1 :
1 = 1 / 1 = 1.
: 4
1 / 4,
4 1 / 4 = 1.
. ,
4 ≡ 1 (mod 7)
k,
4 ≡ 7 k + 1,
k .
,
(mod n) = 1.
1 ≡ (mod n).
, . , 5 14 3, ..
5 3 = 15 ≡ 1 (mod 14).
, 2 14.
1 ≡ (mod n)
, n .
n ,
1 ≡ (mod n)
.
. Î {0, 1, 2, , n - 1}.
(, n) = 1, i (mod n) i = 0, 1, 2, , n 1 {0, 1, 2, , n - 1}.
. = 3 n = 7 ( (3, 7) = 1), 3 i (mod 7)
i = 0, 1, 2, , 6 0, 3, 6, 2, 5, 1, 4, .. {0, 1, 2, , 6}.
, (, n) ¹ 1.
. = 2 n = 6, 2 i (mod 6) º 0, 2, 4, 0, 2, 4
i = 0, 1, 2, , 5.
(, n) = 1, 1,
0 < 1 < n, ,
1 º 1 (mod n).
, i (mod n) 0, 1, 2, , n 1, i ,
i º 1 (mod n).
, 0 n 1 n. , ( > 0) r = a (mod n) 0 n 1.
n. .
. n = 11 . 11: {0, 1, 2, , 10}. : 0. 11 11 1 = 10 .
, n n 1 .
. n = 10. 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. 1, 3, 7, 9 10. 10 {1, 3, 7, 9}. :
|
|
( ), | |
2 | ( ), |
5 | ( ), |
.. . 10, 10 6 = 4. , .
p q = n ( p 1) (q 1) .
. n = p q = 2 5 = 10 (p 1) (q 1) = (2 1) (5 1) = 4.
. 27 = 3 3 18 : {1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26}. , 3 ( ).
: n r n r 1 (n 1) .
n = 3, r = 3 3 3 1 (3 1) = 3 2 2 = 18 .
j(n).
1.3.1 j(n)
n | j(n) |
n n 2 ... n r | n 1 n (n 1) ... n r 1 (n 1) |
p q (p, q - ) ... ... i ei (p i - ) | (p 1) (q 1) ... ... i ei (p i 1) |
, j(n) , n, n.
: n (, n) = 1,
n 1 º 1 (mod n).
: (, n) = 1,
j(n) º 1 (mod n).
n , , , j(n) = n 1, ( )
n 1 º 1 (mod n).