### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:**
**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu
**Bryan Ek**, Rutgers University, bryan [dot] t [dot] ek {at} math [dot] rutgers [dot] edu

Title: Sporadic Apéry-like numbers modulo primes

Speaker: **Amita Malik**, Rutgers University

Date: Thursday, September 14, 2017 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

At the ICM in 1978, R. Apéry's proof of the irrationality of *ζ(3)* was presented. In this proof, he introduced a sequence of integers, now known as Apéry numbers. Apéry-like numbers are special integer sequences, studied by Beukers and Zagier, which are modeled after Apéry numbers. Among their remarkable properties are connections with modular forms, Calabi-Yau differential equations, and a number of *p*-adic properties, some of which remain conjectural.

A result of Gessel shows that Apéry's sequence satisfies Lucas
congruences. We prove corresponding congruences for all sporadic
Apéry-like sequences. While, in some cases, we are able to
employ approaches due to McIntosh, Samol-van Straten and
Rowland-Yassawi to establish these congruences, there are few others
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's
numbers.

This is joint work with Armin Straub.

See: http://sites.math.rutgers.edu/~bte14/expmath/