# Periodicity of Euler Numbers in Modular Arithmetic

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 (modm) is periodic in for any integer . Incidentally, the sequence (modm) is periodic with respect to .

E

n

sech

sech(x)/n!

∞

∑

n=0

n

x

E

n

c

n

E

2n

n

m>1

n

(-1)

E

2n

n