Welcome to my web page. For academic info about me, be sure to check out
my somewhat-up-to-date Curriculum Vitae [pdf file]. For
class or research info, read on!
How To Reach Me
Office Location: Robinson Hall R-1.
Office Phone: (540) 458-8806.
FAX: (540) 458-8239.
Robinson Hall R-1
Washington & Lee University,
204 West Washington Street,
Lexington VA 24450-2116.
e-mail: dresdeng "at" wlu "dot" edu. If you don't hear from me after
a few days, it's probably because your e-mail got deleted by
my automatic spam filter. Try to send it again as a plain-text message
or just call me on the phone.
For winter term 2015-2016 (starting January 11, 2016).
These are subject to change, and you can always make an appointment to see me.
Monday at 12:20,
Tuesday at 1:25,
Wednesday at 12:20,
Friday at 10:10,
... and one "floating" office hour, by appointment, each week.
Fun with differential equations
I had a tremendous amount of fun this last term with my Math 332 class
(Ordinary Differential Equations). We took some rather simple
differential equations and created lovely stream plots that depict the behavior of
the solution over time, given a certain initial condition. Here are two pictures
from computer labs we did in class.
On the left, we see a traditional stream plot for the equation
dx/dt = x2 - t
with various initial conditions, and on the right we see the same
stream plot but run through this special
Mathematica filter that gives it that lovely coffee-toned look. We also did some
animations, although that took up a lot of computer processing time.
This photo (of me and student Scotty Groth) appeared on the Newsweek/Daily Beast website for W&L
(which was ranked #11 on the list of Most Rigorous Schools).
I work in an area of mathematics called number theory, and in
particular on the subject of the Mahler measure of a polynomial.
I also study other topics from both number theory and abstract algebra (on polynomials,
groups, algebraic extensions, etc).
Here are some of the articles I have written:
Orbits of Algebraic Numbers with Low Heights, Math. Comp.
67 (April 1998), 815--820.
(Download the pdf file.)
Two Irrational Numbers From
the Last Non-Zero Digits of n! and nn,
Math. Mag. 74 (October 2001), 316--320.
(pdf as published, and the
revised version that corrects a small problem with one of the proofs.)
In a not-yet-published paper by Grau and Oller-Marcén (with an assist by me), we extended my result on
to any arbitrary base.
Sums of Heights of Algebraic Numbers, Math. Comp.,
72 (2003), 1487--1499.
On the Middle Coefficient of the Cyclotomic Polynomial,
MAA Monthly 111 (June-July 2004), 531--533.
There Are Only Nine Finite Groups of Linear Fractional Transforms with Integer Coefficients,
Math. Mag. 77 (June 2004), 211--218.
Finding Factors of Factor Rings over the Gaussian Integers, with
Wayne Dymacek (W&L),
MAA Monthly 112 (Aug-Sep, 2005), 602--611.
A Combinatorial Proof of Vandermonde's Determinant,
with Art Benjamin (Harvey Mudd College),
MAA Monthly 114 (Apr, 2007), 338--341.
Three Transcendental Numbers From the Last Non-Zero
Digits of nn, Fn, and n!,
Math. Mag. 81 (Apr, 2008), 96--105.
Resultants of Cyclotomic Polynomials,
Rocky Mountain Journal of Mathematics, 42 No. 5 (2012).
Look, There's More to Say about Conway's Look and Say Sequence,
Jacob Siehler (Lexington, VA),
submitted to Mathematics Magazine (Nov. 2009) and provisionally accepted,
subject to revisions, in December of 2013.
Binet-type formulas for r-generalized Fibonacci numbers,
with Zhaohui Du (Shanghai, China), Journal of Integer Sequences, 17 No. 4 (2014).
(pdf). An earlier version of this article has been available for many years
here on arXiv.org (the vast and free depository for science articles),
and so it was referenced many times in other articles before it was actually accepted for publication.
It's funny how that happens. (This result was found independently by me and by Zhaohui Du, so with his permission I put both our names on this paper
when I submitted it to the Journal of Integer Sequences.)
A New Approach to Rational Values of Trigonometric Functions,
I'm rather fond of my three articles that were published in the MAA Monthly,
partly because two of them were jointly written with other mathematicians (which
is a tremendous amount of fun),
and partly because the MAA Monthly
(as seen here)
is the most
widely read mathematics journal in the world.
Thanks to my joint articles, I have an
Erdös number of 3. This means that I'm only three
co-authors away from Paul Erdös, the most prolific
mathematician in history (biographical links
to the MacTutor history of mathematics site
The MathSciNet database gives the chain as Dresden -- Art Benjamin (at Harvey Mudd) --
Phyllis Chinn (at Humboldt State) -- Paul Erdös. How cool is that!
(Some famous people with Erdös number 3 include
Larry Page, Sergey Brin, Kurt Gödel, and John von Neumann.)
Some of my articles have also been referenced in books. For example, my two papers on the last
non-zero digits of various sequences are quoted on page 32 of the book,
"Numbers and Functions: From a classical-experimental mathematician's point of view" by
Victor H. Moll of Tulane University, as seen here.
my Abstract Algebra students and I worked on finding natural
representations for finite rings. Please visit
page for examples.
I've given many presentations on mathematics and
one on teaching mathematics,
at local and national conferences. Also, Art Benjamin (mentioned above)
gave a presentation on our joint work at MIT in December of 2004,
and fellow W&L professor Wayne Dymacek gave a talk on our joint paper here at
I gave a talk at JMU on the Mahler measure and again on the Look-and-Say sequence,
and a student of mine gave a talk at Loyola (in Maryland) on
her senior honors thesis (on the subject of factor rings).
Math Information for Washington & Lee Students (and others)
All students can get their free download of Mathematica by requesting it here. We're currently using Mathematica version 10.
Two extremely cool references on the web are Eric Weisstein's
World of Mathematics,
hosted by the nice folks at Wolfram Research, and also
Wolfram Alpha, which can actually
solve most calculus problems!
The amazing Salman Khan
has recorded over a thousand YouTube videos
of short (10-20 minutes) tutorials in everything from history to statistics,
with hundreds of videos in calculus, linear algebra, and probability. If you're stuck
on something (chain rule, trig substitution, etc) I can guarantee you that
he has a video explaining how to do it!
My Math Department actuary web
page tells you everything you want to know about how to pursue this
lucrative and rewarding career (and yes, we have math classes for that).
Anticipating having trouble with Calculus 101 or 102? You might want to check out
the free nightly tutoring sessions in Robinson 6, Sunday through
Thursday. Check with the department secretary for details.
Prof. Finch and I share teaching one of our favorite courses,
This is an introductory course in number theory, a very
beautiful area of mathematics that dates back to the ancient Greeks, yet
contains many modern applications ranging from deciphering
German codes during WWII
to verifying the account numbers on your ATM card.
Students who took this course during Winter term really enjoyed it.
I'm very excited about a new spring-term course,
Math 171: cryptography and number theory.
This course covers many of the issues concerning the current
encryption methods used by commercial programs and by the military:
how the codes work, why they work, and how to break them.
one can create very secure (but not perfectly secure!) encryption
systems using some fairly easy facts of basic number theory about
prime numbers. An excellent example of an encryption system is
the PGP system by Phil Zimmermann. So secure is Zimmermann's PGP
program that the U.S. government classified it as a restricted
munition, and tried to keep him from sharing it with folks in
foreign (and potentially hostile) nations. Indeed, many fear that
terrorists and criminals can use these systems to encrypt their
files, thus making recovery nearly impossible.
The most exciting part of the course, I think, will be our field trips to
the Marshall Museum here in Lexington and the NSA Museum in Washington, DC.
Both museums have extensive collections of
cryptographic materials and artifacts. In this picture from a few years back, Dr. Dymacek
(on the right)
explains to us (at the Marshall Museum) some of the details about the ENIGMA machine. You'll
recall that the ENIGMA was used by the Germans to encode military
messages during the war.
British and American scientists (led by the brilliant Alan Turing) managed
to break the code, and thus tremendously advanced the Allied war effort.
Did I mention that one of the exams in this course will itself be written in code? So, even
before you can take the exam, you have to decode it. Sounds like fun!
For general mathematical information, here are some of my favorite links:
MAA Online, from the
Mathematical Association of America, has many great links to career and
membership is $30, and well worth it.
The MAA Careers page can help
to answer the question, "What can I do with a math major?"
The page Career Profiles demonstrates
that there are many options beyond just teaching or graduate school. Check out the dozens of
profiles in investment, medical science, sales and marketing, law, engineering,
business management, actuarial science, and more... and all of them are
about former math majors!
The AMS Home Page, from
American Mathematical Society, contains valuable professional information,
along with a searchable index to mathematical articles.
Here is a direct link to the
... and to the
database (to find out who's quoting whom).
If you're thinking of going to graduate school, you might want to look
into getting an
membership, at around $40 or so.
You should also look at the
AMS/YMN math grad student blog,
which includes links to
some really funny math comics!
And speaking of graduate school, the UC Davis Math Department
has lots of great advice from and for grad students, including their page,
Wish I'd Known....
Recent W&L graduates Isaac Lambert and Lu Li are now studying at UC Davis.
UCLA's Terrence Tao is widely considered to be
the most brilliant mathematician alive. His blog is well worth reading.
If you're interested in teaching math in middle or high school, you might want to check out the many organizations that offer free master's degrees (plus stipends), such as the Woodrow Wilson foundation in Indiana, and Math for America in New York, Berkeley, LA, and a few other places.
What happens when two guys in Silicon Valley decide to build a
place for mathematicians to hang out? Well, you get
American Institute of Mathematics,
soon to be a major research center.
Want to find out who your advisor's advisor was? Check out the
Mathematics Genealogy Project,
hosted by the AMS amd the folks at North Dakota State University.
The Mathematics Information
Server is a giant listing of just about every mathematical group
you can imagine, including a world-wide list of on-line
college and university mathematics departments. This is very helpful in
providing a direct link to math professors and programs around the world.
Here's a direct link to the
Number Theory web.
For a (rather rough) ranking of graduate math departments,
this AMS web site.
As you no doubt know, the
Appalachian Trail is only a few minutes
from campus. I don't think I'll ever actually walk the whole thing, but I do
like to get out and hike for a few hours on parts of the trail, accompanied
occasionally by our faithful trail dog, Mr. Puppy. Here's a pair of nice pictures from up on
the Blue Ridge Parkway
(also a few minutes from campus), with the
Peaks of Otter in the background of the first shot and the James River
in the second.
This page was written in HTML by Prof. Dresden, using a template
originally borrowed from
UT-Austin. It's been up, in one form
or another, since 1997. It's likely that some of the information and links here are out of date; I'm
not as diligent in updating things as I should be.