Novel Introductions to Number Theory
Thursday, July 27, 2017, 3 pm
– 4:35 PM, Salon C6
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, inclass 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 inclass 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, inclass 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 

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 inclass 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 

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 

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 ntuple.” 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 

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 

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.