2-color, radius 3/2 rule 39780

Eric Rowland

Hi! I'm a professor in the Department of Mathematics at Hofstra University.

On my YouTube channel, I make videos about discoveries in number theory. You can also watch some of my research talks.

I'm currently writing a book, The Hierarchy of Integer Sequences, for an undergraduate course in combinatorics.

My research is broadly in number theory and combinatorics. I make extensive use of software for conducting experiments, discovering conjectures, and proving theorems. As a graduate student, I proved that a recurrence discovered in 2003 generates primes. (I made a video about it.)

One of my main interests is in studying arithmetic properties of integer sequences that arise in combinatorics. Binomial coefficients are a classic example of an enumeration question producing numbers with interesting properties. I discovered a matrix product for counting binomial coefficients according to the highest power of p dividing each, generalizing a well known result of Fine from 1947. I also found a characterization, in terms of rotational symmetry, of the base-p digits for which binomial coefficients satisfy a Lucas congruence modulo p2. An analogous characterization holds for the Apéry numbers, which arose in Apéry's landmark proof that ζ(3) is irrational.

There are many sequences in combinatorics (for example, the famous Catalan numbers) that, when reduced modulo prime powers, are generated by automata. Here is a quick introduction. Reem Yassawi and I used this to produce wholesale congruence theorems for algebraic sequences and, more generally, diagonals of rational functions. This subsumes many results in the literature on specific sequences, and it allowed us to answer several open questions. Doron Zeilberger and I showed that a similar technique works for constant-term sequences. Moreover, we have bounds on the size of the automaton modulo primes and modulo prime powers (forthcoming); this determines the time complexity for computing the automaton.

p-adic numbers have been instrumental in several projects. Luis Medina, Victor Moll, and I used them to determine the period length of the p-adic valuation of a polynomial sequence. Reem Yassawi and I computed p-adic limits of Fibonacci numbers by working out explicit p-adic interpolations of constant-recursive sequences. I built on this with Nick Bragman to determine the limiting density of residues attained by the Fibonacci sequence modulo powers of p. Another application, in a paper with Émilie Charlier, Adeline Massuir, and Michel Rigo, was in deciding whether the set of integers represented by a given automaton in a linear numeration system is eventually periodic.

In combinatorics on words, I have written papers with Jeff Shallit, Lara Pudwell, Manon Stipulanti, and a team of undergraduates in the Polymath Jr program on extremal integer sequences avoiding repetitions of various kinds. These sequences are self-similar, but the structures are on such large scales that they can only be found with a computer.

I'm a big proponent of communicating mathematics as clearly and accessibly as possible. I've collected some advice on writing and refereeing.

... among other things. Here are more systematic lists:

Here is my CV, but what's not on my CV is that my favorite LaTeX command is \rangle.

I highly recommend A Mathematician's Lament by Paul Lockhart (2002) [also available in paperback] and the introduction by Keith Devlin.