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Experimental Mathematics Seminar

Sporadic Apéry-like numbers modulo primes 

 Amita Malik, Rutgers University 

Location:  HILL 705
Date & time: Thursday, 14 September 2017 at 5:00PM - 5:48PM

Abstract:   At the ICM in 1978, R. Apéry's proof of the irrationality of ?(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 numbers.

This is joint work with Armin Straub. 

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Department of Mathematics

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