Wolfram Demonstrations Project

13,000+ Open Interactive Demonstrations
Powered by Notebook Technology »

Periodicity of Euler Numbers in Modular Arithmetic

Initializing live version
Download to Desktop

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.

Embed Code

Take advantage of the Wolfram Notebook Emebedder for the recommended user experience.

Related Demonstrations More by Author
Browse all topics
Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send