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7 N N i a b a 1 1 b a < b a b b a ii N iii N A 1 A x A x + 1 A A N ii { 5, 6, 7, } {a a 5 } { } { } 1, 1 4, 1 1 8, k k N a b a + b ab a b N a + b ab N N Z N Q a a { a a > a 0 a a < 0 6

8 1.1. iii pn n pn n 1 p1 pk pk , n pn 1 3 iii pn M M {n pn, n N} n pn n M N 1 1 M k M k + 1 M iii M N,1 k k + 1 M N n M n

9 0 Z Z {, 3,, 1, 0, 1,, } Z + Z x y x + y x y Z I i x + y + z x + y + z ii x + y y + x iii x x + 0 x 0 iv x x + y 0 y x II i x y z x y z ii x y y x iii x x 1 x 1 III x y + z x y + x z Z Z G G i a, b G a b G ii a b c a b c iii a G, a e e a a e G 8

10 iv a G, a x x a e x G x a 1 a, b G a b b a Z M a b c a b a c, b c a b a c a M Z Z a b a b 1 a 1 a a 0 a + 1 a 0 a + a 0 a a {1 + 1} a 1 a + 1 a a + 1 a 1 a a 3 a + a 1 a + 1 a {1 + 1} a 0 a 0 a a a 1 a 1 { 1 a} { 1 1} a { 1 1} a a a b 1 a 1 b { 1 1} a b a b Z 9

11 K K K Q + Q Z Q Z iv 0 x x y 1 y y x 1 Z Q N Z Q 1.. Z a b 1 a qb + r q 0 < r < b r q a b r 1 a, b a qb + r, 0 < r < b q, r a b qb < a < q + 1b 1.1 q a < t + 1b t A A N {0} A q q 1 a < t + 1b qb < a a < q + 1b 1.1 q r a qb qb < a < q + 1b 0 < r < b a qb + r a > 0 a qb + r a < 0 a qb r r 0 q q a qb + r 0 < r < b a qb r q 1b + b r 10

12 b r 0 < b r < b q 1 qb r r a qb + r 1 q r q, r q 1, r 1 q, r q 1 q r 1 r q 1 > q, q 1 > q + 1 a > q 1 b > q + 1b q b + b > q b + r a q 1 < q q 1 q r 1 r q a b r r 0 a bq b 0 q a b b a a b b a r a, b a qb + r, b < r < b q, r e < r < e + b Z 1..1 b 1 a a a a b a bq q b a. a b a b 0 11

13 3 a, b, c, 0 a, b, c, least common multiple L.C.M. a, b, c, 1 a, b, c, greatest common measure G.C.M. 1 a b a, b 1, a, b l d ab dl 4 a, b c b bc a c a 1 a, b, c, l, m m l q r m ql + r, 0 < r < l l m a l al, m am r m ql am ql, r a b, c, r a, b, c, l r 0, l, l r 0 m l a, b, c, d, m l d m a m d m d 1 a l b, c, l l a, b, c, d l < d, l d m d < l l d 1

14 d m l d d m m d 3 l a, b a b l ab ba ab a, b 1 ab l ab ml ab ml ma b, ab ml mab a ma, b mb m a, b d d me d a, b a da, b db a da mea, b db meb a ma, b mb ma mea mb meb a ea, b eb l e e 1 l ab aeb, l a b ea b l e ab a b a, b l m d ab dl 4 a, b 1 a, b ab bc a a b bc ab bc ab c a c a, a, b, c, a, b, c, a, b, c, 1 a, b, c, 1 a, b a, b 1 a, b

15 , Eukleides 365? 75? Euclid , 4 1, 1 3 4, a > b > 0, a b r a, b r, b 1 {r n } r 1 a, r b n > r n > 0 r n+1 [r n 1 r n ] r n 0 r n

16 N r N 0 r N+1 0 r N a, b 1 a, b d 1 a a d 1, b b d 1. r a d 1 b d 1 q d 1 a b q r d 1.,d 1 b r, d 1 < b, r d b, r d, b b d, r r d. a b d q + r d d b q + r d < a, b d 1 d 1 d a, b b, r r k > 0 r 1 a > r b > r 3 > > r k > r k+1 > r k+ > > 0 N r N > 0 r N+1 0 r N 1 r N 1 a, b b, r 3 r N 1, r N r N a, b, c, a a, b, c, a b, c, a, b, c, a, b, c, 0 a, b, c, , , , , 7 7, , 34 34,

17 , 391, , 136, , 17, x y x, y, z x + 3y 1, xy x 3y + 1 0, 1 x + 1 y + 1 z 1, x 3y 1 Diophantos 3, x + 4y 3z 5 a b ax + by k 1 3 a 1 x 1 + a x + + a n x n k a b ax + by k. a > b > 0 a < 0 a x + by k a b a > b > 0 16

18 b b 0, a, b 1 a 1 1 x + 0 y k k, 0. 0 < j < b j c > j j c cx + jy k. ax + by k a > b > 0 a bq + r 0 < r < b ax + by bq + rx + by bqx + y + rx a b { b r { bx +ry k X 0, Y 0 qx 0 + y 0 X 0 x 0 Y 0 x 0 Y 0 y 0 X 0 qy 0 x 0, y 0 ax + by k a > b > 0 ax 0 + by 0 ay 0 + bx 0 bqy 0 bx 0 + ry 0 k b,, b b > 0, a > b a ax + by k. i m n m > n ii n n i 0,, b 1 ai b r i. r i 0 b 1 0 < i, j < b 1 r i r j r i r j. ai bq i + r i aj bq j + r j ai j bq i q j a b i j b. 0 b 1 < i j < b 1 0 b i j 0., i j r i r j 0 < i < b 1 b i r i r i 0 < i < b 1 0 b 1 17

19 k b q s r i s i ai bq i + s i q i s k bq ai + bq q i k x, y i, q q i ax + by k x + 13y 1 u 1,,..., 1 37u 13 11, 9, 7, x, y 6, 17 A A { ax + by x, y Z} A, d d ax 0 + by 0 A ax + by d > 0. ax + by dq + r 0 < r < d r ax + by dq ax qx 0 + by qy 0 r A r > 0 d A r 0 A d d nd anx 0 + bny 0 A a a 1 + b 0 a A b A a b d d a b a, b 1 d 1 A k ax + by k. 3 b 0 b 18

20 1 A a, b A a b A A A 0 a a A a 0 a a A a A a, b A a + b a b A A a n na A n 1 n 1a A na n 1a + a A A d A x d q r x dq + r, 0 < r < d dq A r x dq A r 0 d r A d r 0 x dq A d d A 5 a 1, a,, a n, k a 1 x 1 + a x + + a n x n k 1. k a 1 a, a n d a 1 a,, a n J {a 1 x 1 + a x + + a n x n x 1, x,, x n } J J J J e e J n e a k l k 1.3 k1 19

21 l k k 1,, 3,, n d e a 1 a a a n 0 a i J i e e a 1 a,, a n e d e d J d 1. k d a 1 x 1 + a x + + a n x n k x y fx, y 0 px, qx x pt, y qt t t x pt, y qt x pt, y qt fx, y 0 1 x, y, x + 57y + 68z 1 3, 57, x + 57y + 68z 3x y z 3x + y + z + 5y + 4z 4 8x + y + z y + 4z 4{8x + y + z + 6y + z} + y + 0x + y + z l 8x + y + z + 6y + z 8x + 14y + 17z, m y, n x + y + z 1.4 4l + m + 0n 1 0

22 l t, m 1 4t, n s t, s Z 1.4 8x + 14y + 17z t y 1 4t x + y + z s x 11 46t + 17s, y 1 4t, z 6 + 5t 8s t, s 3x + y 1 17x + 5y 1 a b 1 k a b k a b a b ax + by 1 x y k ax + by k a b ax + by 1 1 a > b a bq + r, 0 < r < b ax+by bq +rx+by rx+bqx+y y qx+y ax+by 1 rx + by 1 3 rx+by 1 x 0, y 0 y 0 y 0 qx 0 x 0, y 0 ax+by 1 a b b r 1 sx + y 1 x + ty 1 0, 1 1, 0 ax + by 1 17x + 5y x + 5y 1 1

23 , y x + y, 3x + 5y , x x + y, 3x + 6y , y 3x + y, 5x + 6y , x x + y, 5x + y 1 1 x, y 0, 1 x x + y x, y 1, 1 3 y 3x + y x, y 1, 4 4 x x + y x, y 9, 4 5 y x + y x, y 9,, a b x a b x ax + by c d y c d y cx + dy A, B x X y s u s t x u v z AB X AB X t, A sv tu, v y sx + tz sy + tw w ux + vz uy + vw sx + tzuy + vw sy + twux + vz tzuy + sxvw twux syvz sv tuxw yz s t x y u v z w A A 6 1 a, b Z, a > b > 0 a b q 0 r 1 a q 0 1 b 1 0 b r 1

24 a q 0 b + r 1, b q 1 r 1 + r,, r k 1 r k q k + r k+1 a b q q q k r k r k+1 P k Q k X k Y k q q q k X k P k 1, Y k Q k 1, P k Q k 1 P k 1 Q k 1 k+1 3 a b d a b P n Q n P n 1 Q n 1 d 0 n 4 x 1 n 1 Q n 1, y 1 n P n 1 ax + by d 5 ax + by 1 x 0, y 0 ax + by 1 x, y x 0 bt, y 0 + at, t Z 1 a bq 0 + r 1., a b bq 0 + r 1 b q b r 1 P k Q k X k Y k P k 1 X k 1 Q k 1 Y k 1 P k 1 q k + X k 1 Q k 1 q k + Y k 1 q k P k 1 Q k 1, X k P k 1, Y k Q k 1 3

25 P k Q k 1 P k 1 Q k P k P k 1 Q k Q k 1 q 0 1 q q k k+1 3 a b a bq 0 + r 1., a b r 1, b r 1 a, a b b r 1,, b > r 1 > 0. a b, b r 1, r 1 r,, b > r 1 > r, n, r n 0, r n+1 0., a b r n 0 0 0, r n a b d., n, a q 0 1 q 1 1 q n 1 d b P n P n 1 d 0. Q n Q n 1 4 a b P n P n 1 Q n Q n 1 P n P n 1 Q n Q n 1, 1 0 P n P n n+1 Q n 1 P n 1 Q n Q n 1 Q n P n 4

26 , 1 n+1 Q n 1 Q n P n 1 P n a b 1 0, a{ 1 n+1 Q n 1 } + b{ 1 n P n 1 } 1, x 1 n 1 Q n 1, y 1 n P n 1 ax + by 1. 5 x 0, y 0 ax + by 1, { ax + by 1 ax 0 + by 0 1, ax x 0 + by y 0 0, a b, x x 0 b., x x 0 bt t, y y 0 +at., t, x x 0 bt, y y 0 + at., t, x x 0 bt, y y 0 + at, ax + by ax 0 bt + by 0 + at ax 0 + by 0 1, x, y ax + by x + 5y ,

27 , x 9, y 1., , t,. { x 9 5t n y + 17t n 1 n 1 3 a 1, a,, a n a 1 a k q k a 1 + r k, k, 3,, n a 1 x 1 + a x + + a n x n a 1 x 1 + q a 1 + r x + + q n a 1 + r n x n a 1 x 1 + q x + + q n x n + r x + + r n x n k X 1 x 1 + q x + + q n x n, X k x k k,, n a 1 X 1 + r X + + r n X n k X 1 α 1, X α,, X n α n k a 1 α 1 + r α + + r n α n a 1 α 1 + a a 1 q α + + a n a 1 q n α n a 1 α 1 q α q n α n + a α + + a n α n x 1 α 1 q α q n α n, x k α k k,, n a 1 x 1 + a x + + a n x n k n 1 0 6

28 n > ,, p p + 4 e 1 e e , e e, 3, 5, 7, 13 e 3, 9 e 5 n + 1 n n 1,, 3, 4 5, 17, 57, n n + 1 k n + k + 1 5, , , a > 1 a 1 a 1 a a > 1 a a 0 : 1 : 1 : : 7

29 a a n 1 n 1 a a a a 1 a a b c 1 < b < a, 1 < c < a b c a a a a p 1 p p m q 1 q q n 4 p abc abc p p 1, p,, p m q 1 p 1 q 1, p 1, p 1 q 1 p p m q q n b b < a a n n n p 1 p p r p 1 < p < < p r n q 1 q q s q 1 < q < < q s 8

30 p 1,, p r, q 1,, q s r s r s p i q i i p 1,, p r q 1,, q s p i q j n n p 1 < q 1 m m n p 1 q q s q 1 p 1 q q s m < n m q 1 p 1 p 1 p 1 q 1 p 1 p 1 m m n p 1 q q s p 1 p p r q q s m p 1 m m < n n p p 1 p p p ab a b p 1 a b p ab p a b p 1 a p a p ab p 4 b p b p a p p a b p ab p a b 9

31 1 n m m n n m 3 8 n p 1, p,, p n a p 1 p p n a a p 1, p,, p n 1 p 1, p,, p n a a p 1, p,, p n p 1, p,, p n n 1 nn + 1n + n n n 3 n 4 3 n 3 n nn + 1n n 3 3n + 8n 6 30

32 a, b 1 7a + b 3a + b ps qr 1 p, q, r, s 3 11n 4 3n 13 pa + qb ra + sb n x + 13y + 15z 1 x + 6y + 5z + 7w a p α q β r γ a 1 a p x q y r z 0 < x < α, 0 < y < β, 0 < z < γ, a T a T a 1 + α1 + β1 + γ 3 a Sa Sa pα+1 1 p 1 qβ+1 1 q 1 rγ+1 1 r 1 4 a, bc 5 a T abc T at bt c Sabc SaSbSc a T a 5 5 a 1 a a a 4 Sa a a 31

33 1 n > 1 a n 1 n 1 n 1 a a, a, a, b, b, b, aa a bb b a, b 1 a n, b n 1 a 1, a,, a m b 1, b,, b n a 1 b 1, a 1 b,, a b 1,, a m b n 3 a 1, a,, a n p, q, a k p p α kp α k p > 0 a 1, a,, a n m l m p l p Minα1p, αp,, αnp p Maxα1p, αp,, αnp p Min Max 4 a 1, a,, a n d 1 n n C a 1 a, a 1 a 3,, a n 1 a n d a 1, a,, a n k n C k d k a 1 a a n d n i k, n d k d k 1 ii iii d k d k 1 e k e 1 d 1 e k e k 1 e 1 e e n a 1 a a n iv e n a 1, a,, a n 5 a, b, c, {a, b, c, } {a 1, m, a, m,, a n, m} {a 1, a,, a n }, m 6 l a, b, c, a, b, c, a 0, b 0, c 0, l a 0 b 0 c 0 3

34 7 7 1 p p C k p > k > 0 p k p l, p nc k p n > k > 0 p n l 8 8 n! p [ ] [ ] n n + p p + k1 [ ] n p k [x] x 9 9 m n m > 0, n > 1 n n pα q β m n 0 < x < p α, 0 < y < q β,, s > 0 x, y,, s m n x p α + y q β + ± s k, l k > l {a n }, {b n } a 1 k, b 1 l n > 1 { { b n b n 0 a n b n b n 0 a n+1 b n+1 b n 0 b n 0 a n 1 k 1998, l 185 {a n }, {b n } 5 k, l, n b n > b n+1 b n 0 3 k, l b n 0 n 4 b n 0 n a n k l b n 91 a > b a, b {r n } r 1 a, r b, { rn r n > n 1 r n 1 > 0 3 r n 0 r n 1 0 {f n } f 1 0, f 1, f n f n 1 + f n n > 3 33

35 1 r N > 0, r N+1 0 N N r N+ k > f k k 1,,, N f n+1 > 3 n n 1,, 4 N < + log 3 a n n > 3 S {a 1, a,, a n } S a i, a j a i a j, a j a i S a 1, a,, a n n n > 3 S {a 1, a,, a n } S a i a j a i a j a j a i S 1 a i > 0 i 1,,, n a i < 0 i 1,,, n a 1, a,, a n G i,ii i m, n G m + n G ii m, n G m > n m n G G d G {kd k } p, q 1 x p x qx q x pq p x > pq x a, b x pa + qb a, b 34

36 1 4m + 6n 7 m, n 3m + 5n m, n 3 k ak b rk k, l b 1 k l rk rl 4 am + bn 1 m, n 8 8 a, b, c a, b x 0, y 0 ax 0 + by 0 c 1 l m al + bm c l x 0 + bu, m y 0 au u c ab ax + by c x, y 3 c > ab ax + by c x, y 4 ax + by k, 0 < k < ab x, y k p, q 1 < p < q L L { m, n m, n 0 < m < q 1, 0 < n < p 1 } L Am, n NA mp + nq 1 L A, B NA NB A B L Am, n L A # q m, p n A # A 3 NA < pq p + q NA # > pq p + q 4 NA < pq p + q L A m n x 3m + 5n x 35

37 xy x y a, k a > L : ax + a + 1y k 1 L k aa + 1 x > 0, y > 0 L 3 k > aa + 1 x > 0, y > 0 L x y O0, 0, Aa, b, Ba, b + 1, C0, 1 a, b 1 1 OABC. 1 P 1, P,, P t OP i A i 1,,, t a > xy x y m, n r 5 r n fx πx sin n 1 {fk k } m n { } n 1 fmk k 0 < k < m x + y < x, y x + y 3 + z 6 < 10 0 x, y, z 36

38 k k a m a, m, fk a., 1 S 50. S n n S n n. 3 n 1 < S n < n n fk k1 n d n a k 1 < k < d a 1 1 a d n a k < a k+1 1 < k < d n n 60 n 6 1 a a 6 1 a ,, N 60 N a, b, p, q p + q a pq b a b 1 1 pq b a + b a, b, c a + b c a, b 1 a b c a a + c d d 37

39 1 1 0 a, a fa 1 a f1 1 a ,3,5,15 f a b m a m b fa m+l 1fb a p q a pq fa > p + 1q q 1 p 3 a, b m, n r, s a m r, b n s a, b { fa b fb a r, s r n+1 1, s m

40 a b m a b m a b mod. m mz mz {mx x Z } mz mz a b mod. m a b mz 9 a b m a b m a b m a b mq a b m q 1, q r 1, r a mq 1 + r 1 b mq + r a b mq mq q 1 q m + r 1 r m q q 1 + q r 1 r q q 1 + q 0 m q q 1 + q > m m r 1 r < m q q 1 + q 0 r 1 r 0 m m a b m a mq 1 + r, b mq + r a b mq 1 q a b m 39

41 A a b ab i aa ii ab ba iii ab, bc ac,.1.1 p a, b ab a b p A a A a A a a b a b a b a a a A {a a A} A a a mod. m a b mod. m b a mod. m a b mod. m b c mod. m a c mod. m a b, b c mz a c a b + b c mz Z m Z m m m m m Z Z / mz Z m m m Z m m 40

42 m m {0, 1,, 3, 4, 5, 6} {0, 1,, 3, 3,, 1} {7, 6, 9, 4, 10, 9, 13} 7 10 a a mod. m, b b mod. m a ± b a ± b mod. m ab a b mod. m.1 a a mod. m, b b mod. m, c c mod. m, fx, y, z, x, y, z, fa, b, c, fa, b, c, mod. m. a a mod. m a a m b b mod. m b b m a+b a +b a a +b b m, ab a b a a b+a b b m.1.1 a a mod. m N Na Na mod. m Na α b β c γ Na α b β c γ mod. m.1 Na α b β c γ Na α b β c γ mod. m. Z m a a m a m Z m a + b a + b, a b a b 41

43 Z m c a + b c a + c ab 0 1 Z m m..1. m 6 0 {, 1, 6, 0, 6, 1, } 1 {, 11, 5, 1, 7, 13, } {, 10, 4,, 8, 14, } 3 {, 9, 3, 3, 9, 15, } 4 {, 8,, 4, 10, 16, } 5 {, 7, 1, 5, 11, 17, } Z 6 Z / 6Z {0, 1,, 3, 4, 5} , + 0, , p Z p Z / pz p 0 a a 1 p 1 p a pq + ab 1 b a b ab 1 b a 1 a Z p 4

44 .1. fx fx 0 mod. m x 0, x 1 x 0 10 fx 1 fx 0 0 mod. m m x 0 m, x 0, 1,, m 1 m 1 ax b mod. m a, m 1 a, m d > 1 b d d a, m 1 {x 1, x,, x m } m {ax 1, ax,, ax m } m ax i ax j mod. m, a m x i x j mod. m i j b {x 1, x,, x m } ax i b mod. m x i a, m d > 1 ax b mod. m.3, ax b mn N b ax mn d a, m a da, m dm, b db 43

45 .3 11 a x b mod m.4 a m.4 x m x x 0 mod. m.4 x x 0 + m t t.5 t 1 t x m m t 1 t 0 mod. m t 1 t 0 mod. d.5 t d {0, 1,, d 1}, m.3 d.3 ax + my b 13 m 1, m,, m k, a 1, a,, a k x a 1 mod. m 1 x a mod. m.6 x a k mod. m k x M m 1 m m k x x a 1 + m 1 t.7 a 1 + m 1 t a mod. m m 1 t a a 1 mod. m, m 1 m t t 0 + m s m.7 x a 1 + m 1 t 0 + m 1 m s 44

46 x a 1 + m 1 t 0 mod. m 1 m.6 x x 0 mod. M Chinese Remainder Teorem, Gauss M m 1 m m k M m 1 M 1 m M m k M k.8 M n m n M n t n 1 mod. m n n 1,,, k.9 t n n 1,,, k.6 x a 1 M 1 t 1 + a M t + + a k M k t k mod. M.9 a n M n t n a n mod. m n, M 1,, M k M n m n x a n mod. m n n 1,,, k, x 1 x.6 x 1 x mod. m n n 1,,, k, m 1, m,, m k M x 1 x mod. M 45

47 .1.3 fx a 0 x n + a 1 x n a n fx 0 mod. m x a i m a 0 0 mod. m n 14 p, n fx 0 mod. p.10 n.10 p n n 1 a 0 x + a 1 0 mod. p, a 0, p 1 1, p n 1 n 1.10 x a mod. p fa 0 mod. p 37 fx x af 1 x + fa n n 1 f 1 x fx a k x n k k0 n 1 fx fa a k x n k a n k k0, f 1 x.10 x af 1 x 0 mod. p p x a f 1 x p x a mod. p n 1 f 1 x 0 mod. p n 1.10 n fx 0 mod. m 46

48 m m m p m p e, m p m p e m p e+1 15p m p e m m p e q f, m p e, m q f, e p fx 0 mod. p e+1.11 fx 0 mod. p e.1.11 p e.1 x.11.1 x 0 x x 0 + p e y.13.1 x 0.13 x.11 y fx fx + y fx + yf x + + y k f k x k! + + y n f n x n! f k x x n k k! x x 0 y p e y fx fx 0 + p e y fx 0 + p e yf x 0 + p e y f x 0! + f x 0, f x 0 3 p e+1! fx 0 mod. p e+1 fx 0 p e fx 0 + p e yf x 0 0 mod. p e+1 fx 0 p e + yf x 0 0 mod. p.14 47

49 1 f x 0 0 mod. p.14 p y 0 mod. p.11 x x 0 + p e y 0 mod. p e+1 f x 0 0 mod. p.14 fx 0 p e p fx 0 p e p p y x 0, x 0 + p e, x 0 + p e,, x 0 + p 1p e mod. p e+1.11 f x 0 0 mod. p.1 x 0.13 x fx 0 0 mod. p e+1 y.11 fx 0 0 mod. p e+1.11 p e+1 p.1.3 p, a p x a mod. p ±x 0 x 0 0 mod. p, x 0 x 0 mod. p, fx x a, f x x f ±x 0 ±x 0 0 mod. p 1 x a mod. p e x 0 ±3 x mod. 7 x mod. 49 x 3 + 7y 3 + 7y mod y mod. 49 6y 1 mod. 7 y 1 mod. 7 x 10 mod. 49 x mod

50 16 m m p e q f fx 0 mod. p e.15 fx 0 mod. q f.16 l, l, fx 0 mod. m.17 ll x α mod. p e x β mod. q f.18 α, β, p e, q f, fx 0 x x x 1 mod. 3 x 1 mod. 4 α ±1 mod. 3 β ±1 mod. 4 x 1 mod. 1 } x 1 x 1 x 1 x 1 } x 1 x 1 } x 1 x 1 } mod. 3 mod. 4 x 1, x 7, x 5, x 11 mod φn 1,,, n n x φn φ1 1, x 1 49

51 φ 1, x 1 φ3, x 1, φ4, x 1, 3 φ5 4, x 1,, 3, 4 φ6, x 1, 5 φn φn p φp p 1 p φp e p e p e 1 p e 1 1 p 1 p e p e p 1 p, p,, p e 1 p p e 1 n Z n n Z n {0, 1,, n 1} k n n k x x k + nt t k + nt, n k, n k n n k φn F x a b F ab F af b 17 φn a b φab φaφb.19 a k a b l b A {bx + ay x k a, y l b } 50

52 ab s, t x k+as, y l+bt bx + ay bk + al + abs + t s, t ab A bk + al ab bx + ay ab a b a b a bx + ay, a 1 bx, a 1 x, a 1 k, a 1 k a a k k mod. a k bk + al bk + al mod. ab bk + al bk + al mod. ab bk bk mod. ab a b k k mod. a ab m ab a b bk + al m k, l m ab Z a k Z b l a k a b l b ab m ab φaφb α 1, α,, α m, m φa β 1, β,, β n, n φb a b mn φaφb α i, β j γ α i mod. a, γ β j mod. b.0 γ ab γ ab γ, ab 1.0 α i, β j ab γ α i, β j.19 φn 18 n n p α q β r γ φn n p q r 17 φn φp α q β r γ φp α φq β φr γ 51

53 p α p α 1 q β q β 1 r γ r γ 1 p α 1 1 q β 1 1 r γ 1 1 p q r n p q r.1..1 a 3, b 5 α 1,, β 1,, 3, 4 φ3, φ5 4 γ 1,, 4, 7, 8, 11, 13, 14 φ15 8, 5α + 3β mod. 15 γ α β α + 3β γ d n d n d 1, n d n d n n 1 n 19 φd n. d n n d n 1 n x x, n d φ d x x, n d d, n 1 d 1 n d n d n φ d 1 n x x, n { 1,,, n } { x x, n d, 1 < x < n } d n d n 1 n 1 n n φ n d d n d n n d n. 5

54 .. n 15 d x n φ d 1 1,, 4, 7, 8, 11, 13, 14 8 φ15 3 3, 6, 9, 1 4 φ5 5 5, 10 φ φ n 1 d x n φ d 1 1, 5, 7, 11 4 φ1, 10 φ6 3 3, 9 φ4 4 4, 8 φ φ φ1 1.. F n, Gn F d Gn.3 F n Gn Möbius µn..4 d n 1 n 1 µn 1 k n k 0 n µ1 1, µ 1, µ3 1, µ4 0, µ5 1, µ6 1, n > 1 µd 0 d n 53

55 n > 1 n e n p 1 e e 1 p p k k x x µd µp1 1 x x p p k k d n 0 < x 1 < e 1, 0 < x < e,, 0 < x k < e k x 1, x,, x k 0 µd µ1 + {µp 1 + µp + + µp k } d n +{µp 1 p + µp 1 p µp k 1 p k } + +µp 1 p p k 1 k + k C k C k 1 1 k 0 µn F n, Gn 0 F x, Gx F d Gn d n F n d n µ n Gd d n n µ F δ d d n δ d δ δ d n d, n d n δ n δ δ n F δ δ n δ µδ > 1 0 F nµ1 n µ Gd F n d d n 54

56 n d µ δ F d d n d n δ n δ d δ n δ Gδ µδ Gδ Gn F n φn Gn n 19. n φn φn d n µ n d d φn n p α q β r γ φn n µ d µd n d d d n d n µ1n + µp n p + µqn q + µrn r + +µpq n pq + µpr n n µpqr pr pqr n n p n q n r + + n pq + n pq + n pqr n p q r VISUAL m m n 1, n,, n m N n 1 + n + + n m n 1, n,, n m g m n 1, n,, n m F n 1, n,, n m F n 1, n,, n m N! n 1!n! n m! fn 1, n,, n m 1 pn 1, n,, n m 55

57 1 d q dn 1, n,, n m pn 1, n,, n m q 1 n 1, n,, n m 3 1 q d n 1, n,, n m > 1 d n 1, n,, n m g fn 1, n,, n m d g q d n 1, n,, n m 3 F n 1, n,, n m N d g 1 d q dn 1, n,, n m n1 4 d g q d n 1, n,, n m p d, n d,, n m d 1 1 d d d n 1, n,, n m g 1 d d g d d 3 1 d 1 d N d F n 1, n,, n m > N 1 d q dn 1, n,, n m d g 4 1 d 1 d N d n 1 d, n d,, n m d n 1 d, n d,, n m d d N 1 d fn 1, n,, n m n1 p d, n d,, n m d d g F n 1, n,, n m N 1 d p n1 d, n d,, n m d d g 56

58 a j n j g F d F da 1, da,, da m n1 p d, n d,, n m g d d d p da 1, da,, da m d g d g d d Gd N g dp da 1, da,, da m F g d g Gd g n g Gg d g µ g F d d g fn 1, n,, n m d g p da 1, da,, da m p da 1, da,, da m Gd e d µ d F e e g dn Gd fn 1, n,, n m g d µ F ea 1, ea,, ea m dn e d g e d g N F ea 1, ea,, ea m d 1 µ e d e g d:e d g g N F ea 1, ea,, ea m µ j 1 je e g j g e g 1 N e F ea 1, ea,, ea m µ j 1 j e g j g e µ j 1 j e g g φ e j g e fn 1, n,, n m 1 g φ F ea 1, ea,, ea m N e e g 1 n1 φ d F N d, n d,, n m d d g 57

59 .3 n.3.1 n 1 n x n n cos kπ n kπ + i sin, k 0, 1,, n 1.5 n n 1.4 n n.4.5 α cos π n + i sin π n α k k 0, 1,, n 1 α n 1.5 k n k, n 1.5 kπ n π n α k n 1 1 n n 1 1 n 1 1 n φn cos kπ n kπ + i sin n.6, k n k, n 1.6 kπ n π n α k n 1 α k n β n β x n 1 0 β α l l, n d > 1 n dn, l dl cos lπ n lπ + i sin n cos l π n + i sin l π n, α l n 1, n 1 l, n 1 α k n n k α k φn α l n 58

60 , 1, 1 ± 3i, 1 ± 3i 6 1 ± 3i n n p α q β r γ F n x xn 1x n n pq 1x qr 1 x n n n.7 p 1x q 1 x pqr 1 x n d 1 µd d n.8, F n x 1 n F n x φn 1 µn 1 n 1 F n x 0 1 n n d n d 1 d n n n n n d d d n 1 n F d x x n 1 x F d x d n log F d x logx n 1 d n n d 6 0 log F n x µd logx n d 1 d n F n x d n x n d 1 µd x x x.8 F n x 1 φn F n x 1.3. F 6 x x6 1x 1 x 1x 3 1 x x + 1 F 1 x x1 1x 1 x 6 1x 4 1 x4 x

61 p F p x xp 1 x 1 xp 1 + x p F p ex x pe 1 x pe 1 1 e 1 xp p 1 + x pe 1 p m a m a φm 1 mod. m.9 m p, a, p 1 a p 1 1 mod. p.30 m φm x 1, x,, x φm ax 1, ax,, ax φm a, m 1 x y mod. m ax ay mod. m x ax x m a ax, ax x m ax m {x 1, x,, x φm } {ax 1, ax,, ax φm } m m x 1 x x φm ax 1 ax ax φm mod. m a φm x 1 x x φm mod. m x 1 x x φm, m 1 a φm 1 mod. m 60

62 .9 m p φp p φ mod φ mod φ mod φ mod φ mod mod. 5 a a e 1 4 mod mod. m e a m a m e a k 1 mod. m k e φm e m φm k e q r 1 a k a eq+r a r mod. m 0 < r < e r 0 a r 1 mod. m r e r 0 k e.4. 5 m n n 10, n 1 m n e, e 10 e 1 mod. n e φn, n 10 e 1 n a 61

63 m n ma na ma 10 e 1 ma 10 e + ma ma + 10e 10 3e + m < n ma < na < 10 e m n m e c n e m n c 10 e + c 10 e + c 10 3e + c 10 e 1 m n 10 e 1 na, c ma 10 e 1 mod. n 10, n 1 n u 5 v n u v k k maxu, v 10k m m n n 10, n 1 10 n e m n 1 n e.4. n mod. 7, 10 mod. 7, mod. 7, mod. 7, mod. 7, mod e { }

64 n e { }

65 ,, 3,?. a + dk k 0, 1,, a d a 1, d 4 1, 5, 9, 13, 17, 1, 5, 9, 33, 37, 41, 64

66 a 3, d 4 3, 7, 11, 15, 19, 3, 7, 31, 35, 39, 43, a 1, d 3 1, 4, 7, 10, 13, 16, 19,, 5, 8, 31, a, d 3, 5, 8, 11, 14, 17, 0, 3, 6, 9, 3, 6 a> 0 d> 0 Dirichlet 1837 a 1 7 m 1 + mt t Z 1 4n 1 4n 1 p 3, 7,, p 4n 1 4n a a p 1 a a 1 mod k + 1 4k + 1 4n 1 4n 1 a 4n 1 3 p 4n 1 3, 7,, p 3, 7,, p 4n 1 4n 1 4n + 1 4n + 1 4n + 1 x + 1 4n , + 1 5, , , , , x + 1 p x mod. p 65

67 x 1 mod. p x 4 1 mod. p x x 1 mod. p, x 1 mod. p x mod. p 1 1 mod. p p 4 p 1 4 p 1 4n p 4n + 1 4n + 1 4n + 1 p 4n a a 5 13 p + 1 a q x + 1 4n + 1 q 4n + 1 a p 4n + 1 q 5, 13,, p 4n + 1 5, 13,, p 4n + 1 4n mt + 1 m 1 m m 4 4n + 1 4n + 1 x + 1 F 4 x 1 4 ±i m F m x a F m a ±1 F m a m mt + 1 x m 1 F m xgx Gx x a m 1 F m aga F m a, Ga p F m a a m 1 p a m 1 mod. p 66

68 a e 4 e m m ef m > e x m 1 x e 1 x e 1 F m x F m x m 1 x m 1 x e 1F m xhx.31 Gx x e 1Hx Hx.31 x e 1 x ef 1 + x ef + + x e + 1 F m xhx x a a e 1 mod. p f F m aha 0 mod. p p f m ef m e m a m p 1 p mt + 1 F m a ±1 F m a m mt + 1 a m a m 1 mod. p a, p 1 p m a p mt + 1 F m a ±1 F m a ±1 a a m mt + 1 mt + 1 p mt + 1 m m mp p mpt + 1 p mt + 1 p p mt m 1 x 1 1 x 6 + 1x 6 1 x 6 + 1x 1x 4 + x + 1 F 1 x x 4 + x + 1 a 6 F , 97 1 mod ,, 3, 4 67

69 mod mod mod. 5, mod mod mod. 5 1,, 3, a a a a a a a a a a a a p, a p a p 1 1 mod. p a , a, a,, a a p 1 1 a p p 1 n n 1 1 n p K {0, 1,,, p 1}, p 1 1 p a e 1 e p 1 1 e a p p 1 K p K 0 K p 1 p K, 68

70 8 p r, r, r,, r p a p a 0 mod. p a m a m 1 mod. p a 0 1, a 1,, a m 1.3 x m 1 mod. p.33, 1 < i < m 1 a i a j mod. p a i j 1 mod. p 4 i j m m + < i j < m i j m p 1 a m < p 1 a m p p 1.3 b b n n m b m 1 mod. p b.33 b 1 m, n 1 ab mn ab mn a mn b mn 1 mod. p ab x 1 mod. p ab mx b mx 1 mod. p 4 mx n m, n 1 x n x m x m n m, n 1 mn ab mn mn > m p m m, n d > 1 m n l 6 6 l m 0 n 0, m 0, n 0 1 m 0 m n 0 n a m m 0, b n n 0 m 0, n 0 m 0, n 0 1 a m m 0 b n n 0 m 0 n 0 ln m l > m p m p 1 p 1 r.3, r, r,, r p p 1 k, p 1 1 r k φp φ13 1 4, 6, 7, 11 69

71 .5. 8 r p a 0 r α a mod. p mod. p a α 0 < α < p 1 α r a index Ind r a α α 0 < α < p 1 r s a mod. p s α mod. p 1 s, a p 1 Ind r a s mod. p 1 a b mod. p Ind r a Ind r b mod. p 1 r p p 0 a r α a mod. p α p 1 α p 1 r a Ind r a Ind r a p 1 Ind. a.5. p 13 p a I Ind. a a I p r Ind. ab Ind. a + Ind. b Ind. a n n Ind. a mod. p 1 70

72 Ind. a α, Ind. b β a r α, b r β mod. p ab r α+β mod. p Ind. ab α + β Ind. a + Ind. b mod. p 1 Ind. a n Ind. a a n 1 Ind. a + Ind. a n 1 mod. p 1 Ind. a n n Ind. a mod. p 1 Jacobi Canon arithmeticus Jacobi Cunninghana Jacobi Messenger of methematics, p 13 7x 10 mod. 13 Ind. 7 + Ind. x Ind. 10 mod Ind. x 10 mod. 1 x 7 mod. 13. Ind. x 1 11 mod p, a 0 mod. p x n a mod. p f p 1 n, p 1 a f 1 mod. p x n a mod. p p r n Ind r x Ind r a mod. p

73 n, p 1 e 1.34 Ind r a e Ind r a α α e α eq a r eq mod. p a f r efq r p 1q 1 mod. p a f 1 mod. p r fα 1 mod. p fα p 1 ef α e e n, p 1 K x n a mod. p a p n p 0 n a 0 mod. p a n, p 1 1 a n n, p 1 e > 1, Ind. a e a n p 1 ef 0, e, e,, f 1e n n p p 1 f p 1 e.5.4 n, p 7 e, f 3 1,,, 6 1, 4, a a a n 10 n + a n 1 10 n a a 0 a a 0 + a a n mod 9 a a 0 a n a n mod , ,

74 n, 13 n n 1 n 7 0, 1,, 4 n 5 n 10 3 n n n 4 + n n + 10n a, b a + b c c a, b 3 a, b, c a + b c a, b, c a, b a 11 3a 3 + b 11 4 b x 1 mod m, n d l { x a mod. m, x b mod. n. a b mod. d l 73

75 19 19 n x a i mod. m i, i 1,,, n a i a j mod. m i, m j, i, j 1,,, n m 1, m,, m n x + x mod. 5 x 1 mod α 1 mod. 8 e > 3 x α mod. e e x 0 4 ±x 0, ±x 0 + e xy 0 < x < 10 < y < a, b, c, Φx x a, b, c, [ ] [ ] [ ] x x x Φx [x] a b c [ ] [ ] [ ] x x x ab ac bc [ x abc ] [x] x 74

76 5 5 F n Gn d n F d Gn 6 6 x n φn, x φn, n φn [x] x 19 0 φd, dx [nx] d n φn, x d n µd [ ] x d 7 7 F n x n F n x φn 1 µn 1 n µn 9 9 α 1 n α k k 0, 1,, n 1 n k, n 1 k n a, b 1 a b ab 1 ab 1 a b 1 ab,j a m e a k 3 3 e k, e 75

77 1 fx n p fx p f0, f1,, fp 1 p n n fx n p f0, f1,, fp 1 p n + 1 fx p 3 p p 1! + 1 p [] fx x 1x x p + 1 x p p > p p! 1 p x k x + 1x k x k 1 + x k x + 1 m m! + 1 4n + 1 4n n , p p 13, r 1 Ind. 100 Ind. 1 3 Ind. x 9 x 4 Ind. x 1 x x 1 11x 5 mod

78 x 3 5 mod x + 3x 10 0 mod p p a + b p Ind. 1 p 1 Ind. a Ind. b p 1 mod. p Ind r a Ind r a Ind r r mod. p k p 1 1 k + k + + p 1 k 0 mod. p 4 4 p p 1! 1 mod. p 8 n 3 n n n 19 n + 1 n 1 4n n fx x n + a 1 x n a n 1 x + a n n > 1 1 α fx 0 α k> 1 k f1, f,, fk k fx 0 77

79 n n 9 n N ϕn N N ϕn {n n 1 < n < NgcdN, n 1 } gcda, b a b A A ϕ6 {1, 5}, ϕ15 {1,, 4, 7, 8, 11, 13, 14} 8 1 p q N pq i N n gcdn, n 1 ii ϕn p q N pq N ϕn p q N ϕn 3 N ϕn N pq p > q p q p q, i α cos π 3 + i sin π 3 1 α n n 1 α n 1 α n 1 α 3n 1 α 4n 1 α 5n 1 α1 α 1 α 3 1 α 4 1 α 5 p n z 1, z,, z n a n k 1,,, n z k p 1 z k 1 z 1 z z n 1 78

80 1 a 3 a n+ a n a n+1 3 a n G G z, w zw G 1 n G n G p 1 < r < p 1 r, r p C r p p 1 C r 1, p C r p. p p p. 3 n n p p, n fn, gn fn n 7 7 gn 3f. k1 1 n fn 7 fn. n gn. gn k n 79

81 , 1,, 3, 4 0, 1 1, 4, 3 4 mod. 5, 4 1 mod a b b a 5 0, 1 0, 1 p x a a 0 mod. p a p mod. p a a p a +1 1 p Legendre p Ind. a a a 1 Ind. a 3.1 p p p 1 x p x p 1 mod. p 1,,, 3.1 a a 1 a a mod. p p p abc a b c p p p p 80

82 , p mod. 5, 4 3 mod p a a a p 1 mod. p p a 30 a p 1 1 mod. p a 1 a p 1 1 mod. p p a 1 a p 1 1 mod. p a p 1 p 1 mod. p a p 1 1 mod. p. a p A A r A { 1,, 3,, p 1 } rs a mod. p A s s r a 1 r x a mod. p, r r p r p x a mod. p, p r p r A x a mod. p 14 p p p r p 3 p 3 a p 3 mod. p rp r r a mod. p a 1 p 1! a p 1 mod. p p 81

83 a 1 A p a 1 p 1! a p 1 mod. p p 1 1 p p 1! 1 mod. p a p a p 1 a p a p 1 1 mod. p a ±1 p a p 1 a p 1 1 mod. p p 5 1, 4, 3 3, 4 1 mod , 1, , p , 1, 3 1, mod n n x 1 + x + x 3 + x 4 0 < x 1, x, x 3, x 4 0 8

84 x 1 + x + x 3 + x 4 y 1 + y + y 3 + y 4 3. x 1 y 1 + x y + x 3 y 3 + x 4 y 4 + x 1 y x y 1 + x 3 y 4 x 4 y 3 +x 1 y 3 x y 4 x 3 y 1 + x 4 y + x 1 y 4 + x y 3 x 3 y x 4 y n p n n p > 1 p x + 1 ph 1 p 1 p 1,,, p 1 p 1 p 1 k, k + 1 k k 1 1 k p p p x 1, x x 1 k x k 1 mod. p, mod. p, x 1 + x mod. p x 1 + x + 1 ph p x 1 + x + x 3 + x 4 ph 3.4 x 1, x, x 3, x 4 h x h > 1 x 1, x, x 3, x 4 x 1, x, x 3, x 4 1 < h < h x 1 + x + x 3 + x 4 ph, h h x 1, x, x 3, x 4 h y 1, y, y 3, y 4 x 1 y 1, x y, x 3 y 3, x 4 y 4 mod. h h y i <, i 1,, 3, 4 83

85 y 1 + y + y 3 + y 4 x 1 + x + x 3 + x 4 0 mod. h y 1 + y + y 3 + y 4 hh 3.3 z 1 + z + z 3 + z 4 ph h z 1 x 1 y 1 + x y + x 3 y 3 + x 4 y 4 x 1 + x + x 3 + x 4 0 mod. h z x 1 y x y 1 + x 3 y 4 x 4 y 3 x 1 x x x 1 + x 3 x 4 x 4 x 3 0 mod. h z 3 x 1 y 3 x y 4 x 3 y 1 + x 4 y x 1 x 3 x x 4 x 3 x 1 + x 4 x 0 mod. h z 4 x 1 y 4 + x y 3 x 3 y x 4 y 1 x 1 x 4 + x x 3 x 3 x x 4 x 1 0 mod. h z 1 ht 1, z ht, z 3 ht 3, z 4 ht 4 t 1 + t + t 3 + t 4 ph hh y 1 + y + y 3 + y 4 < 4 h < h h h h y i h i 1,, 3, 4 h x i y i + m i h m i + 1 h i 1,, 3, m h 4 + m + 1 h 4 + m h 4 + m h 4 ph {m m m m } h 4 p m 1 + m m 4 + m 4 + 1h h p h < h 84

86 p q 33 p, q p q 1 : 1 p 1 q 1 q p 1 : 1 p 1 p 3 : 1 p 1 8 p 1 p 1 q 1 1 p 1 q 1 1 q p p 1 mod. 4 q 1 mod. 4 p q q p p q 3 mod. 4 p q 1 p 1 p 1 { 1 p 1 mod. 4 1 p 3 mod p 1, 7 mod. 8 p 1 p 3, 5 mod. 8 p Leonhard Euler Adrien Marie Legendre Karl Friedrich Gauss

87 a p 1 a, a, 3 a,, p 1 a, 3.5 p p n a 1 n p p p p p p, n 3.5 ±1, ±,, ± p p p 1 p 1 1,,, n 1a a 3a p 1 a 1n 1 p 1 mod. p a p 1 1 n mod. p 31 a 1 n mod. p p ±1 p a 1 n p 3..1 a 3, p 7 p 1 3 3, 6, 9 7 3, 6, n ,, 4, 3, 5, 6 86

88 33 a p 1 mod. p p p 1 1 p 1 p a ,, 3,, p 1, n p p 1 p a 3.5, 4, 6,, p 5, p 3, p 1 p n p < k p k < p 1, 3, 5 p p 1 p 1 p 1 1 p n p 1 mod. n 1 p 1 p 1 p + 1 p n 1 p 1 8 mod. xy A p + 1, 0 B 0, q + 1 p + 1 C, q

89 B q L G C H G O H A y q p p x L, q OACB OL p 1 c 1,,,, cq p r x c cq OL P c,, r p p x c P P r cq a q n c x c P c, p 1 OL y 1 GG n OLG G p 1 m m OL x 1 q HH OHH L p q 1 m+n m + n q p LH CG OGG CH H m + n p + 1 OGG CH H OC 4, q p + 1 OGG CH H 4, q p + 1 m + n 4, q m + n p + 1 4, q + 1 s, t 4 p 1 s 1, q 1 p a, p t 1 p 1, q 1 a p 88

90 3.. p mod. 8, , , , , , p α 1 p α cos π p + i sin π p G p 1 k G α k p k1 G

91 r p 8 1, r, r,, r p 30 i r i ±1 p G r G β 0 β 1 p 3 j0 p 3 j0 α rj α rj p 3 j0 α rj+1 p 3 j0 α rj+1 α + α r α rp 3 α r + α r α rp α α p 1 + α p β 0 + β G β 0 β 1 β 0 + β 1 4β 0 β 1 G β 0 β p , 3 4, 3 3, mod. 5 α 1 5 β 0 α + α 3 α + α 4 β 1 α 3 + α 33 α + α 3 β 0 β 1 α + α 4 α + α 3 α 3 + α 4 + α 6 + α 7 α + α + α 3 + α p , 3, 3 3 6, 3 4 4, 3 5 5, mod. 7 α 1 7 β 0 α + α 3 + α 34 α + α + α 4 β 1 α 3 + α 33 + α 35 α 3 + α 5 + α 6 90

92 β 0 β 1 α + α + α 4 α 3 + α 5 + α 6 α 4 + α 6 + α 7 + α 5 + α 7 + α 8 + α 7 + α 9 + α α + α + α 3 + α 4 + α 5 + α α 1 p r p 1 c 1,, c p 1 c 1 α + + c p 1 α p 1 0 c 1 c p 1 0 q 0,, q p F X q 0 X + q 1 X r + q X r + + q p X rp F α F α r F α 1 α 0 c 1 + c α + + c p 1 α p 0 α 1 p p c 1 c p 1 0 α ri rj α ri+j 1, r, r,, r p i j mod. p 1 r i r j mod. p X, X r,, X rp α α ri 1 p 1 i 1 F α F α r α rp 1 α q 0 α + q 1 α r + q α r + + q p α rp q 0 α r + q 1 α r + q α r3 + + q p α rp 1 q 0 q p, q 1 q 0,, q p q p 3 q k q 0 q 1 q p F α q 0 α + α + + α p 1 q 0 91

93 p 5, p 4 β 0 β 1 1 p 4 p 1 mod. 4 p 1 mod. 4 r p 1 1 mod. p k r p 1 1 mod. p r k p 1 k+ + r 0 mod. p β 0 β 1 α α r β 0 β 1 β 0 β 1 β 0 β 1 c 1,, c p 1 c 1 α + + c p 1 α p 1 c 1 c p α + + α p 1 0 α p 1 α p i p 1 p 1 mod. 4 p 1 β 0 β 1 α + α r α rp 3 α r + α r α rp 1 p 1 p 1 p 1 p 1p 3 4 p 1 α + + α p 1 1 p 3 4 β 0 β 1 1 p p p 4 4 ii p 1 p 1 mod. 4 p 1 β 0 β 1 1 p 1 α + + α p 1 1 β 0 β 1 1 p p 4 9

94 G 5 p a n x n + a n 1 x n a 0 a n x n + a n 1 x n a 0 p a n p x pn + a n 1 p x pn a 0 p mod. p 1 < k < p 1 k p C k p p 1 C k 1 p C k p a n x n + a n 1 x n a 0 p {a n x n + a n 1 x n a 0 } p a p n x pn + a n 1 x n a 0 p mod. p a p n x pn + a p n 1 x pn 1 + a n x n + + a 0 p mod. p a p n x pn + a p n 1 x pn 1 p + + a 0 36 q p p ±1 mod. 4 q ± p p q G β 0 β 1 31 G q G β 0 β 1 β 0 + β 1 4β 0 β p ±p p 1 k1 p 1 k1 p 1 q p ±p q 1 G q 1 ±p q 1 ± p q ± p q mod. q ± p G q G mod. q q q k α k p k p k1 p 1 k1 α kq q kq α kq p mod. q q k α kq mod. q 5 p q G p 93

95 { q p q G p ± p q ± p q G mod. q } G 0 mod. q q G α, α, α p 1 ±1 p 0, ± q q ± p p q ± p q q G G G G 3.3. J.P,Serre G.Eisenstein,F m sin mx sin x m 1 4 m 1 j1 sin x sin πj m, cos mx + i sin mx cos x + i sin x m m mc k cos m k x i sin x k k0 m C 0 cos m x m C cos m x sin x + m C 4 cos m 4 x sin 4 x +i m C 1 cos m 1 x sin x m C 3 cos m 3 x sin 3 x + m C 5 cos m 5 x sin 5 x sin mx sin x{ m C 1 cos m 1 x m C 3 cos m 3 x sin x + m C 5 cos m 5 x sin 4 x } 94

96 m m u + 1 mc 1 cos m 1 x m C 3 cos m 3 x sin x + m C 5 cos m 5 x sin 4 x m C 1 cos u x m C 3 cos u x sin x + m C 5 cos u 4 x sin 4 x m C 1 1 sin x u m C 3 1 sin x u 1 sin x + m C 5 1 sin x u sin 4 x 1 u m C 1 + m C 3 + m C 5 + sin u x + 4 m 1 sin x m 1 + sin mx sin x sin x m 1 4 m 1 sin mx 0 sin mx sin x m 1 4 x ± πj m ± sin πj m m 1 j1 m 1 1 < j < m 1 1 < j < sin x sin πj sin x + sin πj m m 33 p q p q q p 1 p 1 q 1 i 1,,, p 1 qi p p n n sin πqi p i 1,,, p 1 34 sin πi p p 1 i1 i 1,,, sin πqi p 1 n p 1 p 1 i1 sin πi p 95

97 34 6 m q, x πi p q 1 n p q p p 1 i1 p 1 i1 sin πqi p sin πi p 4 q 1 4 p 1q 1 4 q 1 j1 sin πi p p 1 q 1 i1 j1 πj sin q sin πi p πj sin q p q p 4 p 1q 1 4 q p 1 q 1 i1 j1 sin πj sin πi q p p 1 q 1 p 1 q 1 i1 j1 sin πj sin πi 1 p 1 q 1 q p p 1 q 1 i1 j1 sin πi p πj sin q p 1 q 1 p 1 q q p p q 1 p 1 q 1 q p p 8k + 1 8k p 96

98 45 45 p 5k ± p p 1k ± p p p 1 mod. 4 p 1 a1 a p 0< b< p b p 0< c< p c 0 p x + y + z n x, y, z n 5 n 8 7 x + y + z n x, y, z 97

99 4 4.1 Z 1 Z a, b ab 0 a 0 b 0 0 a, b ab 0 a b A A Z Z a a a 0, b b qa + r 0 < r < a q, r R a 0 1 R 3x {a + bi a, b Z} R[x] R[x] 98

100 Q[x] C[x] K Q R C K K[x] x fx deg fx fx deg fx 37 fx, gxdeg gx > 1 fx gx qx + rx, deg rx < deg gx qx, rx deg fx < deg gx qx 0, rx fx deg fx > deg gx, deg fx n, deg gx m fx gx n, m a 0 x n, bx m f 1 x fx a 0 b xn m gx, deg f 1 x < deg fx f 1 x gx f k x n k a k f k+1 x f k x a k b xn k m gx l, deg f l x < deg gx f 1 x f l x rx, qx l 1 k0 fx gx a 0 b xn m + f 1 x a k b xn k m gx a 0 b xn m + gx a 1 b xn1 m + f x gxqx + f l x fx gx q 1 x + r 1 x gx q x + r x gx {q 1 x q x} r x r 1 x 4.1 q 1 x q x 0 degr x r 1 x > deg gx deg r 1 x < deg gx, deg r x < deg gx degr x r 1 x < deg gx 4.1 q 1 x q x, r 1 x r x 99

101 fx gx fx qxgx qx fx gx fx gx fx gx fx gx 0 { } 1 x + 3x + x + 1x + {3x + 1} x + 3 K[x] 0 K[x] fx 0 0 fx x + 1, 3x + 1, x fx x 4 4 Q[x] fx x x + Q[x] x, x + R[x] fx x x + x + R[x] x, x +, x + C[x] fx x x + x ix + i C[x] x, x +, x i, x + i K ±1 0 fx gx fx gx fx gx fx gx ax bx 100

102 ax, bx lx dx axbx dxlx 4 ax, bx cx bx bxcx ax cx ax 1 ax, bx, cx, lx, mx mx lx qx rx mx qxlx + rx, deg rx < deg lx lx mx ax lx axlx, mx axmx rx mx qxlx ax{m x qxl x}, rx ax bx, cx, rx ax, bx, cx, lx rx 0, lx, lx rx 0 mx lx fx fx p 1 x e1 p x e p m x em p 1 x, p x,, p m x e 1, e,, e m 39 fx fx fx p 1 xp x p r x fx q 1 xq x q s x deg p 1 x < < deg p r x deg q 1 x < < deg q s x 101

103 p 1 x,, p r x, q 1 x,, q s x r s r s p i x q i x p i x q j x p 1 x,, p r x q 1 x,, q s x p i x q j x fx fx deg p 1 x < deg q 1 x n deg q 1 x deg p 1 x a deg{q 1 x ax n p 1 x} < deg q 1 x gx gx fx ax n p 1 xq x q s x {q 1 x ax n p 1 x}q q s deg gx < deg fx gx q 1 x ax n p 1 x p 1 x p 1 x q 1 x p 1 x p 1 x gx gx fx ax n p 1 xq x q s x p 1 x{p x p r x ax n q x q s x} gx p 1 x gx deg gx < deg fx fx K[x] H {0} fx, gx H fx gx H fx K[x], gx H fxgx H H dx H { dxfx fx K[x]} 10

104 0 fx fx H fx H fx 0 fx H H dx H fx dx fx dxqx + Rx degrx < degdx dx H Qxdx H Rx fx Qxdx H Rx 0 dx Rx 0 H fx dx H { dxfx fx K[x]} 4..1 Z A x Z, a A xa A R A i A ii x R, a A xa A A R px qx pxux + qxvx 1 ux vx H { pxux + vxgx ux, vx K[x]} fx pxu 1 x + qxv 1 xgx pxu x + qxv x H fx gx px{u 1 x u x} + qx{v 1 x v x} H 40 H H dx dx pxu 0 x+ qxv 0 x px px 1 + qx 0 H, qx px 0 + qx 1 H 103

105 px qx dx dx px qx px qx dx H 0 d 0 1 d u 0x, pxu 0 x + qxv 0 x d 1 d v 0x pxux + qxvx 1 fx, gx 4 fx, gx 1 qx fx, gx fx qx gx, gx fx gx rx fx, gx rx, gx 1 fx, gx d 1 x, fx qx gx, gx d x fx d 1 xf 1 x, gx d 1 xg 1 x fx qx gx d xhx, gx d xg x fx qx gx + d xhx d x{qxg x + hx} d x fx gx fx gx d 1 x 38 d x d 1 x fx qx gx d 1 x{f 1 x qx g 1 x} gx d 1 xg 1 x d 1 x fx qx gx gx d 1 x d x d 1 x d x fx gx qx rx 1 fx qx gx + rx fx, gx fx qx gx, gx rx, gx 4.. x 3 + x 4x 8, x + 6x + 4 x 3 + x 4x 8, x + 3x + x 3 + x 4x 8 x + 3x + x 3 + x 4x 8 x + 3x + x 1 3x 6 104

106 x 3 + x 4x 8, x + 6x + 4 3x 6, x + 3x + x +, x + 3x + x + 3x + x + 1x + x 3 + x 4x 8, x + 6x + 4 x x i 1 R { a + bi a, b Z} R α a + bi, β c + di α ± β a + bi ± c + di a ± c + b ± di αβ a + bic + di ac bd + ad + bci R R R, R R α a + bi ᾱ a bi α R R α a + bi Nα αᾱ a + b, α α Nα Nαβ NαNβ R α a + bi, β c + di α β a + bi c + di ac + bd bc ad c + + d c + d i α γ α β, α β β β α 105

107 1 1 α 1 α 1 N α 1 NαN α 1 α Nα > 0 Nα 1 α a + bi a + b 1 a b a, b 1, 0, 1, 0, 0, 1, 0, 1 1, 1, i, i R R 1 α, α β α α, α, iα, iα β ± R α β β 0 α βγ + ρ, 0 < Nρ < Nβ γ, ρ R α β r + si r, s 1 r, s m, n r m <, s n 1 < γ m + ni α 1 1 N β γ < + 1, ρ α βγ Nρ N β α α β γ NβN β γ < Nβ < Nβ γ ρ a bq + r 0 < r < b deg fx gxqx + rx 0 < deg rx < deg gx N α βγ + ρ 0 < Nρ < Nβ 106

108 44 α, β R J J { αx + βy x, y R } J R δ δ J 0 0 δ αx 0 + βy 0 J αx + βy αx + βy δγ + ρ, 0 < Nρ < Nδ ρ αx γx 0 + βy γy 0 R, Nδ ρ 0 J δ δ zδ αzx 0 + βzy 0 J δ δ δ δ α β α β δ α α n π 1 π π r Nπ 1 < Nπ < < Nπ r n ρ 1 ρ ρ s Nρ 1 < Nρ < < Nρ s π 1,, π r, ρ 1,, ρ s r s r s π i ρ i i π 1,, π r ρ 1,, ρ s. π i ρ j α 107

109 α Nπ 1 < Nρ 1 π 1 4 π 1 iπ 1 π 1 iπ 1 ρ 1 π 4 ϵπ 1 Nρ 1 ϵπ 1 < Nρ 1 β O ρ 1 ϵπ 1 β α ϵπ 1 ρ ρ s ρ 1 ϵπ 1 ρ ρ s Nβ Nρ 1 ϵπ 1 Nρ ρ s < Nρ 1 Nρ ρ s Nα β ρ 1 ϵπ 1 π 1 π 1 ρ 1 π 1 π 1 β β α ϵπ 1 ρ ρ s π 1 π π r ϵρ ρ s β π 1 β Nβ < Nα α 46 p p 1 R π R p π π π π p π π p R p ϵπ 1 π π l ϵ π 1, π,, π l p Nπ 1 Nπ Nπ l Nπ l p 1 l 1 p ϵπ 1, p p, l Nπ 1 π l π l p 108

110 , π l π p π π π π π ±π, ±iπ π x + iy π π π π π x 0 p y π ±iπ y ±x p x p p N1 + i 1 + i1 i i i i, 1 i, 1 + i, 1 i 1 i i1 + i, 1 + i i1 + i, 1 i 1 + i 4. 7 α a + bi λ 1 i a b mod. a, b α a + bi λ a + bi 1 i a + bi λ a + bi 1 a 1 b a + bi 1 λ α a + bi λ λ 1 i 0, 1, i, 1 + i i 1 i 8 α λ 1 i α α λ 1 i 7 α 1 α i α + 1 α + i α 1 α α p 47 p p 1 mod. 4 p a + bia bi p 3 mod. 4 p 109

111 p p a + bia bi a + b p a b p 1 mod. 4 p 1 mod. 4 1 p 1 x mod. p x x + 1 p x + 1 x + ix i p p x + i, x i p p 1 mod. 4 p 3 mod x 4 + y 4 z xyz 0 x, y, z 0, 0, 0 α, β, γ α 4 + β 4 γ α β δ γ δ 4 γ δ 4 α + δ 4 β γ δ δ α δ, β δ, γ δ α β α γβ γ α β α, β λ 1 i 8 α 4 1, β α 4 + β 4 γ 8 γ 8η γ 1 + 4η γ γ λ 1 i γ λµ γ iµ µ µ λ 110

112 γ iµ 8 iµ 1 iµ i µ i 4 8 µ 1 µ i 1 + i 4 α β λ 1 i α β α λ k x, k x, β, γλ λ 4k x 4 + β 4 γ x, β, γ ϵλ 4k x 4 + β 4 γ, ϵ 4.3 x, β, γ 4.3 γ + β γ β ϵλ 4k x 4 γ + β, γ β d β γ d γ + β, γ β γ, β β γ d γ + β, γ β λ 1 i γ + β γ β + β γ + β λ i γ β λ 4.3 λ i d γ + β λ λ 3 γ β λ 4k β iβ 4.3 γ + β ϵ 1 λ y 4, γ β ϵ λ 4k z 4 ϵ 1, ϵ, y z, y λ, z λ λ i β ϵ 1 λ y 4 ϵ λ 4k z 4 β iϵ 1 y 4 + iϵ λ 4k 4 z 4 iϵ 1, iϵ ϵ 1, ϵ β ϵ 1 y 4 + ϵ λ 4k 4 z k > 1 k λ β λ k > k > 1 β ϵ 1 y 4 λ 4 4 β y λ 8 β + 1 β 1 4 y β iβ 1 4 β 1 4 ϵ ϵ 1 ϵ ϵ β y 4 + ϵ λ 4k 1 z

113 4.3 k k x, β, γ k k 1 k 1 k > x, β, γ 48.1 x 4 + y 4 z 4 x 4 + y 4 z x 4 + y 4 Z , 13, 65, 5, 50, x + y z, x, y 1 x, y m n, nm, z m + n m, n 1, m > n > 0 m n p, q, r 3 x 3 + px + qx + r 0 u, v α u + vi α F x x 3 + 5x 3x + 7, Gx x 3 F x GxQx + r x Qx r F x x Gx x a a i F x GxQ 1 x + F 1 x x Q 1 x, F 1 x, F 1 x F x ii F x GxQx + r x Qx r F x 11

114 3 F x x a F a 0 F x x aqx x Qx 4 F x x n F x 0 n fx gx fxpx gx px dx fxgx dx fx gx dx fxgx dx fxgx fx gx 0, 1 fx fx 0 fx gx fx fxgx 3 fx 0 gx fx rx dx rx, fx dx fxgx fx x 1, gx x pxfx + qxgx 1 px, qx px px Qx P x Qx {P x} Qx Qx AxBxCx {Ax} + {Bx} {Cx} Ax Bx Cx fx x gy y xy 1 fxgy 1 fx, gy 113

115 p a b, a + bi p i m fx x 3 + 8x + mx a 0 b fa + bi 0 m i 1 m fx p, p 4k Q a + 4bc p 3 a, b, c 3 Q 3 a, b, c 3 iiiiii : i a < b c a + c, c, b a c ; ii b c < a < b b a, b, a b + c ; iii a > b a b, a b + c, b 1 a, b, c Q i a < b c i a + c, c, b a c Q Q a, b, c a b c a b 3 Q a, b, c p 4k + 1 a, b, c k 1 4 Q a, b, c Q 3 Q 3 3 iiiiii Q 5 p 4k + 1 a, b p a + b 114

116 b 0 a b r a qb + r 0 < r < b a b q b r a b r 0 b a b qb + r q + 1 r b 1 b r + 0 q a b, b r 0 a b ω aω + b c d cω + d a b ω ω d c d c { } { a b e f a 1 ω c d g h c 1 0 ω ω 0 1 ka kb a b 3 ω ω kc kd c d 4 u a c b d ω a c b d b d 1 u ω e g f h } ω 115

117 ω x [x] x ω i q 0 [ω] ii ω q 0 + u 0 < u < 1 iii ω 1 1 u 1 < ω 1 ω q q 0ω ω 1 ω 1 {w n } {k n } k 1,, q ω 1 ω 0 ω, q 0 [ω], 1 ω k, q k [ω k ] ω k 1 q k 1 ω k ω k q 0 1 q k 1 1 ω ω k ω k ω k+1 1, q k ω k 1 ω k ω k ω q ω 1 q q ω 1 q q q + q

118 k > 1 q q q k P k Q k P k 1 Q k 1 P k, Q k k > 1 P k+1 P k q k + P k 1 Q k+1 Q k q k + Q k 1 ω, ω k 0 1 k 1 1 k n 1 ω ω ω > h 0 1 h 1 1 h m 1 ω ω > n > m k 0 h 0, k 1 h 1,, k m h m k 1 1 X 1 0 h 1 1 Y 1 0 k n h m ω ω X > 1, Y > 1 k 0 1 ω X k X h 0 1 Y h Y, k 0 h 0 X Y k 0 h 0, k 1 h 1,, k m h m a b a q 0 1 q 1 1 q n 1 1 b q 0 1 q 1 1 q n

119 P k Q k P k 1 Q k 1 r k r k+1 q q k q 1 1 q k q n a b P k Q k P k 1 Q k 1 r k r k+1 k 1,,, n a b P k Q k P k 1 Q k 1 r k k 1,,, n r k+1, ω a b, ω k r k r k ω ω k + 1 ω k ω 49 ω ω k q 0 1 q k 1 1 ω P k P k 1 Q k Q k 1 ω k ω k 1 P 1 Q 1 < P 3 Q 3 < < P k 1 Q k 1 < < ω < < P k Q k < < P 4 Q 4 < P Q P n lim ω n Q n 3 P n Q n {P n }, {Q n } 1 P n Q n P n 1 Q n 1 q q n n 118

120 P n P n 1 1n, Q n Q n 1 Q n Q n 1 P n+1 Q n+1 P n Q n 1n+1 Q n Q n+1 P n+1 Q n+1 P n 1 Q n 1 1 n Q n+1 Q n 1 Q n 1 Q n Q n+1 Q n+1 Q n q n + Q n 1 Q n 1 < Q n < Q n+1 P n+1 P { n 1 < 0 n Q n+1 Q n 1 > 0 n P n Q n P n 1 Q n 1 1 ω 1 n Q n 1 Q n P n 1 P n ω Q n 1ω P n 1 Q n ω + P n ω n > 0 Q n Q n 1 < 0 ω P n 1 Q n 1 ω P n Q n < 0 P 1 Q 1 q 0 < ω { P n < ω n Q n > ω n N P N Q N < ω < P N+1 Q N+1 N 0 < ω P N Q N < P N+1 Q N+1 P N Q N P N+1Q N Q N+1 P N Q N Q N+1 1 < 1 Q N Q N+1 Q N 0 > ω P N Q N > P N+1 Q N+1 P N Q N P N+1Q N Q N+1 P N Q N Q N+1 1 > 1 Q N Q N+1 Q N 1N+1 Q N Q N+1 1N+1 Q N Q N+1 119

121 lim N Q N 0 < ω P N < 1 lim N Q N Q N ω P N 0 Q N 3 P n Q n 1 Q n P n 1 1 n P n, Q n ±1 P n, Q n P n Q n P n ω ω P q < Q Q n Q p q ω P Q < ω p q P ω Q 50 n P n Q n A, B, C, D A B, C AD BC 1 D 1 BD X Y C D < X Y < A B DX CY > 0, AY BX > 0 { x, y x > 0, y > 0 Ax + Cy X Bx + Dy Y x DX CY, y AY BX X > A, X > C, Y > B, Y > D 10

122 p q P k 1 Q k 1 < p q < P k Q k P k Q k 1 Q k P k 1 1 k 1 q > Q k 1, Q k P k 1 Q k 1 < p q < ω ω < p q < P k P k 1 Q k 1, P k Q k Q k p q Q k 1, Q k 58 x y x Dy ±1 x ωy ω x, y ω y ωx y x 58 ω ωx y < 1 x x, y 49 P N ωq N < 1 Q N x Q N, y P N N x Q N, y P N 5.. ω θ ad bc ±1 a, b, c, d a b ω θ c d, ω θ a b d b ω θ, θ ω c d c a 51 a b ω θ ω θ, θ > 1, c > d > 0 ω c d θ a b ω θ, ad bc e ±1 c d, a c a q 0 1 q n 1 1 c

123 n, e 1 n+1 n q 0 1 q n 1 P n+1 P n Q n+1 Q n a c P n+1 Q n+1 P n Q n 1 0, a P n+1, c Q n+1 P n+1 Q n Q n+1 P n 1 n+1 e aq n cp n e ad bc e, ad Q n cb P n a c, d Q n c c > d > 0 c Q n+1 > Q n > 0, d Q n < c d Q n a b a b P n, c d c a b q 0 1 q n 1 c d ω q q n θ θ > 1 ω < 7 5 < < < < 17 1 < 3 1

124 Gauss H.Minkowski xy x y Minkowski n, s s F F F k s F xy m n F x my n 1 F s s s 1 s s s s s s, s s k s < k 13

125 F P 1, P,, P k x, y P 1 P 1, P,, P k 5 F 4 O O F 1 F F 1 F Px, y P x, y x x, y y F O P Q x, y F x F P Q F M x y y, F P P M O OM M M F, Mx x, y y 54 α, β, γ, δ αδ βγ 0 h, k hk { αx + βy < h γx + δy < k x y 0 F F x, y x, y F 0 < αx + βy < h, 0 < γx + δy < k 4 { { αx + βy h αx + βy 0, γx + δy 0 γx + δy k δh, γh, βk, αk F 4 δh αk γh βk αδ βγhk 4 4 F 54.1 ω ωx y < 1 x x, y 14

126 54 α ω, β 1, γ 1, δ 0 1 h 1 n, k n ωx y < 1 n, x < n x, y n n ωx y < 1 x x y ω xy Aa, b, Bc, d OA OB m, n OP m OA + n OB P OA, OB OA, OB e 1 1, 0, e 01, a, b, c, d ad bc ±1 OP u e 1 + v e u, v u, v Z { u ma + nc v mb + nd { m ±ud vc n ± ub + va P OA, OB m OA + v OB OA, OB q 0 1 ω 1 0 ω q k 1 1 ω k 1 0 P k P k 1 Q k Q k 1 ω k P k+1 Q k+1 P k q k + P k 1 Q k q k + Q k 1 15

127 A k Q k, P k A k xy y A y ωx y ωx ω A 1 1, 0, A 0 0, 1 x 1, y ωx y A 1 1, q 0 A 0 0, 1, OA 1 l 1 : OA 0 + t OA 1 t, tq A 1 tq > ωt 1 ω q 0 > t A 0 A 1 t q 1 A x 1 ω q 0 > t q 1 ω t A q 1, q 0 q q q q 0 q q 0 q 1 1 P q 0 q 1 + 1, Q q 1 A k 1, A k l k : OA k 1 + t OA k Q k 1 + tq k, P k 1 + tp k A k ω ω ω P k Q k P k 1 Q k 1 ω k ω ω kp k + P k 1 ω k Q k + Q k 1 ωq k 1 + tq k P k 1 + tp k > 0 ω kp k + P k 1 tp k + P k 1 > 0 ω k Q k + Q k 1 tq k + Q k 1 ω k tp k Q k 1 P k 1 Q k 1 k ω k t > 0 ω k k k t l k ω A k 16

128 y A k L B A k 1 A k+1 x t 1 A k t t A k 1 A k+1 A k 1 A k+1 ω ω A 1 A 1 A 3 A 5 A 0 A A 4 A 6 A k A k ω x A k x ω 55 ω, A x 0 < x < A 1 ωx y x, y Q n 1 < A n 1 k A k A k+ x A y ωx x, y Q n < A n k A k A k+ x A 3 ωx y x y x Q n, y P n, P n, Q n ω P n A Q n Q n < A < Q n+1 1 ωx y ω x, y y x, y ω 1 3 A k 1 A k+1 B, A k 1 A k+1 ω L OA k A k 1 A k+1 A k, A k 1, B, A k+1 ω OA k, LA k 1, LB, LA k+1 BA k+1 OA k LB > OA k B ω A k ω A n x A 17

129 x ωy x y y 0 ω P n Q n ω P n < 1 Q n 56 Q n ω P n < 1 Q n Q n y A L N B C AQ, P BQ, P OACB Q > Q B A ω AM > BN AL > BL LAM > LBN OAM + OBN < OBA 1 M x OAM OBN 1 4 OAM 1 QQω P, OBN 1 Q Q ω P QQω P < 1, Q Q ω P <

130 a, b, c a > 0, D b 4ac < 0 0, 0 ax + bxy + cy < D π α, β i αx βy 0 ii α β a 1 + a a 1, a, a 3, a 4 a a 4 a a + 1 a 3 a 1 a a 3 + a 1 + a 3 p a a q αq βp 157x 68y n a n b n n a n + b n 1 n > a n b n a n 1 b n 1 a n b n a. a 0 a. a 0, a 0 k 0 a 0 k a 1 19

131 a 1. a n, a n, a n k n a n+1. a n k n + 1 a n+1, {P n } n 0, 1,,,, {Q n } n 0, 1,,,. P 0 1, P 1 k 0, P n+1 P n 1 + k n P n n 1,, Q 0 0, Q 1 1, Q n+1 Q n 1 + k n Q n n 1,,. 1 P n Q n 1 P n 1 Q n 1 n n 1,, n > 1 P n Q n 1. 3 a P n 1 + P n a n n 1,, Q n 1 + Q n a n 4 a P n < 1 n 1,, Q n Q n x, y x, y a, b, c, d Aa, b, Ba + c, b + d, Cc, d, O0, 0 OABC S 1 ad bc 1, S ad bc, S xy x y ABC 1 AB, AC BC AB, AC 3 ABC 8 130

132 ω a, b a 0 aω + b 3 Ap, qbr, s O OAB p, q, r, s 1 131

133 x Dy ±1 D J.Pell i ii iii D 1999 x Dy ±1 P, Q P Q ! A 1 x n y n A n 1 0 n 1,, 3, 1 x n y n 13

134 a + 3, b 3 a n b n x n, y n P 1 x 1, y 1, P x, y, P 3 x 3, y 3,, P n x n, y n, ab A 1 x x 1 1 A 1 x x 3y 1 x 1 3y1 1 y y 1 y x 3y x 1 1 x, y A 1 x y 1 y y > y 1 > 0 a n 3 {a n }, {b n } A n 1 n 1,, 0 b n + 3 n a n + b n 3 n 1,, 4 x 3y 1 x, y 3 a n, b n n 1,, i a b +1 a b 1 ii a + b > 0 a, b g a + b G. 1 G u. 1 u n G g, gu n G 3 G g m, g u m

135 1 x n+1 αy n+1 βx n αy n α, β x n+1 αy n+1 x n + 3y n αx n + y n βx n αy n α β, 3 α αβ β α 3 α ± 3 β 3 { xn+1 3y n+1 3x n 3y n x n+1 + 3y n+1 + 3x n + 3y n { xn 3y n 3 n 1 x 1 3y 1 3 n x n + 3y n + 3 n 1 x 1 + 3y n x n + 3 n + 3 n y n + 3 n 3 n 3 x n an + b n, 3y n an b n x n 3y n a n b n 1 P n, n 1,, x 3y 1 A n A + pa + qe 0 p q A n+ + pa n+1 + qa n 0 A n x 1 y x y 3 1 x 1 y 1 x y { x x1 + 3y 1 y x 1 + y 1 { x1 x 3y y 1 x + y 1 x 1 + 3y 1 3x 1 + y 1 4 3x 1 + 1x 1 y 1 1x 1 y y 1 x 1 3y 1 134

136 y y 1 y x + y x y x y x + y 1 + 3y y x + y 1 + y x + y > 0 y 1 x + y 4y x x + y 4y 1 + 3y x + y y 1 x + y > 0 y > y 1 > n 1 n k a k b k A k 1 0 a k, b k + a k+1 b k+1 3 k a k + b k 3 A k a k b k a k + 3b k a k + b k + 3 k+1 + 3a k + b k 3 a k + 3b k + a k + b k 3 a k+1 + b k+1 3 k + 1 n 4 x 3y 1 x, y { x1 x 3y y 1 x + y y 1 > 0 x 1 x 3y 4x 9y x + 3y 4x 3x 1 x + 3y x + 3 x + 3y > 0 135

137 x 1, y 1 x 3y 1 x, y, x 3, y 3, y > y 1 > y y k > 0 n y n 0 x n x n 1 A n x 1 y 0 y n x A n 1 y 0 a n b n u u a + b a b ±1 a + ba b 1 u a + b > 1 a b < 1 a b < 1 a b > 1 a + b > 1 a > 0, b > 0 a 1, b 1 G g a + b g a + b gg G gg aa + bb + ab + a b aa + bb ab + a b a a b b a b ±a b gg G > 0 G G G G gu n G 3 g a + b u > 1 u m m u m+1 > g > u m m u > g u m > 1 g u m G u g u m 1 g u m { } x n A x n, y n A n 1, n N 0 y n B {x, y x 3y 1, x, y N} A B A B 136

138 ±1, 0 ±1, D x Dy ±1 x, y S { S x, y x Dy ±1, x, y Z, x + } Dy > 0 S 1, 0 S x + Dy > 1 x + Dy p, q 1 x 1, y 1, x, y S n s, t s, t S x 1 + Dy 1 x + Dy n s + Dt S n p + Dq n x n + Dy n x n, y n x 1 p, y 1 q { S x n, y n p + Dq n x n + } Dy n, Z 3 S A p Dq q p { S x n, y n x n y n A n 1 0, Z } 1 x 1 + Dy 1 x + Dy x 1 x + Dy 1 y + x 1 y + x y 1 D, x 1 x + Dy 1 y Dx 1 y + x y 1 x 1x + Dx 1 x y 1 y + D y1y Dx 1y Dx 1 y x y 1 Dx y1 x 1 Dy1x Dy ±1 137

139 , n 1., 1 x + x Dy Dy ±1 ±x Dy x + Dy > 0, 1 x + Dy > 0, ±x D±y ±1, n 1, x 1 + Dy 1 x + Dy ±1 s + Dt s, t, s, t S., x 1, y 1, x, y, n. s, t S, s + Dt > 1. p + Dq, n.,, 1 p + Dq n < s + Dt < p + Dq n+1 1 < s + Dt p + Dq n < p + Dq s + Dt p + Dq n u + Dv u, v, u, v S., p + Dq,, n,. u + Dv 1 s + Dt p + Dq n, s + Dt < 1,, 1 s + Dt, 1 s + Dt > 1 1 s + Dt p + Dq n 138

140 , s + Dt p + Dq n, n, s + Dt p + Dq n., p + Dq n x n + Dy n x n, y n, 1 S, S {x n, y n }. p Dq 3 A q p x n+1 + Dy n+1 x n + Dy n p + Dq x n p + Dy n q + Dx n q + y n p,., x n y n x n+1 y n+1 A n 1 A n 1 x 1 y 1 p q p q Dq p A n 1 Dq p x n y n p q 1 A n x Dy ±1 ±1, 0 P.G.Dirichlet n k n k n k q r r > 0 1 q a 1, a,, a n 1 < a 1, a,, a n < n a 1, a,, a n 1n 139

141 1 q qk n qk r > 0 q a 1, a,, a n 1n n 1 a 1, a,, a n a 1, a,, a n a 1, a,, a n 1n ω x ωy < 1 y x, y i n 0 < y < n, x ωy < 1 n x, y a < b a b [a, b [0, 1 n [ 0, 1, n [ 1 n, [ n 1,, n n, 1 y 0, 1,, n y ωy x n < ωy x < 1 n ωy x ωy 1 x 1 ωy x y 1 y ωy 1 x 1 ωy x < 1 n y 1 > y x x 1 x, y y 1 y x, y ωy x < 1 n 140

142 ii n 0 < y < n ωy x < 1 x, y n n x, y ωy x ωy 0 x 0 1 n < ωy 0 x 0 n n n ωy x < 1 n x, y ωy x < 1 n < ωy 0 x 0 x 0, y 0 1 n < 1 y ωy x < 1 y x, y D D x Dy 1 X, Y, X > 0, Y > 0 58 x Dy < 1 y x, y x > 0, y > 0 1 y < x Dy < 1 y x + Dy < 1 y + Dy x Dy < 1 y x Dy < 1 y + D < 1 + D 141

143 x, y x Dy 1 + D 1 + D x, y l x Dy l l l x, y l x, y x, y s, t, u, v { u s + kl v t + hl tu sv kt hsl Y kt hs s Dt l, u Dv l l s Dt u Dv su Dtv Dsv tu su Dtv DY l su Dtv l su Dtv l su Dtv Xl Xl DY l l X DY 1 X, Y x Dy l px + qx + r

144 ω px + qx + r 0 p x + q x + r 0 u p pu { pω + qω + r 0 puω + q ω + r 0 u q uqω + r ru 0 ω q qu, r ru ω ω ω ω 1 aω 0 + b cω 0 + d ω 1 px +qx+r 0 D D > 0 D p aω0 + b + q cω 0 + d aω0 + b + r 0 cω 0 + d pa + qac + rc ω0 +{pab + qad + bc + rcd}ω 0 + pb + qbd + rd 0 D a, b, c, d pt + qt + r 0, p + qt + rt 0 pa + qac + rc 0, pb + qbd + rd 0 ω 0 D D {pab + qad + bc + rcd} 4pa + qac + rc pb + qbd + rd q ad bc 4prad bc q 4pr D

145 11 px + qx + r 0 D p, q, r pr < 0 4pr q D < 0 q < D q q 4pr q D < 0 p, r 4 p r 61 ω k + 1 ω P k P k 1 Q k Q k 1 ω k ω k+1 N ω N > 1, 1 < ω N < 0 ω ω ω > ω i k 1 < ω k < ω k+1 ω P k Q k P k 1 Q k 1 ω k+1 ω k+1 P k Q k P k 1 Q k 1 1 ω 1 k+1 Q k 1 Q k P k 1 P k ω Q k 1ω P k 1 Q k ω P k k P k 1 lim ω, k Q k 1 Q ω P k 1 k 1 Q k 1 Q k ω P k Q k P k lim ω k Q k ω k+1 < 0 ω P k 1 Q k 1 < 0, ω P k Q k < 0 ω k+1 ω k ω k+1 ω k+1 < 0 ω k+1 q k ω k+1 q ω k+ k+1 ω k+ q k+1 > 1 1 < ω k+ < 0 > 1 144

146 1 k 1 < ω k < 0 ω P k P k 1 Q k Q k 1 ω k+1, ω k+, ω k+3, ω k+1, ω k+, ω k+3, N j ω 1, ω, ω N, ω N+1,, ω N+j ω N N N + j ii N, j j j k ω N+j ω N j k a b a b x fx fx + a fx, fx + b fx fx a + b fx a fx a + a fx j k r j km + r r j km k r 0 k N ω N > 1, 1 < ω N < 0 N M N < M M ω M ω M+j ω M+j ω M > 1, 1 < ω M < 0 ω k+1 ω M 1 ω M+j 1 q M 1 1 ω M 1 0 q M+j 1 1 ω M+j 1 0 q M+j ω M ω M 1 ω M+j 1 q M 1 q M+j 1 ω M 1 ω M+j 1 ω M 1 p + D, ω M+j 1 t + D r r ω M 1 ω M+j 1 p r t r ω M 1 ω M+j 1 145

147 ω M 1 ω M+j 1 < 1 q M 1 q M+j 1 < 1 q M 1 q M+j 1 0 ω M 1 ω M+j 1 1 < ω M < 0 k 1 < ω k < 0 1 < ω k < 0 N ω N > 1, 1 < ω N < 0 k 6..1 ω D 9 ω ω ω x + 6x < ω + 4x 10x < < ω x x ω x 9 + 4x ω 5 + 4x x ω x 10x ω 7 ω ω ω ω 7 k x Dy ±1 6 1 D D k 1 k i m x, y P mk 1, Q mk 1 x Dy 1 ii m x, y P mk 1, Q mk 1 x Dy 1 k i m x, y P mk 1, Q mk 1 x Dy 1 146

148 D q x 1 > 1, 1 < x 1 < 0 x 1 x 1 x 1 > 1 q 0 D x D q 0 D + q0 61 D k D q q q P k Q k x 1 P k 1 Q k 1 q q x 1 P k Q k q k q k P k 1 Q k 1 x 1 x k+1 x 1 x 1 P mk Q mk P mk 1 Q mk 1 x 1 q0 1 D P mk Q mk 1 0 P mk 1 Q mk 1 q x 1 x 1 D P mk 1 q k q 0 m P mk Q mk D P mk q 0 P mk 1 D P mk 1 Q mk 1 Q mk 1 Q mk q 0 Q mk 1 D p q D D r s p D + q r D + s D rd q + s p D 0 rd q 0, s p P mk 1 DQ mk 1 p Dr p ps rq r P mk Q mk P mk 1 Q mk 1 1 mk mk q s q 0 147

149 1 k i m x, y P mk 1, Q mk 1 x Dy 1 ii m x, y P mk 1, Q mk 1 x Dy 1 k i m x, y P mk 1, Q mk 1 x Dy 1 A P k 1 Q k 1 P k q 0 P k 1 Q k q 0 Q k 1 D A D A P k 1 DQ k 1 Q k 1 P k 1 A m P mk 1 Q mk 1 P mk 1 Q mk 1 P mk q 0 P mk 1 Q mk q 0 Q mk 1 DQ mk 1 P mk 1 P k 1 + Q k 1 D m P mk 1 + Q mk 1 D p q Dq p 6.. x 13y ±

150 k 5 x, y P 4, Q 4 x Dy 1 x, y P 9, Q 9 x Dy P 4, Q 4 18, 5 P 9, Q 9 x P 5 P 4 x 1 x 1 x Q 5 Q x P 5 P 4 x 1 x Q 5 Q x 1, P 10 P 9 x 1 x Q 10 Q 9 P 9, Q 9 649, 180 P 4, Q 4 P 9, Q 9 P 4 + Q 4 D P 9 + Q 9 D x 13y ± P mk 1, Q mk 1 63 x Dy ±1 x + Dy > 1 D P mk 1, Q mk 1 k D 149

151 x+ Dy > 1 x 1, y 1 x 1 Dy1 x 1 Dy 1 ±1 y 1 x 1 ±1 x 1 + Dy 1 > 1 57 x 1 > 0, y 1 > 0 q 0 1 D θ 1 0 θ > 1, 0 > θ > 1θ θ x 1 Dy 1 D Dx1 + Dy 1 y 1 x 1 Dy1 + x 1 D x 1 Dy 1 y 1 x 1 q θ q θ 0 1 x 1 Dy 1 q 0 1 θ θ 1 q 0 y 1 x y 1 x 1 q 0 1 θ x 1 q 0 y 1 Dy 1 q 0 x q 0 y 1 + x 1 y 1 θ D q 0 y 1 x 1 q 0 y 1 q 0 y 1 + x 1 y 1 p q D q 0 y 1 x 1 q 0 y 1 r s x 1 Dy 1 ±1 ps qr ±1 ps qr e y 1 x 1 ϵ rθ + s, ϵ rθ + s r D q 0 y 1 > 0 s x 1 q 0 y 1 > x 1 Dy 1 ϵ > 1 θ, θ p t r q s ± 1 x 1 + Dy 1 > 1 θ > 1 t rt + s pt q 0 ϵϵ r θθ + θ + θ rs + s r q r s p rs + s r qr s + ps + s e ±1 ϵ < 1s > rθ + s > r + s s > ϵ > r + s i ϵϵ 1 e 1 1 > ϵ > 0 r > s > 0 ii ϵϵ 1 e 1 0 > ϵ > 1 r > s > 0 150

152 1 r > s > 0 51 p r q s θ θ r s e 1 ps qr 1 p qr 1r > 0 p q 1, r 1 θ q + 1θ + q θ s 0 e 1 ps qr 1, qr 1 r > 0 q r 1 θ pθ + 1 p 1 θ θ 1 0 p q θ θ r s q 0 1 q 0 1 p q D θ θ r s q θ D q 0 1 p q 1 0 r s θ x 1 Dy 1 y 1 x 1 q θ x 1 q 0 + Dy 1 x 1 y 1 q 0 + x 1 y 1 θ h x 1 q 0 + Dy 1 x 1 y 1 q 0 + x 1 y 1 q 0 1 D 1 0 θ m h mk P h Q h P h 1 Q h 1 P h Q h P h 1 Q h 1 x 1 P mk 1, y 1 Q mk 1 θ h k x + Dy > 1 p, q p P k 1, q Q k k m x, y P mk 1, Q mk 1 x Dy 1 x Dy Q D Q D 1 Q D 151

153 6..3 D x Dy ±1 x + Dy > 1 1 D D q 0 x 1 1 D q0 3 P 1 1, P 0 q 0, Q 1 0, Q 0 1 n > 1 i x n ii 1 x n 1 q n 1 x n x n q n { P n q n P n 1 + P n Q n q n Q n 1 + Q n 4 x k+1 x 1 k x P k 1, y Q k 1 UBASIC BASIC 1/ , x k+1 x 1 UBASIC BASIC // UBASIC x k+1 x 1 x n S + T D P n Q n D D 1999 x Dy ±1 10 open "pell.txt" for create as #1 0 for D to if DisqrtD^ then 0 else Q0isqrtD 50 S1Q0//D-Q0^:T11//D-Q0^ 60 XS1+T1*sqrtD:PAQ0:QA1:PB1:QB0:SS1:TT1 70 QintX:PBPA*Q+PB:QBQA*Q+QB 80 KK+1 90 S0S 15

154 100 SQ-S//T^*D-Q-S^:TT//T^*D-Q-S0^ 110 XS+T*sqrtD 10 if SS1 and TT1 then goto 190 else QintX:PAPB*Q+PA:QAQB*Q+QA 140 KK S0S 160 SQ-S//T^*D-Q-S^:TT//T^*D-Q-S0^ 170 XS+T*sqrtD 180 if SS1 and TT1 then goto 00 else print #1,"&",PA,"&",QA,"&",K:goto print #1,"&",PB,"&",QB,"&",K 10 K0 0 next D 30 end 153

155 D D 1999 D P Q k

156 D P Q k P Q k

157 D, D D, D 3 A D 3 x 3y ω x 19y ± x 46y ± x 3y 1 1 x, y a, b, c, d a + b 3 c + d 3 a cb d x, y x, y x, y x, y 1 x, y + 3 n x n + y n 3 x n, y n n 0, 1,, x 0 1, y 0 0, x 1, y x, y, x 3, y n x n y n n 3 n x n 3y n 156

158 x n, y n 1 x, y 0 x 1y 0 3 x 0, y 0 x n 1, y n 1 x n, y n n > 1 x n + y n n x n 1 + y n x n y n x 0, y 0 x 1, y 1 x, y,x n, y n y 1 < y < y 3 < x, y x > 0, y > 0 x + y 3 3 x 3y + y x 3 x x 3yy y x x, y x > x > 0y > y > 0 x, y x, y 3 x 0, y 0 x, y x n, y n n 0, 1,, x ny z nt xz + nyt nxt + yz x y 1 x, y x > xy C + C C + : x y 1 x > 0, y > 0 C : x y 1 x > 0, y > 0 Px, y Qu, v u x + y, v x y Px, y x, y 1 Px, y C + Qu, v C Px, y C x y 1 Qu, v C + Px, y C + C y 1 0 < v < y n x n + y n xn, y n n Px n, y n C + C 157

159 4 C + C Px n, y n n y n+1 y n 5 lim n x n+1 x n 158

160 , 3, 5, 7, 11, 13, 17, 19, 3, 9, 31, 37, 41, 43, 47 53, 59, 61, 67, 71, 73, 79, 83, 89, , 593, 597, 5939, 5953, 5981, N N mn m < n N mn > m , n p < n p , 3, 5, 7, 11, 13, 17, 19, 3, 9, 31, 37, 41, 43 N N N Eratosthenēs km,