WOLFRAM|DEMONSTRATIONS PROJECT

Periodicity of Euler Numbers in Modular Arithmetic

​
m
41
The Euler numbers
E
n
are integers that arise in the series expansion of the hyperbolic secant function
sech
around the origin:
sech(x)
∞
∑
n=0
n
x
E
n
/n!
. The plot above indicates that the sequence
c
n

E
2n
(modm)
is periodic in
n
for any integer
m>1
. Incidentally, the sequence
n
(-1)
E
2n
(modm)
is periodic with respect to
n
.