arminstraub.com

# Talk: Supercongruences for polynomial analogs of the Apéry numbers (Integers 2018)

Supercongruences for polynomial analogs of the Apéry numbers (Integers 2018)

Date: 2018/10/05
Occasion: Integers Conference 2018
Place: University of Augusta

## Abstract

The Apéry numbers $$A(n)$$ are the famous sequence which underlies Apéry's proof of the irrationality of $$\zeta(3)$$. Together with their siblings, known as Apéry-like, they enjoy remarkable properties, including connections with modular forms, and have appeared in various contexts. One of their (still partially conjectural) properties is that these sequences satisfy supercongruences, a term coined by Beukers to indicate that the congruences are modulo exceptionally high powers of primes. For instance, \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes $$p \geq 5$$. In this talk, we introduce and discuss a $$q$$-analog of the Apéry numbers. In particular, we prove a supercongruence for these polynomials.