I mostly work in an area of mathematics called **number theory**, and in particular on polynomials and on Fibonacci numbers. 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. I’m quite proud that some of my co-authors are students, both undergraduates from here at W&L and also high-school students from Pioneer Academics.

I’m pretty excited about my latest article on epitrochoids and hypotrochoids. It’s about how these red figures (called epitrochoids)

… can fit together with these blue figures (called hypotrochoids).

Read the article to learn more!

**Epitrochoids and Hypotrochoids Toget****her Again**, to appear in the*College Math Journal*(2024 or 2025). (pdf).**Periodic Weighted Sums of Binomial Coefficients**, with high-school student Yike Li,*Journal of Integer Sequences***26**No. 8 (2023). (pdf and on arXiv).**On the Brousseau Sums**,*INTEGERS***22**(2022), article #A105. (pdf and on arXiv).**Convolutions of Sequences with Similar Linear Recurrence Formulas**, with student Michael Tulskikh of Columbia University.*Journal of Integer Sequences***25**No. 3 (2022). (pdf).**A General Convolution Identity**, with UCLA student Yichen Wang.*Mathematics Magazine***97**No. 2 (2024), 98-109. (pdf).**Sums and Convolutions of k-Bonacci and k-Lucas Numbers**, with high-school student Yichen Wang.*INTEGERS***21**(2021), article #A56. (pdf).**Chebyshev Polynomials, Sliding Columns, and the k-step Fibonacci Numbers.***Mathematics Magazine***96**No. 1 (2023). (pdf and on arXiv).**Tilings of 2 x n boards with dominos and L-shaped trominos**, with high-school student Michael Tulskikh,*Journal of Integer Sequences***24**No. 4 (2021). (pdf).**Tetranacci Identities via Hexagonal Tilings**, joint work with student Ziqian (Alexa) Jin of the University of Chicago.*Fibonacci Quarterly***60**no. 2 (May 2022), 99-103. (pdf).**Weighted Sums of Fibonacci and Lucas Numbers through Colorful Tilings**, written with high-school student Yu (Angelica) Xiao of Shenzhen, China.*Fibonacci Quarterly***60**no. 2 (May 2022), 126-135. (pdf).**Fault-Free Tilings of the 3 x n Rectangle With Squares and Dominos**, joint work with high-school student Oluwatobi Alabi (from Abuja, Nigeria),*Journal of Integer Sequences***24**No. 1 (2021). (pdf). This was my first publication written with a high-school student (but certainly not my last).**Polynomial Roots with Common Tails**, joint work with W&L students Saimon Islam, Prakriti Panthi, Anukriti Shrestha, and Jiahao Zhang over the summers of 2017 and 2018.*MAA Monthly***127**no. 4 (2020), 316-329. (pdf).**Cubic Polynomials, Linear Shifts, and Ramanujan Simple Cubics**, written with W&L 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 W&L students Prakriti Panthi, Anukriti Shrestha, and Jiahao Zhang from our summer research in 2017.*Rocky Mountain Journal of Mathematics*,**49**no. 4 (2019), 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.)**Resultants of Cyclotomic Polynomials**,*Rocky Mountain Journal of Mathematics*,**42**No. 5 (2012). (pdf).**Three Transcendental Numbers From the Last Non-Zero Digits of n**^{n}, F_{n}, 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 W. Dymacek,*MAA Monthly***112**(August-September 2005), 602–611. (pdf).**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 used to give the chain as Dresden — Art Benjamin (at Harvey Mudd) — Phyllis Chinn (at Humboldt State) — Paul Erdös, but sometimes it replaces Phyllis Chin with Paul Chartrand (Western Michigan), who co-authored with Art Benjamin and with Paul Erdös. (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 over 100 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!

My name appears over 300 times in the Online Encyclopedia of Integer Sequences; some of my favorite entries are A334396, A052980, and A332647.

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.)

I’ve given many presentations on mathematics at local and national conferences. Also, Art Benjamin (mentioned above) gave a presentation on our joint work at MIT in December of 2004. 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 factor rings.

Over the years, I’ve submitted a number of original problems to math magazines. One recent problem was on finding the shaded areas of special kinds of polar-coordinate graphs; shown here are the graphs for r=sin(4 theta/3) and r=sin(6 theta/5). The answer was both simple and surprising: it’s always pi/2 for odd denominators (pdf).

This appeared in print as problem #1221 in the March, 2022 issue of the *College Math Journal*.

Here’s a complete list of all my published problems.

- In the
*College Mathematics Journal,*- #758 on polar graphs, Sep 2003. (pdf).
- #1151 on matrix powers, May 2019. (pdf).
- #1186 on continued fractions (with ZhenShu Luan), Nov 2020. (pdf).
- #1221 on areas in polar coordinates, Mar 2022. (Extended solution pdf).
- #1274 on an area of the loops of an epitrochoid, Mar 2024. (pdf.)
- #1276 on a variation of Zeckendorf’s (with Yunqi Liu), May 2024. (pdf.)

- In
*Crux Mathematicorum,* - In
*Elemente der Mathematik,* - In
*La Gaceta de la RSME,*- Problem on self-tangent epitrochoids, under review (2024).

- In
*Math Gazette,*- #2023.3 on limits of cotangents, Jul 2023. (pdf).

- in
*Mathematics Magazine,*- #2035 on decimal tails (with Prakriti Panthi, Anukriti Shrestha, and Jiahao Zhang), Dec 2017. (pdf).
- #2058 on Galois groups (with Saimon Islam and Jiahao Zhang), Dec 2018. (pdf).
- #2090 on star-shaped graphs, Feb 2020. (pdf).
- #2120 on normalizers (with Kathleen McNeill and Jackson Gazin), Apr 2021. (pdf).
- #2160 on quadrilaterals inside triangles, Dec 2022. (pdf).
- #2191 on the area between cosines (with John Chapman and Yvonne Cheng), Mar 2024. (pdf).

- in
*Math Horizons*,