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

.