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, 1-1.. 3. 4. 5. 6. 7. n 8. 9. 10. 11. 1. 13. 14. 15. 16. 17.

1 5 1.1....................................... 5 1........................................ 7 1.3......................................... 8 9.1................................ 9.................................. 11.3......................................... 13 3 14 3.1............................... 14 3........................... 18 3.3......................................... 5 4 8 4.1........................................... 8 4.......................................... 31 5 34 5.1.......................................... 34 5...................................... 37 5.3.................................... 41 5.4......................................... 46 6 ϕn 47 6.1 ϕn.................................. 47 6................................... 51 7 n 54 7.1 n....................................... 54 7.......................................... 56 8 58 8.1.................................... 58 8................................... 59 8.3......................................... 63 8.4......................................... 66 9 67 9.1.......................................... 67 9............................................ 69 9.3......................................... 7 3

10 73 10.1............................ 73 10............................ 75 10.3.................................. 77 10.4.................................. 8 10.5................................ 87 10.6......................................... 90 11 91 11.1....................................... 91 11.......................................... 95 1 97 1.1................................. 97 1.......................................... 105 13 106 13.1.................................... 106 13.................................. 107 13.3......................................... 109 14 110 14.1........................... 110 14.......................................... 113 14.3....................................... 116 14.4......................................... 117 15 119 15.1.................................... 119 15...................................... 11 15.3......................................... 15 16 17 16.1................................ 17 17 131 17.1................................. 131 17.................................. 135 18 139 18.1...................................... 139 18....................................... 163 4

1 1.1 1 3 3 3 3 3 1 1 N 1 A 1 A x A x + 1 A A N 3 a 0 a qb + r, 0 < r < b q, r a qb < a < q + 1b q i bi < a a < bi i b i < a < bi a 5

i i N {i, i + 1,, 0} M i N M M { t a < t + 1b t M } M q q 1 a < q 1 + 1b qb < a r a qb qb < a < q + 1b 0 < r 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 r [ 0] a b [x] x, a q b a, b bx a a x b Z a Z a b b b a q a bq b 0 a b, b a Z N, Z, Q, R, C 1.1 b 1 [ ] 50 a 50 50 4 1 + q 4 r 1 [ 50 a 50 50 5 1 + 10 q 5 a 5 5 1 1 + 7 q 1 [ 5 1 ] r 10 1 ] r 7 6

1. x deg fx fx 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 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 1 q 1 x q x, r 1 x r x 7

1.3 1 1 96 1 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 3 F x x a F a 0 F x x aqx x Qx 4 F x x n F x 0 n 8

.1 4 180 4 4 180 4 3 1 90 1 4 30 7 4 180 4 3 6 37 a, b, c, 0 least common multiple L.C.M. a, b, c, 1 greatest common measure G.C.M. 3 1 3 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 9

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 d m l d d m m d 3 l 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 me d a, b a da, b db a da mea ma, b db meb mb a ea, b eb l ab aeb, l a b ea b l e ab a b l a, b e l e 1 m d ab dl 4 a, b 1 a, b ab bc a a b bc ab bc ab c a c a 10

a, b, c, a, b, c, a, b, c, ±1 a, b, c, 1 a, b a, b 1 a, b. 4 1 a > b > 0, a b r a, b r, b {r n } r 1 a, r b n > r n > 0 r n+1 r n 1 r n r n 0 r n+1 0 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 11

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,.1 6188, 4709. 6188, 4709 4709, 1479 1479, 7 7, 119 119, 34 34, 17 17 69, 391, 55 119, 136, 55 119, 17, 17 17.1 1 n 1 nn + 1n + n + 3 4 n n 3 n 4 3 n 3 n 1 4 4 nn + 1n + 1 6 5 n 3 3n + 8n 6. a, b 7a + b 1 3a + b 1

pa + qb ps qr 1 p, q, r, s ra + sb 3 11n 4 3n 13 n.3 98 k, l k > l {a n }, {b n } a 1 k, b 1 l n > 1 { { b n b n 0 a n+1 b n+1 b n 0 a n a n b n b n 0 b n b n 0 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 3 3 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 1 r N > 0, r N+1 0 N N r N+ k > f k k 1,,, N + 1 3 f n+1 > 3 n n 1,, 4 N < + log 3 a 13

3 3.1 a, b, c, k ax + by + cz k x 0, y 0, z 0 Diophantos, x + 13y 1 1 ax + by 1 a b 3 a 1 x 1 + a x + + a n x n k ax + by 1 a b a > b > 0 3 b b 0, a, b 1 a 1 1 x + 0 y 1 1, 0. 0 < k k k a ax + ky 1. ax + by 1 a > b > 0 a bq + r 0 < r < b ax + by bq + rx + by bqx + y + rx a b b r { { bx + ry 1 qx 0 + y 0 X 0 x 0 Y 0 X 0, Y 0 x 0 Y 0 y 0 X 0 qy 0 14

x 0, y 0 ax + by 1 a > b > 0 ax 0 + by 0 ay 0 + bx 0 bqy 0 bx 0 + ry 0 1 b,, b b > 0, a > b a ax + by 1. i 0,, b 1 ai b r i. A {r 0,, r b 1 }, B {0,, b 1}. r i 0 b 1, A B. B A. A 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. i j 0., 0 b 1 < i j < b 1 0 i j r i r j, A. na nb A B A B 1 A. ai bq i + 1 i. ai bq i 1, x, y i, q i ax + by 1. 3.1 37x + 13y 1 u 1,,..., 1 37u 13 11, 9, 7, 1 37 6 13 17 + 1 376 + 13 17 1 x, y 6, 17 15

A { ax + by x, y : }. A d. d ax 0 + by 0 a, b d > 0. a dq 1 + r 1 b dq + r 0 < r 1 < d 0 < r < d r 1 a dq 1 a x 0 + by 0 q 1 a1 q 1 x 0 + b q 1 y 0 r 1 A d A r 1 0 r 0 d a b a, b 1 d 1 1 ax 0 + by 0 ax + by 1 x 0, y 0. 3 b 0 b 5 a 1, a,, a n, k a 1 x 1 + a x + + a n x n k k a 1 a, a n d a 1 a,, a n a i d d k d J {a 1 x 1 + a x + + a n x n x 1, x,, x n } J e e n a k l k k1 l k k 1,, 3,, n 16

a 1, a, a n e > 0. a 1 eq 1 + r 1 a eq + r a n eq n + r n 0 < r 1 < e 0 < r < e 0 < r n < e j j 1,, 3,, n n r j a j eq j a j a k l k q j k1 a 1 l 1 q j + a l q j + a j 1 l j q j + a n l n q j J e r j 0 j 1,, 3,, n e a 1, a,, a n e < d J d e d, d < e k d k dn e d d d dn dn n a k l k k1 n a k l k n k1 l k n α k k n a k α k k1 α 1, α,, α n J {dn n } {dn n } J J {dn n } 17

6 fx, gx, kx, hx fxhx + gxkx 1 fxhx + gxkx {fxhx + gxkx hx, kx } 0 0 tx, fx fx 1 + gx 0 fx fx tx rx rx fx txqx rx tx tx 0, rx 0, tx fx, gx tx, tx fx, gx tx fxhx + gxkx, 0 fxhx + gxkx c, hx/c, kx/c hx, kx fxhx + gxkx 1 3.1 3 A A a, b a + b A, a b A A 3. a 1 x 1 + a x + + a n x n k 18

x y fx, y 0 t pt, qt t, x pt, y qt fx, y 0, x pt, y qt fx, y 0 3x + 57y + 68z 1 3, 57, 68 3 57 3 1 + 5 68 3 + 4 3x + 57y + 68z 3x + 3 1 + 5y + 3 + 4z 3x + y + z + 5y + 4z 4 8x + y + z + 4 6 + 1y + 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 3 4l + m + 0n 1 l t, m 1 4t, n s t, s Z 3 8x + 14y + 17z t y 1 4t x + y + z s x 11 46t + 17s, y 1 4t, 6 + 5t 8s t, s 19

3x+y 1 17x+5y 1 ax + by k 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 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 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 1 1 17x + 5y 1 17 5 + 3, y x + y, 3x + 5y 1 3 5 3 + 6, x x + y, 3x + 6y 1 4 3 6 3 + 5, y 3x + y, 5x + 6y 1 5 6 5 1 + 1, 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,, 17 9 + 5 1143 + 1144 1 0

a b x a b x ax + by c d y c d y cx + dy x A, B X y AB X AB X s t, A sv tu, u v s t x y sx + tz sy + tw u v z w ux + vz uy + vw sx + tzuy + vw sy + twux + vz tzuy + sxvw twux syvz sv tuxw yz s t x u v z A A 7 1 a, b Z, a > b > 0 a b q 0 r 1 a b q 0 1 1 0 b r 1 y w a q 0 b + r 1, b q 1 r 1 + r,, r k 1 r k q k + r k+1 a q 0 1 q 1 1 q k 1 b 1 0 1 0 1 0 Q k Y k r k r k+1 P k X k q 0 1 q 1 1 q k 1 1 0 1 0 1 0 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 P n P n 1 b Q n Q n 1 d 0 n 1

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

3.1 17x + 5y 1 17 5 1 1 0 1 1 0 1 1 0 1 1 0 5 1 1 3 3 1 0 1 0 6 1 3 1 6 1 0 1 0 5 1 3 1 1 1 5 1 0 1 0 1 0 1 1 3 1 1 1 5 1 1 0 1 0 1 0 1 0 1 0, 1 1 0 5 1 5 1 1 3 1 1 1 1 0 1 0 1 0 4 3 5 1 1 1 1 0 3 4 17 6 1 5 9 5 1 1 0, x 9, y 1., 17 9 + 5 1143 + 1144 1, t,. { x 9 5t y + 17t n n 1 n 1 3 a 1, a,, a n a 1 a k q k a 1 + r k, k, 3,, n 4

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 4 4 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 4 4 0 0 n 1 0 3. 4 1 5x + 13y + 15z 1 x + 6y + 5z + 7w 1 3.3 4 4 89 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 > 1 5

5 5 91 xy x y m, n r 5 r 6 6 93 1 α, β i αx βy 0 ii α β a 1 + a + 1 1 a 1, a, a 3, a 4 a 3 + 1 a 4 a 1 + 1 a + 1 a 3 a 1 a a 3 + a 1 + a 3 p a a 3 + 1 q αq βp 157x 68y 3 7 7 a, b, c a, b x 0, y 0 ax 0 + by 0 c 1 l m al + bm c u l x 0 + bu, m y 0 au 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 8 8 00 xy x y a, k a > L : ax + a + 1y k 6

1 L k aa + 1 x > 0, y > 0 L 3 k > aa + 1 x > 0, y > 0 L 9 9 00 m n x 3m + 5n x 10 10 00 a, b 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 11 11 90 n fx πx sin n 1 {fk k } m n { } n 1 fmk k 0 < k < m 1 1 0 p, q 1 pq p + q 4 NA < pq p + q L A 7

4 4.1 1 n > 6 6 + + 7 + + 3,, 0 + 5 + 13 100 103 3 10 119 17 3 11 13 17 19 9 31 p p + 4 e 1 e e 3 3 1 7, e 5 31 11 1 047 3 89 e, 3, 5, 7, 13 e 3, 9 e 5 n + 1 n n 1,, 3, 4 5, 17, 57, 65537 n 5 5 + 1 641 6 n + 1 k n + k + 1 5, 4 + 1 17, 6 + 1 37, 10 + 1 101 1 7 a > 1 a 1 a 1 a a > 1 a a 8 0 : 1 : 1 : : 1 1 8

8 4 4 a a a b c 1 < b < a, 1 < c < a b c a a a p 1 p p m q 1 q q n 3 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 9 n p 1, p,, p n a p 1 p p n + 1 p 1, p,, p n 1, p 1, p,, p n, a p 1, p,, p n p 1, p,, p n 4.1 5 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, 9

a T a 3 a Sa T a 1 + α1 + β1 + γ Sa pα+1 1 p 1 qβ+1 1 q 1 4 a, b c T abc T at bt c rγ+1 1 r 1 Sabc SaSbSc 5 a a T a 4. 6 a 1 a a a 4.1 a Sa a 1 n > 1 a n 1 n 1 n 1 a 4.3 7 1 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 k α k > 0 a 1, a,, a n m l m l p p Minα 1, α,, α n p p Maxα 1, α,, α n Min Max 30

4 a 1, a,, a n d 1, a 1 a, a 1 a 3,, a n 1 a n d a 1, a,, a n k d k a 1 a a n d n i k, n d k d k 1 d k ii e k e 1 d 1 e k e k 1 d k 1 iii 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 0, b 0, c 0, a, b, c, 4.4 8 l a 0 b 0 c 0 1 p pc k p > k > 0 p k p l, p nc k p n > k > 0 p n l 4.5 9 n! p [ ] [ ] n n [ ] n + p p + p k [x] x m 4.6 10 m > 0, n > 1 n n n p α q β m n 0 < x < pα, 0 < y < q β,, s > 0 x, y,, s m n x p α + y q β + ± s k1 4. 13 13 88 k k a m a, m, fk a., S n n fk k1 31

1 S 50. n S n n. 3 n 1 < S n < n 14 14 97 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 1 n 6 + 1 1 a 3 a 6 a 15 15 98 1 81 1,,81 378 3 N 60 N 3 16 16 98 a, b, p, q p + q pq a b a b 1 1 pq b a + b 17 17 99 a, b, c a + b c a, b 1 a b c a a + c d d 18 18 0 a, a fa 1 a f1 1 a 15 15 1,3,5,15 f15 4 3

1 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+1 1 33

5 5.1 a b m, a b m a b mod. m a b mq a b m q 1, q r 1, r a b mq a mq 1 + r 1 b mq + r 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 a mod. m a b mod. m b a mod. m a b mod. m, b c mod. m a c mod. m A A 3 m, m 3 1 3 1 n m r 0 < r < m 1 n mq + r, r 0, 1,, m 1 34

n m 0, 1,, m 1, m, m a mk + a, k : 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 5 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 6 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 5 5 a a mod. m N Na Na mod. m Na α b β c γ Na α b β c γ mod. m 5 Na α b β c γ Na α b β c γ mod. m 6 ab ac, a 0 b c 35

11 ac bc mod. m c, m 1 a b mod. m c, m d, m dm a b mod. m d 1 m dm, c dc m c ac bc m N ac bc mn a bc m N m c a b m a b mod. m 5.1 11 a a a n 10 n + a n 1 10 n 1 + + a 1 10 + a 0 a a 0 + a 1 + + a n mod 9 a a 0 a 1 + + 1 n a n mod 11 5. 1 10 6, 10 100, 3 100 5.3 13 1 65 + 1 11 n, 13 n + 6 7 3 3 15 3 15 15 1. 4 100 1 99 100 5.4 14 n 1 n 7 0, 1,, 4 36

n 5 n 10 3 n n 1 8 4 n 4 + n 3 + 11n + 10n 4 5.5 15 1 a, b a + b c c a, b 3 a, b, c a + b c a, b, c 1 5 5.6 16 a, b a 11 3 a 3 + b 11 4 b 11 5. fx fx 0 mod. m x x 0, x 1 x 0 10 fx 1 fx 0 mod. m m x 0, x 0, 1,, m 1 m 1 ax b mod. m a, m 1 a, m d > 1 b d d, m i a, m 1 {x 1, x,, x m } m {ax 1, ax,, ax m } 37

m ax i ax j mod. m, a m x i x j mod. m i j b {x 1, x,, x m } x i ii a, m d > 1 ax i b mod. m ax b mod. m 7, ax b mn N b ax mn d a, m a da, m dm, b db 7 11 a x b mod m 8 a m 8 x m x x 0 mod. m 8 x x 0 + m t t 9 t 1 t x m m t 1 t 0 mod. m t 1 t 0 mod. d 9 t d {0, 1,, d 1}, m 7 d 7 ax + my b 38

13 m 1, m,, m k, a 1, a,, a k x a 1 mod. m 1 x a mod. m 10 x a k mod. m k x M m 1 m m k x x a 1 + m 1 t 11 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 11 x a 1 + m 1 t 0 + m 1 m s x a 1 + m 1 t 0 mod. m 1 m 10 x x 0 mod. M Chinese Remainder Teorem 3 5 3 7 16 Gauss 39

M m 1 m m k M m 1 M 1 m M m k M k 1 M n t n 1 mod. m k n 1,,, k t n n 1,,, k 10 x a 1 M 1 t 1 + a M t + + a k M k t k mod. M 1 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 10 x 1 x mod. m n n 1,,, k, m 1, m,, m k M x 1 x mod. M 5.7 17 6x 1 mod. 57 5.8 18 3 1 5 7 3 5.9 19 m, n d l { x a mod. m, x b mod. n. a b mod. d l 40

5.10 0 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 5.3 fx a 0 x n + a 1 x n 1 + + a n fx 0 mod. m x a i m a 0 0 mod. m n 14 p, n fx 0 mod. p 13 n p n n 1 a 0 x + a 1 0 mod. p, a 0, p 1 5, p n 1 n 1 13 x a mod. p fa 0 mod. p 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 13 x af 1 x 0 mod. p 41

p x a f 1 x p x a mod. p n 1 f 1 x 0 mod. p n 1 13 n fx 0 mod. m m m p m p e, m p 15 m m p e q f, m p e, m q f, 16 15 fx 0 mod. p e 14 fx 0 mod. p 15 x 0 15 1 f x 0 0 mod. p fx 0 mod. p e x x 0 mod. p mod. p e f x 0 0 mod. p fx 0 mod. p e x x 0 mod. p fx 0 mod. p e+1 p fx 0 mod. p e+1 x x 0 mod. p e 4

e 1 e fx 0 mod. p 16 16 15 16 x p 15 16 x x x 0 + py 17 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! 17 16 fx fx 0 + py fx 0 + pyf x 0 + p y f x 0! + 0 mod. p f x 0, f x 0 3 p! 17 16 fx 0 + pyf x 0 0 mod. p y fx 0 p fx 0 p + yf x 0 0 mod. p 18 1 f x 0 0 mod. p 18 y y 0 mod. p py py 0 mod. p 17 x x 0 + py 0 mod. p 16 f x 0 0 mod. p 18 fx 0 p p fx 0 p y p x 0, x 0 + p, x 0 + p,, x 0 + p 1p mod. p 16 17 x 16 p p e 14 x 0 e + 1 x 0 43

mod. p mod. p x x 0 + p e y fx 0 mod. p e+1 A f x 0 0 B f x 0 0 mod. p p e+1 mod. p p e+1 p p fx 0 0 mod. p e+1 5.1 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 mod. 7 x 0 ±3 x mod. 49 x 3 + 7y 16 3 + 7y mod. 49 9 + 4y mod. 49 6y 1 mod. 7 y 1 mod. 7 x 10 mod. 49 x 10 39 mod. 49 m m p e q f 44

fx mod. p e 19 fx mod. q f 0 l, l, ll fx 0 mod. m 1 x α mod. p e x β mod. q f α, β, p e, q f, fx 0 x 1 19 0 1 x 19 0 1 5. 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. 1 5.11 1 1 x + x + 1 3 mod. 5 x 1 mod. 39 45

5.4 19 19 8 n 3 n+1 + 4 n 1 13 0 0 n 19 n + 1 n 1 4n 3 7 1 1 8 n fx x n + a 1 x n 1 + a n 1 x + a n n > 1 1 α fx 0 α k> 1 k f1, f,, fk k fx 0 01 n n 9 n 3 9 3 3 α 1 mod. 8 e > 3 x α mod. e x 0 ±x 0, ±x 0 + e 1 46

6 ϕn 6.1 ϕn 1,,, n n x ϕn ϕ1 1, x 1 ϕ 1, x 1 ϕ3, x 1, ϕ4, x 1, 3 ϕ5 4, x 1,, 3, 4 ϕ6, x 1, 5 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 ϕn Z n Z n k {x x k mod. n} k k k mod. n Z n k Z n k Z n k n Z n n Z n 1, Z n,, Z n n Z 1 { 3, 1, 1, 3, } Z { 4,, 0,, 4, } 5 n 1,,, n 47

Z n k n n Z n k x x k + nt t k + nt, n k, n Z n k n n Z n k ϕn ϕn n F x a b 17 F ab F af b ϕab a b ϕab ϕaϕb 3 a Z a k b Z b l A {bx + ay x Z a k, y z b l} ab s, t x k+as, y l + bt bx + ay bk + al + abs + t s, t ab A Z ab bk + al bx + ay ab a b a b a bx + ay, a 1 bx, a 1 x, a 1 k, a 1 Z a k 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 Z ab m a b bk + al m k, l Z ab m Z a k Z b l a Z a k b Z b l ab Z ab m ϕaϕb 3 5 13 48

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 4 γ ab γ ab γ, ab 1 4 α i, β j ab γ α i, β j 3 ϕn 18 a b n 17 n p α q β r γ ϕn n 1 1 1 1 1 1 5 p q r ϕn ϕp α q β r γ ϕp α ϕq β ϕr γ 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 1 1 p 1 1 q 1 1 r 6.1 a 3, b 5, ab 15 α 1,, β 1,, 3, 4 ϕ3, ϕ5 4 γ 1,, 4, 7, 8, 11, 13, 14 ϕ15 8, γ 5α + 3β α 1 1 1 1 β 1 3 4 1 3 4 γ 1 7 13 4 11 8 14 d n d n 1 d 1, n d n 19 n 1 n ϕd n 6 d n 49

d n 1 n x, x, n d x x, n d d, n 1 d 1 n d n n d ϕ d 1 n x x, n { 1,,, n } d n { x x, n d, 1 < x < n } d n 1 n 1 n n ϕ n d d n n d n n 6 d 6. 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 ϕ3 15 15 1 ϕ1 15 6.1 1 1 151 151 6. 3 xy 0 < x < 1, 0 < y < 1 6.3 4 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 50

6. F x, Gx F d Gn 7 d n F x Gx Möbius µn 1 n 1 µn 1 k n k 0 n 6.3 µ1 1, µ 1, µ3 1, µ4 0, µ5 1, µ6, 1 n > 1 µd 0 d n n > 1 n n p 1 e 1 p e p k e k µd d n x µp1 x 1 p x p k x k 0 < x 1 < e 1, 0 < x < e,, 0 < x k < e k x 1, x,, x k 0 µd µ1 + {µp 1 + µp + + µk} d n +{µp 1 p + µp 1 p 3 + + µp k 1 p k } + +µp 1 p p k 1 k + k C k C 3 + + 1 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 51

n n µ F δ d d n δ d δ d, n d n δ δ d F δ δ n δ µδ n 1 δ > 1 0 F nµ1 d n µ n Gd F n d n F d d µ Gδ δ d n d n δ d δ n δ n δ µδ Gδ Gn F n ϕn Gn n 19 6 n ϕn ϕn n µ d d d n 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 pr + n + + µpqr pqr n n p n q n r + + n pq + n pq n + n pqr 1 1 1 1 p q 1 1 r 5

6.4 5 F n Gn F d d n Gn 1 6.5 6 x n ϕn, x ϕn, n ϕn [x] x 19 0 ϕd, dx [nx] d n ϕn, x d n µd [ ] x d 53

7 n 7.1 n 1 n n cos kπ n x n 1 0 8 kπ + i sin, k 0, 1,, n 1 9 n n 1 8 α cos π n + i sin π n, k 0, 1,, n 1 9 α k k 0, 1,, n 1 x n 1 x α k x n 1 x n 1 Qxx 1x α x α n 1 Qx 1 n 8 α n 1 9 k n kπ k, n 1 9 n π n αk n 1 1 n n 1 1 n 1 1 n ϕn cos kπ n kπ + i sin n, k n kπ k, n 1 1 n π α k n 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 54

7.1 1 6 1, 1, 1 ± 3i, 1 ± 3i 1 ± 3i 6 3 1 1 1 n n p α q β r γ F n x xn 1x n n pq 1x qr 1 x n n n 30 p 1x q 1 x pqr 1 x n d 1 µd d n 31, F n x 1 n F n x ϕn 1 µn 1 n 1 F n x 0 n 1 n n d d d 1 n n d n n n 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 d n µd logx n d 1 F n x d n x n d 1 µd x x x 31 F n x 1 ϕn 31 30 1 F n x 1 7. F 6 x x6 1x 1 x 1x 3 1 x x + 1 55

F 1 x x1 1x 1 x 6 1x 4 1 x4 x + 1 p F p x xp 1 x 1 xp 1 + x p + + 1 F p ex x pe 1 x pe 1 1 e 1 xp p 1 + x pe 1 p + + 1 7.1 7 F n x n 1 +1 7. 8 F n x ϕn 1 µn 1 n µn 7.3 9 α 1 n α k k 0, 1,, n 1 n k, n 1 k n 7.4 30 1 a, b 1 a b ab 1 ab 1 a b 1 ab 7. 4 4 70 i α cos π 3 + i sin π 3 n 1 α 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 π 5 5 α n n 1 x n 1 0 1, α, α,, α n 1 x n 1 0 3 1 + α1 + α 1 + α n 1 56

4 m n 1 + α m 1 + α m 1 + α n 1m 6 6 01 p n z 1, z,, z n a n k 1,,, n z p k 1 z k 1 z 1 z z n 1 1 a 3 a n+ a n a n+1 3 a n 7 7 01 0 G G z, w zw G 1 n G n G 1 57

8 8.1 4n+1 3 m a m a ϕm 1 mod. m 3 m p, a, p 1 a p 1 1 mod. p 33 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 x 1 x x ϕm, m 1 a ϕm x 1 x x ϕm mod. m a ϕm 1 mod. m 3 m p ϕp p 1 33 58

8.1 1 ϕ5 4 13 4 1 mod. 5 13 4 1 5 571 ϕ5 4 11 4 1 mod. 5 11 4 1 5 98 ϕ11 10 10 1 mod. 11 10 1 11 93 ϕ1 5 5 5 1 mod. 1 5 5 1 1 5 ϕ60 60 1 1 1 1 1 1 16 7 16 1 mod. 60 7 16 1 7 17 + 1 3 5 7 + 17 4 + 17 8 + 1 13 4 1 mod. 5 11 mod. 5 a a e 1 mod. m e a m 4 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 < 1 r 0 a r 1 mod. m r e r 0 k e 8.1 31 1 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,. 8. 3 a m e a k e k, e 8. 1,, 3,? 4 59

a> 0 d> 0 Dirichlet 1837 a 1 5 m mt + 1 1 4n 1 4n 1 p 4n 1 4 1 a a 43 7 11 p 1 a p 4n 1 a a 4 1 4 1 1 4k + 1 4k + 1 4n 1 4n 1 a 4n 1 3 p 4n 1 p 4n 1 4n 1 4n + 1 4n + 1 4n + 1 x + 1 4n + 1 1 + 1, + 1 5, 3 + 1 5, 4 + 1 17, 5 + 1 13, 6 + 1 37, x + 1 p x + 1 0 mod. p x 1 mod. p x 4 1 mod. p 60

x 4 4 1 x 1 mod. p, x 1 mod. p x + 1 0 mod. p 1 1 mod. p p 4 p 1 4 p 1 4n p 4n + 1 4n + 1 4n + 1 p 4n + 1 1 a a 5 13 p + 1 a a 45 13 p + 1 4n + 1 q 4n + 1 a p 4n + 1 q p 4n + 1 p 4n + 1 4n + 1 3 mt + 1 m 1 m m 4 4n + 1 4n + 1 x + 1 19 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 a e 1 e m m ef 61

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 Gx x e 1Hx Hx 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 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 1 6 6 4 6 + 1 161 13 97 13 1, 97 1 mod. 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 + 1 J. 8.3 33 x k+1 + 1 x + 1x k x k 1 + x k a + 1 n n! + 1 4n + 1 4n + 1 6

8.3 6 m n n 10, n 1 m n e, e 10 e 1 mod. n e ϕn, n 10 e 1 n a m n ma na ma 10 e 1 ma 10 e + ma ma + 10e 10 3e + m < n ma < na < 10 e m n e m e n c 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 10 k m m k k maxu, v n n 10, n 1 10 n e e n 1 m n 8. n 7 10 1 3 mod. 7, 10 mod. 7, 10 3 6 mod. 7, 10 4 4 mod. 7, 10 5 5 mod. 7, 10 6 1 mod. 7 10 7 e 6 10 7 63

10 1 3 + 7 1 10 + 7 14 10 3 6 + 7 14 10 4 4 + 7 148 10 5 5 + 7 1485 10 6 1 + 7 14857 1 7 14857 10 6 1 14857 { 10 6 1 + 1 10 6 + 1 } 10 1 + 0. 1485 7 7 3 7 7 6 7 4 7 5 7 101 7 1 0. 4857 1 10 7 103 7 104 7 105 7 14 0. 8571 4 14 0. 85714 148 0. 5714 8 1485 0. 7148 5 6 7 6 7 + 1 7 1 0. 9 0. 9 0. 1485 7 0. 85714 8.3 n 13 10 7 10 1 10 + 13 0 10 9 + 13 07 10 3 1 + 13 076 10 4 3 + 13 0769 10 5 4 + 13 0769 10 6 1 + 13 07693 10 7 10 + 13 076930 10 8 9 + 13 0769307 10 9 1 + 13 07693076 64

10 10 3 + 13 076930769 10 11 4 + 13 076930769 10 1 1 + 13 0769307693 10 13 e 6 1 13 07693 10 6 1 07693 { 10 6 1 + 1 10 6 + 1 } 10 1 + 0. 0769 3 10 13 9 13 1 13 3 13 4 13 101 13 0 0. 7693 0 10 13 103 13 104 13 105 13 07 0. 6930 7 076 0. 9307 6 0769 0. 3076 9 0769 0. 30769 10 1 7 + 13 1 10 5 + 13 15 10 3 11 + 13 153 10 4 6 + 13 1538 10 5 8 + 13 15384 10 6 + 13 153846 7 13 5 13 11 13 6 13 8 13 13 101 13 10 13 103 13 104 13 105 13 105 13 1 15 153 1538 15384 153846 0. 53846 1 0. 38461 5 0. 84615 3 0. 46153 8 0. 61538 4 0. 15384 6 1 8.4 34 n 91 7 13, 91 90 91 65

8.4 8 8 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 1 + 1 4 p > p p! 1 p 66

9 9.1 13 0 1 a 1 3 4 5 6 7 8 9 10 11 1 a 4 9 3 1 10 10 1 3 9 4 1 a 3 8 1 1 8 8 5 5 1 1 5 a 4 3 9 1 9 9 1 3 3 a 5 6 10 11 4 7 a 6 1 1 1 1 1 1 a 7 11 7 6 a 8 9 3 3 9 a 9 5 5 8 8 a 10 10 4 4 10 a 11 7 11 6 a 1 1 1 1 1 1 1 1 1 1 1 1 1 p, a p a p 1 1 mod. p a 1 1 6 11 1 1, a, a,, a 11 13 p 0 a p 1 1 a p p 7 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, K, 7 p r, r, r,, r p 67

a p a 0 mod. p a m a m 1 mod. p a 0 1, a 1,, a m 1 34 x m 1 mod. p 35, 1 < i < m 1 a i a j mod. p a i j 1 mod. p 4 i j m m + m p m m, n d > 1 m n l l m 0 n 0, m 0, n 0 1 m 0 m n 0 n d d d 1 d 1 m 0 md 1 d, n 0 nd d 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 l n m l > m p m p 1 p 1 r 34, r, r,, r p p 1 k, p 1 1 r k ϕp 1 68

9.1 ϕ13 1 4, 6, 7, 11 9.1 35 p 41 9. 7 r p a 0 mod. p a r α a mod. p 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 a b Ind r a s mod. p 1 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 9. p 13 p a I Ind. a a 1 3 4 5 6 7 8 9 10 11 1 I 0 1 4 9 5 11 3 8 10 7 6 36 69

8 p r Ind. ab Ind. a + Ind. b mod. p 1 Ind. a n n Ind. a 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 1839 1000 Jacobi Cunninghana Jacobi Messenger of methematics, 46 1916 9.3 p 13 7x 10 mod. 13 Ind. 7 + Ind. x Ind. 10 mod. 1 11 + Ind. x 10 mod. 1 x 7 mod. 13. Ind. x 1 11 mod. 1 9 p, a 0 mod. p x n a mod. p f p 1 n, p 1 a f 1 mod. p 70

x n a mod. p p r n Ind r x Ind r a mod. p 1 37 n, p 1 e 1 37, Ind r a e Ind r a α, α e α eq a f 1 a r eq mod. p a f r efq r p 1q 1 mod. p 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 9.4 n, p 7 e, f 3 1,,, 6 1, 4, 3 9. 36 9. p 13, r 1 Ind. 100 Ind. 1 3 Ind. x 9 x 4 Ind. x 1 x 9.3 37 9. x 1 11x 5 mod. 13 71

x 3 5 mod. 13 3 5x + 3x 10 0 mod. 13 9.4 38 p Ind. 1 p 1 9.5 39 p a + b p Ind. a Ind. b p 1 mod. p 1 9.6 40 Ind r a Ind r a Ind r r mod. p 1 9.7 41 k p 1 1 k + k + + p 1 k 0 mod. p 9.8 4 p p 1! 1 mod. p 9.3 9 9 95 n fn, gn. fn n 7 7 gn 3f k1 k n 1 n fn 7 fn. n gn. gn 7

10 10.1 5 0, 1,, 3, 4 0, 1 1, 4, 3 4 mod. 5, 4 1 mod. 5 3 5 1 4 5 3 5 0, 1 0, 1 p x a mod. p a p a 0 mod. p a a p a +1 1 p Legendre p Ind. a a a 1 Ind. a 38 p p p 1 p 1 mod. p 1,,, 38 a a 1 a a mod. p p p abc a b c p p p p x p x, p 5 3 1 mod. 5, 4 3 mod. 5 30 a a p 1 mod. p p 73

a 9 a p 1 1 mod. p a 1 a p 1 1 mod. p p a 1 a p 1 p 1 mod. p a p 1 1 mod. p a p 1 1 mod. p a p A A r p x a A { 1,, 3,, p 1 } rs a mod. p A s s r a 1 r x a mod. p, r r p r mod. p, p r p r A x a mod. p 11 p p p r p 3 p 3 a p 3 mod. p rp r r a mod. p a p 1 p 1! a p 1 mod. p a 1 A p 1 1 p a 1 p 1! a p 1 mod. p p p 1! 1 mod. p a 1 a p 1 1 p a p 1 a p 1 1 mod. p 74

a 10.1 ±1 p a p 1 a p 1 1 mod. p 10.1 p 5 4, 3 3, 4 1, 4 mod. 5 5 1 3 4 1, 1, 1, 1 5 5 5 5 p 1 1 3 4 1, 1, 3 1, 4 1 mod. 5 5 5 5 5 10. 31 n 0 n x 1 + x + x 3 + x 4 0 < x 1, x, x 3, x 4 x 1 + x + x 3 + x 4 y 1 + y + y 3 + y 4 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 x 3 y x 4 y 1 39 n p n 1 + 1 n p > 1 p x + 1 ph 75

1 p 1 p 1,,, p 1 p 1 p 1 k, k + 1 k k 1 1 k + 1 1 1 1 p p p x 1 k x k 1 x 1, x mod. p, mod. p, x 1 + x + 1 0 mod. p x 1 + x + 1 ph p x 1 + x + x 3 + x 4 ph 40 x 1, x, x 3, x 4 h x 4 39 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 1 40 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 1 y i <, i 1,, 3, 4 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 39 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 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 x 1 y 4 + x y x 3 y x 4 y 1 x 1 x 4 + x x x 3 x x 4 x 1 0 mod. h 76

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 40 i 1,, 3, 4 m 1 + 1 h 4 + m + 1 h 4 + m 3 + 1 h 4 + m 4 + 1 h 4 ph {m 1 + 1 + m + 1 + m 3 + 1 + m 4 + 1 } h 4 p m 1 + m 1 + + m 4 + m 4 + 1h h p h < h 10.3 113 113 p q 3 p, q p q 1 : 1 p 1 q 1 q p 1 : 1 p 1 p 77

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. 4 3 1 p 1, 7 mod. 8 p 1 p 3, 5 mod. 8 p Leonhard Euler 1707 83 Adrien Marie Legendre 175 1833 3 Karl Friedrich Gauss 1777 1855 33 a p 1 a, a, 3 a,, p 1 a, 41 p p n a 1 n p p 1 10. a 3, p 7 3 3, 6, 9 7 3, 6, 3 n 1 1 7 1,, 4, 3, 5, 6 7 78

p p p p p, n 41 ±1, ±,, ± p 1 41 p 41 41 41 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 30 a 1 n mod. p p ±1 p a 1 n p 3 a 1 1 1 p 1 mod. p p p 1 1 p 1 p a 1 41 1,, 3,, p 1, n p 1 1 1 p 1 p 79

a 41, 4, 6,, p 5, p 3, p 1 p n p < k p < p p 1, 3, 5 p 1 p 1 p 1 1 p n 1 + 3 + + 1 + + 3 + + p 1 mod. n 1 p 1 p 1 p + 1 p 1 8 1 n 1 p 1 8 mod. p + 1 xy A, 0 B 0, q + 1 p + 1 C, q + 1 B q L G C H G p 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 1 m OL x q HH OHH L p q 1 m+n m + n q p LH CG OGG CH H m + n 80

p + 1 OGG CH H OC 4, q + 1 4 p + 1 OGG CH H 4, q + 1 4 p + 1 m + n 4, q + 1 4 p + 1 m + n 4, q + 1 s, t 4 p 1 q 1 p 1 s 1, t 1, q 1 a p a, p 10.3 p 3 1 1 1 1 3 1 3 7 mod. 8, 3 3 3 1 1, 3 3 3 4 1 3 3 5 3 1 1 1 1, 3 5 5 5 5 6 3 1 3 3 3 7 3 1, 3 7 7 3 8 1 3 3 9 3 1 3 3 10 5 1 3 3 3 11 3 1 1 3 11 11 17 3 6 3 3 17 17 3 17 3 17 1, 17 3 3 10.1 43 365 1847 10. 44 p 8k + 1 8k + 3 1 p 81

10.3 45 p 5k ± 1 5 1 p 10.4 46 p 1k ± 1 3 1 p 10.5 47 p p 1 mod. 4 p 1 a1 a p b p 0< b< p 0< c< p c 0 p 10.4 3 1 p α 1 p α cos π p + i sin π p G p 1 k G α k p k1 p G 1 p p 0 G 31 3 1 3 r p 7 1, r, r,, r p 9 i r i ±1 p 8

G r G α rj p 3 j0 p 3 j0 α rj+1 β 0 β 1 p 3 j0 p 3 j0 α rj α rj+1 α + α r +... + α rp 3 α r + α r3 +... + α rp α α p 1 + α p + + 1 0 β 0 + β 1 + 1 0 G β 0 β 1 β 0 + β 1 4β 0 β 1 G β 0 β 1 10.4 p 5 3 5 3 3, 3 4, 3 3, 3 4 1 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 4 1 1 5 4 10.5 p 7 3 7 3 1 3, 3, 3 3 6, 3 4, 3 5 5, 3 6 1 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 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 10 3 + α + α + α 3 + α 4 + α 5 + α 6 1 + 7 4 1 83

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 p 5, 7 34 1 + p β 0 β 1 4 1 p 4 p 1 mod. 4 p 1 mod. 4 84

r p 1 1 mod. p r p 1 1 mod. p k 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 1 1 + α + + α p 1 0 α p 1 α p i p 1 p 1 mod. p p 1 β 0 β 1 α + α r +... + α rp 3 α r + α r3 +... + α rp p 1 p 1 1 p 1 p 1p 3 4 p 1 α + + α p 1 p 3 1 4 β 0 β 1 1 p 1 + 1 p 3 1 + p 4 4 ii p 1 p 1 mod. p p 1 β 0 β 1 α + + α p 1 1 β 0 β 1 1 p 1 4 1 p 1 p 1 4 G 3 p a n x n + a n 1 x n 1 + + a 0 a n x n + a n 1 x n 1 + + a 0 p a n p x pn + a n 1 p x pn 1 + + a 0 p mod. p 85

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 1 + + a 0 p {a n x n + a n 1 x n 1 + + a 0 } p a n p x pn + a n 1 x n 1 + + a 0 p mod. p a n p x pn + a n 1 p x pn 1 + a n x n + + a 0 p mod. p a n p x pn + a n 1 p x pn 1 + + a 0 p 35 q p p ±1 mod. p q p ± p G β 0 β 1 G β 0 β 1 β 0 + β 1 4β 0 β 1 1 1 p ±p q 30 ±p q 1 ± p q mod. q G q p 1 k1 p 1 k1 p 1 q p k p G q 1 ±p q 1 ± p ± p G q G mod. q q q k α k p k1 α kq p 1 k1 q mod. q q k α kq mod. q 3 p q kq α kq p { q p q G p ± p q q G p ± p q G mod. q } G 0 mod. q 86

q ± p G α, α, α p 1 ±1 p q 0, ± q q ± p p q q G G G G 1 p 1 17 10.5 J.P,Serre G.Eisenstein,F. 1845 33 4 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 } 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 87

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 m 1 sin x sin x 4 m 1 x ± πj m sin mx 0 ± sin πj m sin mx sin x m 1 4 m 1 j1 m 1 1 < j < m 1 1 < j < sin x sin πj sin x + sin πj m m 88

3 p q p q 1 p 1 q 1 q p i 1,,, p 1 p qi p n n sin πqi p i 1,,, p 1 33 sin πi p p 1 i1 i 1,,, sin πqi p 1 n 33 q 1 n p p 1 p 1 i1 sin πi p 4 m q, x πi 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 89

p 1 q 1 p 1 q q p p q 1 p 1 q 1 q p 10.6 30 30 98 1 x + y + z n x, y, z n 5 n 8 7 x + y + z n x, y, z 90

11 11.1 i 1 R { a + bi a, b : } 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 β γ α β, α β β α 1 1 1 α 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 91

R R 1 α β, α β α α, α, iα, iα, ±1 36 R α β γ, ρ R α β α βγ + ρ, ρ 0 Nρ < Nβ r + si r, s 1 r, s m, n r m <, s n 1 < γ m + ni N α β γ < 1 +, ρ α βγ α Nρ N β β γ NβN 1 1 α β γ < Nβ < Nβ a bq + r 0 < r < b deg fx gxqx + rx 0 < deg rx < deg bx N α βγ + ρ 0 < Nρ < Nβ 37 α, β R J J { αx + βy x, y R } J R δ δ J 0 0 δαx 0 + βy 0 J αx + βy αx + βy δγ + ρ, 0 < Nρ < Nδ 9

ρ αx γx 0 + βy γy 0 R, Nδ ρ 0 δ δ δ δ α β α β δ 1 38 0 39 p p π p π π, π π p π π p R p ɛπ 1 π π l ɛ π 1, π,, π l p Nπ 1 Nπ Nπ l Nπ l p 1 l 1 p ɛπ 1, p p, l Nπ 1 π l π l p, π l π p π π π π π ±π, ±iπ πx + iy x 0, p y, y ±x, p x, p p N1 + i 1 + i1 i i 3 1 + i ±1 ± i p 93

40 p p 1 mod. 4 p a + bia bi p 4 mod. 4 p p p a + bia bi a + b p a b p 1 p 1 mod. 4 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. 4 4 1 11.1 48 5, 13, 65, 5, 50, 13 G G G G 1 a, b G a b G a b c a b c 3 a G, a e e a a e G 4 a G, a x x a e x G x a 1 94

, a b b a 11.1 N, Z, Q, R, C N, Z, Q, R, C 0 N, Z, Q, R, C M a b c a b a c, b c a b a c a M 11. 1 Z x x 3 x 4 K K K 11.3 1 11. 31 31 x + y z, x, y 1 x, y m n, nm, z m + n m, n 1, m > n > 0 m n 3 3 01 p a b, a+bi p i 33 33 0 1 5 4 1 p, p 4k + 1 1 Q a + 4bc p 3 a, b, c 3 Q 3 a, b, c 3 i ii iii : 95

i 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 i ii iii Q 5 p 4k + 1 a, b p a + b 96

1 1.1 x Dy ±1 D J.Pell 1610-1385 1 3 95 85 D 1999 x Dy ±1 P, Q P 40770139938913808695911951306886478800 Q 900846650300460494303744455049 k 84! 1.1 95 3 A 1 x n y n A n 1 0 n 1,, 3, 97

1 x n y n 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 1 1. 95 3 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,, 1.3 85 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. 1.1 98

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 1 + 3 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 1 b n 1 1 P n, n 1,, x 3y 1 1.1 A n p q A + pa + qe 0 A n+ + pa n+1 + qa n 0 A n 1. 1 x 1 y 1 3 1 1 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 99

1 x 1 + 3y 1 3x 1 + y 1 4 3x 1 + 1x 1 y 1 1x 1 y 1 + 9 1y 1 x 1 3y 1 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 > 0 3 1 n 1 a k n k A k 1 a k, b k + b k 0 3 k a k + b k 3 a k+1 b k+1 A k+1 1 0 3 1 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 100

y 1 > 0 x 1 x 3y 4x 9y x + 3y 4x 3x 1 x + 3y x + 3 x + 3y > 0 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 y A n 1 0 a n b n 1.3 1 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 gg G aa + bb ab + a b a a b b a b ±a b 1 + 1 1 + > 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 101

{ } x n A x n, y n A n 1, n : 0 y n B {x, y x 3y 1, x, y } A B A B ±1, 0 ±1, 0 1985 41 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 x 1 + Dy 1 x + Dy n s + Dt s, t s, t S 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, n } 10

p Dq 3 S A q p { S x n, y n x n y n A n 1 0 }, n 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 y 1y Dx 1y Dx 1 y x y 1 Dx y 1 x 1 Dy 1x Dy ±1, n 1., 1 x + x Dy Dy ±1 {±x + D y } x + Dy > 0, 1 x + > 0 Dy, ±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., p + Dq n < s + Dt < p + Dq n+1 1 < s + Dt p + Dq n < p + Dq, 1 s + Dt p + Dq u + Dv u, v, n u, v S 103

., p + Dq,, n,., s + Dt < 1,, 1 s + Dt,, u + Dv 1 s + Dt p + Dq n 1 s + Dt > 1 1 s + Dt p + Dq n 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+1 y n+1., x n A n 1 y n 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 0 1 0 1.1 49 D, D 3 1 D, D 3 A D 3 x 3y 1 104

1. 34 34 98 1 x ny z nt xz + nyt nxt + yz x y 1 x, y x > 100 35 35 01 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 3 + 1 n x n + y n xn, y n n Px n, y n C + C 4 C + C Px n, y n n y n+1 y n 5 lim n x n+1 x n 105

13 13.1 x Dy ±1 ±1, 0 P.G.Dirichlet 1805-59 n n + 1 4 ω x ωy < 1 y x, y i n 0 < y < n, x ωy < 1 y x, y a x ωy 1 x 1 ωy x < 1 n x x 1 x, y y 1 y ωy x < 1 n 1 n < 1 y 106

ωy x < 1 y ii n 0 < y < n ωy x < 1 y x, y n x, y ωy x ωy 0 x 0 1 n < ωy 0 x 0 n n ωy x < 1 n x, y ωy x < 1 n < ωy 0 x 0 x 0, y 0 4 13.1 91 50 m n m > n 1 1 5 1 1 9 1 13. 107

43 D D x Dy 1 X, Y, X > 0, Y > 0 4 x Dy < 1 y x, y x > 0, y > 0 1 y < x Dy < 1 y x + D < 1 y + Dy x Dy < 1 y x Dy < 1 y + D < 1 + D 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 1 108

X, Y x Dy l 13.3 36 36 99 A, B, C, D, E { A z z + z } {, B z z + z } { 3 C z z z } {, D z z z } i 3i E A C A D B C B D E 17 F F E 37 37 97 n n a k 1 < k < n i, ii i a k 1 < k < n 1 n ii s j 1 < a k < n j a k s j 1 < j < n k1 1 s n n a k 1 < k < n 109

14 14.1 7 a qb + r 0 < r < b a b q 1 1 0 b r a b a b qb + r q b r + 1 b b r b q + 1 r b 1 + 0 r a q b, b r a b aω + b ω c d cω + d a b ω ω d c d c { } { } a b e f a b e f 1 ω ω c d g h c d g h 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 1 u ω 110

ω ω 1 ω q 0 ω q 0 + u 0 < u < 1 3 ω 1 1 u 1 < ω 1 ω q 0 + 1 ω 1 q 0ω 1 + 1 ω1 q 0 1 1 0 ω 1 q 0 1 q k 1 ω k+1 ω 1 1 0 ω k + 1 q 0 1 q k 1 ω ω k+1 1 0 1 0 ω k+1 ω k+1 1 1 ω k+1 1 1 0 ω k+ 1, q k+1 ω k+1 1 ω 1 ω q 0 + 1 ω 1 q 0 + 1 q 1 + 1 ω 1 q 0 + 1 q 1 + 1 q + q 3 + P 0, Q 0 q 0, 0 q 0 1 q 1 1 q k 1 1 0 1 0 1 0 P k Q k P k 1 Q k 1 111

P k, Q k n > P k P k 1 q k + P k Q k Q k 1 q k + Q k ω, ω ω k 0 1 1 0 h 0 1 1 0 n > m k 1 1 1 0 h 1 1 1 0 k n 1 1 0 h m 1 1 0 ω ω > 1 ω ω > 1 k 0 h 0, k 1 h,, k m h m k 1 1 k n 1 X 1 0 1 0 h 1 1 h m 1 Y 1 0 1 0 X > 1, Y > 1 k 0 1 ω X k 0 + 1 1 0 X h 0 1 Y h 0 + 1 1 0 Y, k 0 h 0 X Y ω k 0 h 0, k 1 h,, k m h m a b a q 0 1 q 1 1 q n 1 1 b 1 0 1 0 1 0 0 q 0 1 q 1 1 q n 1 1 1 1 1 1 0 1 0 1 0 1 0 0 ω a P k P k 1 b Q k Q k 1 r k r k+1, k 1,,, n 11

a b P k Q k P k 1 Q k 1 r k, k 1,,, n 1 r k+1, ω a b, ω k+1 r k r k+1 14. ω ω k + 1 ω k ω 44 ω ω k + 1 ω q 0 1 1 0 P k Q k P k 1 Q k 1 q k 1 1 0 ω k+1 ω k+1 1 P 0 Q 0 < P Q < < P k Q k < < ω < < P k+1 Q k+1 < < P 3 Q 3 < P 1 Q 1 P n lim ω n Q n 3 P n Q n 1 P n Q n P n 1 Q n 1 1 n+1, P n Q n 1 Q n P n 1 1 n+1 P n Q n P n 1 Q n 1 1n+1 Q n Q n 1, P n+1 Q n+1 P n Q n 1n Q n Q n+1 Pn+1 Q n+1 P n 1 Q n 1 1 n+1 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 113

P n Q n P n 1 Q n 1 1 ω 1 n+1 Q n 1 Q n P n 1 P n ω Q n < 0 Q n 1 Q n 1ω P n 1 Q n ω + P n ω n+1 > 0 ω P n 1 Q n 1 ω P n Q n P 0 q 1 < ω Q 0 < 0 { P n > ω n Q n < ω n P N N < ω 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+ Q N Q N+1 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+1 P n, Q n 114

P n ω Q ω P n Q q < Q p q ω P Q < ω p q P Q ω 45 P n n Q n A A, B, C, D B, C AD BC 1 D 1 BD X Y C D < X Y < A B DX CY > 0, AY BX > 0 { Ax + Cy X Bx + Dy Y x, y x DX CY, y AY BX x > 0, y > 0 X > A, X > C, Y > B, Y > D p q P k Q k Q k, Q k+1, P k < p Q k q < ω ω < p q < P k+1 p Q k+1 q Q k, Q k+1 P k P k+1, Q k Q k+1 13 4 x y 4 4 x Dy ±1 x ωy ω x, y ω y x ωx y 115

46 ω ωx y < 1 x x, y 44 P N ωq N < 1 Q N, x Q N, y P N N x Q N, y P N 14.3 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 47 a b ω θ ω θ, θ > 1, c > d > 0 ω c d θ ω a c b d θ, ad bc e ±1, a c a k 0 1 k n 1 1 c 1 0 1 0 0 n, e 1 n+1 n k 0 1 k n 1 1 0 1 0 P n Q n P n 1 Q n 1 a c P n Q n P n 1 Q n 1 1 0 116

, a P n, c Q n P n Q n 1 Q n P n 1 e aq n 1 cp n 1 e ad bc e, ad Q n 1 cb P n 1 a c, d Q n 1 c c > d > 0 c Q n > Q n 1 > 0, d Q n 1 < c d Q n 1 b P n 1 a b a, c d c a c b d k 0 1 1 0 k n 1 1 0 ω k 0 1 1 0 k n 1 1 0 θ θ > 1 ω 14.1 1 1 1 1 1 + 1 1 0 1 1 0 1 1 1 3 1 + 1 + 1 1 0 1 0 1 3 1 1 7 3 + 1 + 1 1 1 0 5 7 3 1 17 7 + 1 5 1 0 1 5 + 1 1 1 < 7 5 < < < < 17 1 < 3 14.1 51 7 5 14.4 38 38 00 a. a 0 a. a 0, a 0 k 0 a 0 k 0 + 1 a 1 117

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 118

15 15.1 xy x y Gauss 1777-1855 H.Minkowski 1864-1909 Minkowski n 48 4 49 s s F F F k s F xy m n F x m y n 1 F s s s 1 s s s s 119

s s, s s k s < k F P 1, P,, P k x, y P 1 P 1, P,, P k 48 F 4 O 1 O F F F 1 F Px, y P x, y x x, y y F O P Q x, y F x P Q F M x y y, F P P M O OM M M F, Mx x, y y F 50 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, γx + δy 0 δh, γh, βk, αk 4 δh αk γh βk { αx + βy 0 γx + δy k F 4 αδ βγhk 4 F 10

ω 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 4 15.1 5 a, b, c a > 0, D b 4ac < 0 ax + bxy + cy < D π 0, 0 15. xy Aa, b, Bc, d OA OB m, n OP m OA + n OB P OA, OB e1 1, 0, e 0 1, a, b, c, d ad bc ±1 u, v { u ma + nc v mb + nd { m ±ud vc n ± ub + va OA, OB 11

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

A k, A k 1 l k : OA k + t OA k 1 Q k + tq k 1, P k + tp k 1 A k ω ω ω P k 1 Q k 1 P k Q k, ω k ω kp k 1 + P k ω k Q k 1 + Q k ωq k + tq k 1 P k + tp k 1 > 0 ω kp k 1 + P k tp k 1 + P k > 0 ω k Q k 1 + Q k tq k 1 + Q k ω k tp k 1 Q k P k Q k 1 1 k ω k t ω k k k t l k ω A k y t 1 A k t t A k 1 A k B N A k x A k A k A k A k ω ω A A 0 A A 4 A 1 A 1 A 3 A 5 A k A k ω x A k x ω 51 ω, A 0 < x < A 1 ωx y x, y Q n < A n k A k A k+ x A y ωx x, y Q n 1 < A n 1 k A k A k+ x A 13

3 ωx y x y x Q n, y P n, P n, Q n ω A Q n Q n < A < Q n+1 1 ωx y ω x, y y x, y ω 1 3 A k A k B, A k A k ω L OA k 1 A k A k A k 1, A k, B, A k ω OA k 1, LA k, LB, LA k B ω A k 1 ω A n x A P n x ωy x y y 0 ω Q n ω P n < 1 Q n Q n 5 ω P n < 1 Q n Q n P n 14

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 1 QQω P, OBN 1 Q Q ω P 1 OAM OBN 4 QQω P < 1, Q Q ω P < 1 15.3 39 39 x, y x, y 1 1 40 40 a, b, c, d Aa, b, Ba + c, b + d, Cc, d, O0, 0 OABC S 1 ad bc 1, S ad bc, S 41 41 9 xy x y ABC 15

1 AB, AC BC AB, AC 3 ABC 8 4 4 1 3 3 3 5 16

16 16.1 px + qx + r 0 5 ω px + qx + r 0 p x + q x + r 0 u p pu q uqω + r ru 0 ω q qu, r ru ω ω ω 53 1 1 ω 1 aω 0 + b cω 0 + d ω 1 px + qx + r 0 D p aω0 + b + q cω 0 + d aω0 + b + r 0 cω 0 + d pa + qac + rc ω 0 +{pab + qad + bc + rcd}ω 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 17

1 6 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 q 54 N ω N > 1, 1 < ω N < 0 ω ω ω > ω i ω k + 1 ω P k Q k P k 1 Q k 1 ω k+1 ω P k Q k P k 1 Q k 1 ω 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 Q ω P k 1 k 1 Q k 1 Q k ω P k Q k P k 1 lim ω, k Q k 1 k ω k+1 < 0 P k lim ω k Q k ω P k 1 Q k 1 < 0, ω P k Q k < 0 ω k+1 ω k ω k+1 > 1 18

ω k+1 < 0 ω k+1 q k+1 1 1 0 1 ω k+1 q ω k+ k+1 ω k+ q k+1 > 1 1 < ω k+ < 0 ii ω P k Q k P k 1 Q k 1 ω k+1 ω k+1, ω k+, ω k+3, 6 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 iii 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 iv N ω N > 1, 1 < ω N < 0 N ω N > 1, 1 < ω N < 0 M N < M ω M ω M+j ω M+j M q M 1 1 ω M 1 ω M 1 0 q M+j 1 1 q M+j 1 1 ω M+j 1 ω M+j ω M 1 0 1 0 19

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 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 N < 0 N ω N > 1, 1 < ω N < 0 k 16.1 ω 1 3 + 9 D 9 ω ω ω 1 3 + 9 + 9 5 x + 3x 5 0 3 + 9 < 1 1 9 + 5 9 3 ω + 4x 5x 1 0 1 < 9 + 5 < 0 9 5 4 4 4 4 9 + 3 9 ω 3 1 + 5x 9 + 3 6x 4 0 9 3 5 5 5 5 9 + 9 3 ω 4 1 + 5x 9 + 4x 5 0 9 5 5 5 5 9 + 3 9 5 ω 5 + 4x 9 + 3 6x 5 0 9 3 4 4 4 4 ω 6 9 + 5 10 + 9 5 x 10x 4 0 9 + 5 9 5 1 9 + 5 ω 7 ω 9 5 4 ω ω ω 7 k 5 16.1 53 ω 1 4 + 13 130

17 17.1 x Dy ±1 55 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 k, x Dy 1 D q 0 1 1 0, x 1 > 1, 1 < x 1 < 0 x 1 x 1 x 1, x 1 > 1, q 0 D x 1 1 1 D q 0 D + q0 54, D k D q 0 1 x 1 1 0 q 0 1 q 1 1 q k 1 1 0 1 0 1 0 q 0 1 q 1 1 q k 1 x 1 1 0 1 0 1 0 P k P k 1 Q k Q k 1 x 1 P k P k 1 Q k Q k 1 x k+1 x 1 P mk P mk 1 Q mk Q mk 1 1 x 1 D P mk Q mk P mk 1 Q mk 1 0 1 D x 1, x 1 1 q 0 P mk 1 P mk q 0 P k 1 D D Q mk 1 Q mk q 0 Q k 1 131

p q D 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 ps rq P mk Q mk p r P mk 1 Q mk 1 1 mk+1 1 1 mk q s 0 1 1 q 0 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 17.1 x 13y ±1 13 3 1 13 + 3 1 0 4 3 1 1 1 13 + 1 1 0 1 0 3 4 3 1 1 13 + 1 1 1 0 3 7 4 1 1 13 + 1 1 1 0 4 11 7 1 1 13 + 3 3 1 0 18 11 6 1 13 + 3 5 3 1 0 4 119 18 13 + 3 33 5 4 13

k 5 x, y P 4, Q 4 x Dy 1 x, y P 9, Q 9 x Dy 1 13 + 3 P 4, Q 4 18, 5 P 9, Q 9, x 1 4 1 3 1 119 18 P 5 P 4 x 1 x 1 x 1 1 0 33 5 Q 5 Q 4 13 3 1 x 1 1 0 119 18 P 5 P 4 x 1 x 1 33 5 Q 5 Q 4 119 18 0 1 119 18 x 1, 33 5 1 3 33 5 4165 649 P 10 P 9 x 1 x 1 15 180 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 18 + 5 13 649 + 180 13 x 13y ±1 18 13 5 1 649 13 180 1 56 x Dy ±1 x + Dy > 1, D P mk 1, Q mk 1 k D 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 1 41 x 1 > 0, y 1 > 0 q 0 1 D θ 1 0 133