Date: 2026/03/07
Occasion: Southern Regional Number Theory Conference
Place: Louisiana State 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 |
|---|---|---|---|
| 2026gessel-lucas-lsu.pdf | 1.10 MB | Slides (PDF, 50 pages) | 5 |