Talk: Lucas congruences (South Alabama)

Lucas congruences (South Alabama)

Date: 2015/10/16
Occasion: Pure Mathematics Seminar
Place: University of South Alabama


It is a well-known and beautiful classical result of Lucas that, modulo a prime \(p\), the binomial coefficients satisfy the congruences \begin{equation*} \binom{n}{k} \equiv \binom{n_0}{k_0} \binom{n_1}{k_1} \cdots \binom{n_r}{k_r}, \end{equation*} where \(n_i\), respectively \(k_i\), are the \(p\)-adic digits of \(n\) and \(k\). Apéry-like numbers are special integer sequences, going back to Beukers and Zagier, which are modelled after and share many of the properties of the numbers that underlie Apéry's proof of the irrationality of \(\zeta(3)\). Among their remarkable properties are connections with modular forms and a number of \(p\)-adic properties, some of which remain conjectural. A result of Gessel shows that Apéry's sequence satisfies Lucas-type congruences. We prove corresponding congruences for all sporadic Apéry-like sequences. While, in several cases, we are able to employ approaches due to McIntosh, Samol-van Straten and Rowland-Yassawi to establish these congruences, there is one sequence in particular, often labeled \((\eta)\), for which we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist-Zudilin numbers are periodic modulo 8, a special property which they share with the Apéry numbers.

This talk is based on joint work with Amita Malik.


1.27 MB Slides (PDF, 62 pages) 1212