Date: 2025/10/05
Occasion: AMS Fall Southeastern Sectional Meeting, Special Session on Modular Forms in Combinatorics and Number Theory
Place: Tulane University
Abstract
It is a well-known and beautiful classical result of Lucas that, modulo a prime \(p\), the binomial coefficients satisfy $$ \binom{n}{k} \equiv \binom{n_0}{k_0} \binom{n_1}{k_1} \cdots \binom{n_r}{k_r}, $$ where \(n_i\) and \(k_i\) are the \(p\)-adic digits of \(n\) and \(k\), respectively. For each of the \(15\) known sporadic Apéry-like sequences, we prove congruences modulo \(p^2\) that are natural extensions of these congruences. This extends a result of Gessel for the numbers used by Apéry in his proof of the irrationality of \(\zeta(3)\). We review the connection of these sequences with modular forms and highlight a recent proof of Lucas congruences by Frits Beukers, Wei-Lun Tsai and Dongxi Ye which is based on this modularity.Download
| Link | Size | Description | Hits |
|---|---|---|---|
| 2025gessel-lucas.pdf | 1.11 MB | Slides (PDF, 51 pages) | 22 |