Periodicity of Euler Numbers in Modular Arithmetic
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 .
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.