I work in an area of mathematics called number theory, and in particular on Fibonacci numbers and polynomials. I also study other topics from both number theory and abstract algebra (on sequences, groups, algebraic extensions, etc). Here are some of the articles I have written, with the most recent ones on top:
- Polynomials with Common Tails, joint work with students Saimon Islam, Prakriti Panthi, Anukriti Shrestha, and Jiahao Zhang over the summers of 2017 and 2018. To appear, MAA Monthly (2020). (pdf).
- Cubic Polynomials, Linear Shifts, and Ramanujan Simple Cubics, written with students Prakriti Panthi, Anukriti Shrestha, and Jiahao Zhang in the summer of 2017. Math. Mag. 92 no. 5 (Dec 2019), p. 374-381 (pdf). The cover art (“Ramanujan’s Butterfly”) on this issue of Math. Mag. was by David Riemann and was inspired by our article. It also appeared in the Mathematical Art exhibition at the Joint Math Meeting (January 2020, Colorado).
- Finite Subgroups of the Extended Modular Group, again with students Prakriti Panthi, Anukriti Shrestha, and Jiahao Zhang from our summer research in 2017. Rocky Mountain Journal of Mathematics, 49 no. 4, 1123-1127. (pdf.)
- When is a^n + 1 the sum of two squares, a collaborative effort with three W&L students (Saimon Islam, Aaron Schmitt, and Pan Yue) and a team at Wake Forest (students Kylie Hess, Emily Stamm, Terrin Warren, and professor Jeremy Rouse) from our work in the summer of 2016. Involve, 12 No. 4 (2019). (pdf).
- Finding cycles in the kth power digraphs over the integers modulo a prime, with Wenda Tu (W&L class of 2014), Involve, 11 No. 2 (2018). (pdf). This article was from my summer research with Wenda back in 2013 (see this page for more details on my summer work with students)
- 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.)
- Look, There’s More to Say about Conway’s Look and Say Sequence, with Jacob Siehler (Gustavus Adolphus College), provisionally accepted at Mathematics Magazine subject to revisions, in December of 2013. (pdf to appear).
- Resultants of Cyclotomic Polynomials, Rocky Mountain Journal of Mathematics, 42 No. 5 (2012). (pdf).
- Three Transcendental Numbers From the Last Non-Zero Digits of nn, Fn, and n!, Math. Mag. 81 (Apr, 2008), 96–105. (pdf).
- A Combinatorial Proof of Vandermonde’s Determinant, with Art Benjamin (Harvey Mudd College), MAA Monthly 114 (Apr, 2007), 338–341. (pdf.) This was a really fun paper to work on, and it all started over dinner at a local restaurant when I asked Art, “So, what interesting problems are you working on?” We eventually found a nice proof of this theorem using a card-counting technique. In an interview published in the AMS Notices (January 2017), Art Benjamin was asked, “What theorem are you most proud of”, and he said it was this one! (link to interview here). This also lowered my Erdös number to 3 (see below).
- Finding Factors of Factor Rings over the Gaussian Integers, with Wayne Dymacek (W&L), MAA Monthly 112 (Aug-Sep, 2005), 602–611. (pdf.) This was my first joint paper, and it gave me an Erdös number of 4 (Dresden — Dymacek — Bailey — Kennedy — Erdös).
- There Are Only Nine Finite Groups of Linear Fractional Transforms with Integer Coefficients, Math. Mag. 77 (June 2004), 211–218. (pdf.)
- On the Middle Coefficient of the Cyclotomic Polynomial, MAA Monthly 111 (June-July 2004), 531–533. (pdf).
- Sums of Heights of Algebraic Numbers, Math. Comp., 72 (2003), 1487–1499. (pdf.)
- Two Irrational Numbers from the Last Nonzero Digits of n! and n^n, Math. Mag. 74 (October 2001), 316–320. (pdf.) I discovered a few years later that not only was there a mistake in one of my proofs, but also that my Theorem 1 had already been proven a decade earlier in the Canadian problem journal Crux Mathematicorum! These kinds of things do happen from time to time. I’ve updated the pdf of my article with details.
- Orbits of Algebraic Numbers with Low Heights, Math. Comp. 67 (April 1998), 815–820. (pdf.)
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 Wikipedia here and to the MacTutor history of mathematics site here). 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.)
According to my Author Page on the math indexing site http://www.ams.org/mathscinet/, I have just over 60 citations (that’s where someone referenced one of my publications in their own article). It’s not as high as the citation counts for some of my colleagues, but on the other hand it’s nice to know that people are actually reading and referencing my articles!
Some of my articles have been referenced in AT&T’s On-Line Encyclopedia of Integer Sequences. For example, check out 0, 0, 1, 0, 1, -1, … (reference number A094754) and 1, 1, 2, 6, 4, 2, … (reference number A008904), among others. Thanks to Jacob Siehler for providing the Mathematica code for those references.
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 in Note 1.8.10 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. (Note that I’ve actually proved that these are transcendental numbers and not just irrational numbers.)
Along with Professor Siehler, my Abstract Algebra students and I worked on finding natural representations for finite rings. I’ve got some nice pictures on that topic, and at some point I will post them here.
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 W&L. Recently, 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).