# Periodicity of Euler Numbers in Modular Arithmetic

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The Euler numbers are integers that arise in the series expansion of the hyperbolic secant function around the origin: . The plot above indicates that the sequence is periodic in for any integer . Incidentally, the sequence is periodic with respect to .

Contributed by: Oleksandr Pavlyk (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

As of August 27, 2009, Stan Wagon informed the author, quoting Herbert Wilf, that the following proof is well known and published in the book by S. K. Lando, *Lectures on Generating Functions*, Providence, RI: AMS, 2003.

The formal generating function of Euler numbers has the simple continued fraction

;

likewise, the formal generating function for the absolute value of the Euler numbers is

.

Hence, the formal generating functions of the sequences and have a terminating continued fraction expansion, and thus are rational functions in . It is well known that a rational generating function gives rise to a periodic sequence in modular arithmetic.

## Permanent Citation