My Favorite Number Theory Proof
Saturday,
August 3, 2019, 9 AM – 11:20 AM, Duke Energy Convention Center, Rooms 207 &
208
Session Summary:
This session invites presenters to share favorite
number theory proofs suitable for introduction to proofs courses or
undergraduate number theory courses, but not graduatelevel number theory
courses. While nonstandard proofs for the Fundamental Theorem of Arithmetic or
the Chinese Remainder Theorem may be submitted for consideration, standard
versions of these proofs and other typical proofs such as the irrationality of
the square root of two, for example, will not be considered. Presenters must do
the full proof, discuss how the proof fits into the course, provide information
regarding prerequisite topics for the proof, and discuss associated areas with
which students experience difficulty and how such concerns are addressed so
that students understand the proof. Presenters are invited to discuss how they
have modified the proof over time, share historical information related to
“classic” proofs, and discuss explorations/demonstrations which they use to
help students comprehend related theorems and topics. Abstracts should include
the theorem to be proved/discussed as well as brief background information.
Summary for Abstracts Booklet:
Presenters share favorite proofs suitable for
introductory proofs or undergraduate number theory courses, giving the complete
proof, discussing how the proof fits into the course, providing information
regarding prerequisite topics, areas of difficulty, and making the proof
accessible for students. Modifications to the proof over time, historical
information, and explorations/demonstrations used to make related theorems/topics
comprehensible for students are discussed.
Organizer: Sarah L. Mabrouk, Framingham State University
9 AM – 9:15 AM 
Divisibility,
Modular Arithmetic, and Induction, Oh My! Martha
H. Byrne, Sonoma State University 

For any
natural number n, n^{3} – n is
divisible by 6. Actually, this is true of any integer as well, and the proof
techniques used don't change. It is well situated in an introduction to
proofs course, as it can be proved in a variety of ways. Many novice proof
writers struggle to find the “right” way to prove a claim to their detriment.
In this talk, the presenter will go over several different proofs and talk
about how to draw out student discussion about proof techniques that bridge
the different approaches and support student understanding of multiple
proving paths. 


9:20 AM – 9:35 AM 
Various
Teaching Strategies to Prove that a Certain Conjecture is Equivalent to
Goldbach’s Conjecture Kristi
Karber, University of Central Oklahoma 

One of the
most famous conjectures in Number Theory authored by Goldbach states, “Every
even integer greater than 2 can be expressed as the sum of two primes.” It
can be shown that the conjecture, “Every integer greater than 5 is the sum of
three primes,” is equivalent to Goldbach’s conjecture. In this talk we
provide various strategies that can be used to help students understand or
construct a proof which verifies the aforementioned conjectures are indeed
equivalent. Possible strategies include an exploration, as well as an
emphasis on beginning proof writing skills such as utilizing definitions and
proof techniques. A proof will be provided while demonstrating the various
approaches. 


9:40 AM – 9:55 AM 
Fermat’s
Bracelets and Wilson's Polygons: Seeing Two Foundational Theorems
Geometrically Adam
J. Hammett, Cedarville University 

Nearly
every undergraduate course in number theory will include a presentation of
both Fermat's little theorem and Wilson's theorem. Let p be a prime. The
former states that for any given integer n > 0 we must have p  n^{p}
– n, and the latter that p  (p – 1)! + 1. Most textbooks include proofs
that, while elegant and brief, tend to labor in the realm of the purely
analytical. However, George Andrews in his Number Theory text [pp.
3640, Dover Publications, Inc., New York, 1994; MR0309838]
gives remarkably simple, selfcontained proofs of these results that rely
only on symmetries of certain colored bracelets (in the case of Fermat) and
special polygons (Wilson). In this talk, we will endeavor to present both of
these gems in their entirety. 


10 AM – 10:15 AM 
Euler’s
Criterion Scott
Williams, University of Central Oklahoma 

If a
is an integer relatively prime to m, then we say a is a
quadratic residue modulo m if a is a perfect square modulo m
(i.e., if has a solution). For p an odd prime,
the Legendre symbol takes a value of 1 if a is a
quadratic residue modulo p and 1 otherwise. When initially exposed to
quadratic residues, bruteforce methods are generally what a student attempts
to use to compute . Euler's Criterion
gives them their first real computational approach to finding Euler's
Criterion: Let p be an odd prime
and let a be an integer not divisible by p. Then One,
possibly more standard, approach to proving Euler's Criterion involves the
use of Fermat's Little Theorem as well as primitive roots. There is, however,
an alternative proof which makes use of various other results discussed in an
Elementary Number Theory course rather than primitive roots; e.g., the
existence of modular inverses and Wilson's Theorem. In this talk we will
discuss how the latter approach provides an excellent opportunity to tie
together numerous concepts which are presented throughout such a course. 


10:20 AM – 10:35
AM 
Seeding
Polynomials for Quadratic Congruences Modulo Prime Powers Larry
Lehman, University of Mary Washington 

The number of
solutions of a quadratic congruence can appear to be unpredictable when p^{2}
divides , the discriminant of f. We will
prove that these solutions can be counted and constructed in terms of those
of a “seeding polynomial” for , having discriminant . The proof is one that can be
visualized, and is appropriate for a first course in number theory. As a
consequence of this approach, we also establish a formula for the number of
solutions of an arbitrary quadratic congruence modulo any prime power. 


10:40 AM – 10:55
AM 
The
Exact Power of p Dividing n! Scott
Zinzer, Aurora University 

The Legendre
formula gives the prime factorization of n!.
For a prime p, each factor of p appearing in n! must
arise from a factor of p in one of the integers between 1 and n.
The proof of the Legendre formula for p requires a careful counting of
the number of integers between 1 and n that are divisible by each
power of p. By using a measurement model for division on the number
line, we demonstrate how the greatest integer function provides this count.
We provide some initial motivation previewing why the sum of values of the
greatest integer function successfully counts the exact number of times p
appears as a factor among all of the integers between 1 and n.
Finally, we use geometric sums to rewrite the Legendre formula for p
in terms of the basep expansion of n. 


11 AM – 11:15 AM 
A
Silver Version of Dirichlet's Bronze Approximation Theorem Andrew
J. Simoson, King University 

Given a
positive irrational number and a reduced fraction of positive integers, we say that is
a bronze, silver, or gold approximation to if, respectively, and differ by less than, , or . Johann Peter Gustav Lejeune Dirichlet
proved that there are an infinite number of bronze winning fractions for in about 1840. Fifty years later, Adolf
Hurwitz showed that there are an infinite number of gold winning fractions
for. We shall do the same with respect to
silver by using Farey sequences, named after John Farey (1766–1826), and Ford
circles in the xyplane, named after Lester R. Ford (1886–1967). Briefly, as
follows from Euclid’s greatest common divisor algorithm, any reduced fraction
is the mediant of two unique reduced
fractions and, called Farey neighbors, where and . The Ford circle associated with is a circle with center and radius. A fun – and elementary – geometric
argument shows that two Ford circles are tangent if and only if their
associated fractions are Farey neighbors, which is enough to give us a recursion
generating a sequence of only silver winning fractions for. 
This page was created and is maintained by S.
L. Mabrouk, Framingham State University.
This page was last modified on Wednesday, June 12, 2019.