
4 PM – 4:15 PM	Some Interesting Infinite Families of Primitive Pythagorean Triples David Terr, UC Berkeley
	In this paper we investigate families of primitive Pythagorean triples of the form (a, b, c), where mc − nb = t, mc − na = t or mb − na = t for some fixed positive coprime integers m and n, and t a fixed nonzero integer. A few of these cases are especially interesting since the solutions may be simply written in terms of Fibonacci and Lucas numbers.
4:20 PM – 4:35 PM	Arithmetical Structures on Graphs Darren Glass, Gettysburg College Presentation File
	In this talk, we will discuss the concept of an arithmetical structure on a finite connected graph, first introduced by Dino Lorenzini. While these structures are typically defined in terms of linear algebra and have interesting applications in algebraic geometry and algebraic combinatorics, they also can be framed in terms of elementary number theory in a way that provides an entry point to students to think about open questions related to divisibility and congruence.

This page was created and is maintained by S. L. Mabrouk, Framingham State University.

This page was last modified on Monday, August 14, 2017.