Research

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 high-school students from Pioneer Academics.

I’m pretty excited about this: my recent article on epitrochoids and hypotrochoids won the George Pólya award from the MAA. The article is about how these red figures (called epitrochoids)

 

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

Read the article (look down the list) to learn more!

  • Unexpected Tribonacci Tilings, with high-school student Peiyang Ye, submitted to the College Mathematics Journal in late 2026.
  • Tilings and Self-Convolutions, with W&L student Guanzhang Zhou, to appear in the College Mathematics Journal, hopefully in 2026. (pdf).
  • Self-Convolutions of Generalized Narayana Numbers, with W&L student Guanzhang Zhou and high-school student Yuechen Xiao, to appear in the Fibonacci Quarterly in 2026 or 2027. (pdf).
  • Further Extensions of Sury’s Identity, with high-school student Xiaoya (Lexie) Gao, Journal of Integer Sequences 29 No. 3 (2026), article 26.3.2 (pdf).
  • Areas Within Cyclic-Harmonic Curves, with W&L student Saad Ghori, to appear in Mathematics Magazine (2026, probably). (pdf).
  • A Tiling Proof for Triple Convolutions, with high-school student Xiaoyuan Wang, to appear in Mathematics Magazine (2026, I hope). (pdf).
  • Areas Between Cosines, with high-school student Muhammad Adam Dombrowski (now at Brown), Journal of Integer Sequences 28 No. 2 (2025), article 25.2.3 (pdf).
  • Epitrochoids and Hypotrochoids Together Again, College Math Journal 56 No. 3 (May 2025), 203-213 (pdf). The editors liked my article so much that they used my artwork for the cover. The article also won the George Pólya Award from the MAA.
  • 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)
  • A Simplified Binet formula for k-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 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 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 150 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).

polarpic

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 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.)
    • #1311 on self-tangent hypotrochoids with five loops, Nov 2025. (pdf.)
    • #1314 on the areas between cos^2, and cos^4, Nov 2025. (pdf.)
    • #1319 on parabolas, cubics, and circles, Jan 2026. (pdf.)
    • #1328 on the areas of triangles tangent to x^(1/s) + y^(1/t) = 1, May 2026. (pdf.)
    • #1335 on the lines of intersection between a plane and a surface, Sep 2026. (pdf.)
  • In Crux Mathematicorum,
    • #4721 on diagonals in Pascal’s Triange (with Myanna Nash), Mar 2022. (English pdf, French pdf).
  • In Elemente der Mathematik,
    • #1441 on self-intersecting parametric curves, Oct 2023. (English pdf, German pdf).
  • In La Gaceta de la RSME,
    • #492 on self-tangent epitrochoids, Aug 2024. (English pdf, Spanish pdf).
  • 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). This was a beautiful little problem, and it won me an Honorable Mention for the Mohammad K. Azarian award from the MAA in 2025.
    • #2191 on the area between cosines (with John Chapman and Yvonne Cheng), Mar 2024. (pdf).
    • #2215, on groups of integers that end in 5, Feb 2025. (pdf of my version, and the editors liked it so much that they modified it to make it a much more general problem).
  • in Math Horizons,
    • #P441, on circles and not-circles, Sep 2022. (pdf).
    • #P478, on cyclic-harmonic curves (with Neil Sriram), Feb 2025. (pdf).
    • #P490, on stars inside of rotated intersecting circles (with Anshika Patel), Nov 2025. (pdf).
    • A problem on consecutive powers of the supergolden ratio, to appear in 2026. (pdf).
    • #P498, on ellipses and not-ellipses, Apr 2026. (pdf).
  • In the MAA Monthly,
    • #12613, on tangency with two epitrochoids (with Alex Bousman, Anshika Patel, Ben Rubin, and Sam Wizda), June-July 2026. (pdf).