5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1P

p P 1 n n n 1 φ(n) φ φ(1) = 1 1 n φ(n), n φ(n) = φ()φ(n) [ ] n 1 n 1 1 n 1 φ(n) φ() φ(n) 1 3 4 5 6 7 8 9 1 3 4 5 6 7 8 9 1 4 5 7 8 1 4 5 7 8 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 19 0 1 3 4 5 6 7 19 0 3 5 6 19 0 3 5 6 8 9 30 31 3 33 34 35 36 φ(abc ) = φ(a)φ(b)φ(c) a, b, c, [ ] 3 p n p n 1 [ ] p n px x 1 x p n 1 x px p p n 1 4 φ(p n ) = p n p n 1 [ ] 3 1/1

5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1Pj ) P j 1 j (P j 1) 1 φ(009) [ ] φ(009) = φ(7 41) = 7(7 1)(41 1) = 1680 3 y = 0, x = 6, y = x [ ] (x,y) y x x y > 0 y φ(x) x 1 + φ(1) + φ() + φ(3) + φ(4) + φ(5) + φ(6) =1 + 1 + 1 + + + 4 + =13 /1

1 0 1 (0, 0) (1, 0) φ(1) 1 (0, 0) (1, 1) 1 y 6 O 6 x 6 [ ] p q 1 3 p + 1 (1) q p () q q p q q p. 3 (1) () li φ() = a = φ() a 1, a, a 3, ( 1) [ ] 3/1

(1) p() φ() p() 6 φ() () 5 φ() li p() = n = ) (1 1Pj p j j 1 φ(p) p = 1 1 p 1 1 a, b a b n n (congruent odulo n) a b ( od n ) 7 a n b, c n b c ab ac [ ] b c n a(b c) n a n 8 n a [ ] n n a φ(n) 1 ( od n ) n a r 1, r, r 3,, r φ(n) (1) ar 1, ar, ar 3,, ar φ(n) () 4/1

n i j = ar i ar j ar i ar j a(r i r j ) 0 r i r j n r i r j () (1) a φ(n) r 1 r r 3 r φ(n) r 1 r r 3 r φ(n) ( od n ) a φ(n) 1 9 p a a p 1 1 ( od p ) a 4 009 009 3 [ ] φ(1000) = φ( 3 5 3 ) = 5 ( 1)(5 1) = 400 009 009 9 009 ( od 1000) 9 400 5+9 9 9 (81 ) 9 561 9 71 9 489 n 1 p = 7, a = 4 4 6 1 ( od 7) 4 3 = 64 1 ( od 7) 6 3 1 p 1 p a 1 p 1 5/1

n = 1 = 3 7 a φ(1) = a 1 1 ( od 1) a φ(3) = a 1 ( od 3) a φ(7) = a 6 1 ( od 7) (a ) 3 1 3 7 a 6 1 ( od 1) 1 6 1 10 n P j j (P i,, P k, j 1) n [ ] ( j + 1) 11 n P j j (P i,, P k, j 1) n j P i j = i=0 P j+1 j 1 P j 1 P j+1 j 1 (P j 1)( j + 1) [ ] 5 009 [ ] (1 + 7 + 7 )(1 + 41) = 57 4 = 394 394 6 = 399 6/1

1 n n nφ(n) n n n n {n φ(n) + 1} n {n φ(n) + 1} {n φ(n)} [ ] n n 0 n = 10 0 4 5 6 8 10 0 n 0 0 1 n n φ(n) + 1 n {n φ(n) + 1} n {n φ(n) + 1} {n φ(n)} n(n + 1) n n nφ(n) n 7/1

0 0 n n n 0 n 6 [ ] 13 009 009 009 {009 φ(009) + 1} {009 φ(009)} = 009 {009 1680 + 1} {009 1680} = 47355 47 n n S n 3 S 0 ( od n) [ ] 1 S = nφ(n) 5 n 3 φ(n) p (p 1) φ(n) S n 14 n a, b n = ab a φ(b)+1 a ( od n) b = p p [ ] a p a ( od n) a φ(b)+1 a a(a φ(b) 1) ( od n) a φ(b) 1 b a(a φ(b) 1) 0 ( od n) a n n φ(b) a φ(b) 1 0 (od n) a ak b (ak) φ(b)+1 (ak) ( od n) 7 1 009 + 009 + 3 009 + + 009 009 1 8/1

[ ] 1 od 1 009 = 7 41 1 = 3 7 1 1 1,, 4, 5, 8, 10, 11, 13, 16, 17, 19, 0 φ(1) = 1 ( 5 φ(3 7) = (3 1)(7 1) = 1 ) 1 a a 1 1 009 = 1 167 + 5 a a 009 a 5 a 1,, 4, 5, 8, 10, 11, 13, 16, 17, 19, 0 (3) a 5 1, 11, 16, 17, 8, 19,, 13, 4, 5, 10, 0 (4) (3) a 5 1 + 11 + 16 + 17 + 8 + 19 + + 13 + 4 + 5 + 10 + 0 0 13 009 = 1 95 + 14 (5) 1 1995 1 a a 009 1 0 1 a a 009 1 0 1996 009 1 a a 009 3 7 1 a a 009 (4) 1996 009 1 a a 009 1 + 11 + 16 + 17 + 8 + 19 + + 13 87 3 (6) 3 009 3 14 3 009 + 6 009 + 9 009 + 1 009 + 15 009 + 18 009 3 5 + 6 5 + 9 5 + 1 5 + 15 5 + 18 5 3 5 (1 5 + 5 + 3 5 + 4 5 + 5 5 + 6 5 ) 9/1

(4) 3 5 (1 5 + 5 + 3 5 + 4 5 + 5 5 + 6 5 ) 1(1 + 11 + 1 + 16 + 17 + 6) 1 0 0 (5) 1 1995 3 009 1 0 1996 009 009 3 5 (1 5 + 5 + 3 5 + 4 5 ) 1(1 + 11 + 1 + 16) 1 19 18 (7) 7 009 14 7 009 + 14 009 7 + 14 0 1 009 7 1 0 (6),(7) 3 + 18 0 [ ] 1 1 a 7 a 3 a 7 a 7 a 3 a 1 a 0 1,,6 6 a a 7 a 1 009 + 009 + 3 009 + + 009 009 1 5 + 5 + 3 5 + + 009 5 5 10/1

(1 + + 3 + + 009) 5 = (1005 009) 5 (18 14) 5 0 5 = 0 15 n a ax b ( od n) (8) x a φ(n) 1 b ( od n) [ ] (8) ax ny = b φ(n) a φ(n) 1 n 16 n n φ(p i ) = p n i=1 [ ] 4 = p n p n 1 + p n 1 p n + + p 1 + 1 = p n 17 n d n φ(d) = n [ ] n = P j j (P i,, P k, j 1) 11/1

d n φ(d) =(φ(p 1 1 ) + φ(p 1 1 1 ) + + φ(1))(φ(p ) + φ(p 1 ) + + φ(1)) (φ(p k k ) + + φ(1)) =P 1 1 P P k k =n 18 n d n 1,, 3,, n (x, d) = d x ( n ) φ d [ ] x = dx, n = dn x 1,, 3,, n n (x, n ) = 1 x φ(n ) 1/1