(a, m) = 1
a(m) 1 (mod m)
cfw_x Z+ : ax 1 (mod m)
m
a
ordm (a) (m)
ordm (a)
(a, m) = 1
m
a
m
Z
m
Z
m
m
ordm (a) = (m)
m
ax 1 (mod m)
(a, m) = 1
ordm (a) | x .
r
q
x = nq + r
n = ordm (a)
r=0
ar
r=0
0r<n
= ar
p
p
p a
p
a
x2 a (mod p)
p a
p
p
x2 a (mod p)
a
x x0 (mod p)
0
p
x0
x0
x2 a (mod p)
1
x2 x2 (mod p)
1
0
p
p | (x1 x0 )(x1 + x0 )
x1 x0 (mod p)
p
(a)
p
p a
x1 x0 (mod p)
p | (x2 x2 )
f :ZC
n
(1) = 1
Z
f
(n)
(n)
n>1
n
1
2 3
4
10 16
20 23 48
(n)
1
1 2
2
4
8
8
22 16
p
(p) = p 1 .
(p ) = p p1
1
p
1, 2, 3, . . . , p
kp
p p1
(m, n) = 1
n
m
p
p
f
f (mn) = f (m)f
m
b
a
m a
m | (a b)
a b(mod m)
0<m
m b
a b(mod m)
m
Z
m
a : a a (mod m) ,
a, b : a b (mod m) b a (mod m) ,
a, b, c : a b (mod m) & b c (mod m) a c (mod m) .
a
[a]m
aZ
a
[a]
m
[a]m = cfw_b Z : a b
b a
a=0
b
a
a b
b
a
x Z : b = ax
a b
b
a
a|b
c
a|b
b|c
a|b
a|c
a|b
b|a
a|b
a
a|b
a|bc .
a|c
x, y Z : a|(bx + cy) .
ab
a = b
b
a|b1 , a|b2 , . . . , a|bn
n
a|
bi xi .
i=1
x1 , x2 , . . . , xn Z
0<a