# Periodicity of Euler Numbers in Modular Arithmetic

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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

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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.

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