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3 0 Gauss a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x ( ), D = b 2 4ac f, ax + by = c, Diophantus ax + by = c a b d c, x = α, y = β ( ) t x = α + ( b )t, y = β ( a )t d d x 2 + y 2 = z 2, Fermat( ) 4k + 1 ( ) p, x 2 + y 2 = p, 3k + 1 p x 2 + 3y 2 = p, ax 2 + bxy + cy 2 19 Gauss( ) Disquisitiones Arithmeticae Legendre Gauss,,,, ( ) Gauss 1 y

4 0 2 Gauss,, 21,, 22, ( ),, 31,,,, 32,, 33,,, D,, 41, 42,,,,

5 D, D,, 51,, D 52,,, m ξ ξ = k 0, k 1,, k n 1, ξ n (n = 1, 2,, m 1) ξ ξ 0 = ξ, ξ 1, ξ 2,,ξ m 1 f(x, y) = ax 2 + bxy + cy 2 n f(x, y) = n x, y n f n, n

6 1, 1, 4, 2, 6,,, 1,, 4, i i 2 = 1 N = {1, 2, 3, 4, } Z = {0, ±1, ±2, ±3, } } Q = a, b Z, a 0 { b a R = C = {a + ib a, b R} x x 11 a, b, c, 11 ( ) a b a = bq + r, 0 r < b q, r q a b, r a b a, b, c a = bc a b, b a, b a, b a 4

7 1 5 a b a, b a b a c a b, c, a c b c c a, b (a, b) (0, 0), a, b 0 a, b a, b (a, b) (0, 0) = 0 (a, b) = 1 a b n a, b n a b a b (mod n), n (modulus) a b 13 (Legendre ) p, a p ( ) a x 2 a (mod p) a p = 1 ( ) p a p = 1 p ( ) a p a = 0 p ( ) Legendre, p ( ) ( ) ( ) ab a b = p p p 14 (Euler ) a p ( ) a p a p 1 2 (mod p) 15 ( ) ( 1 p ) = { 1, p 1 (mod 4) 1, p 3 (mod 4)

8 ( ) ( ) { 2 1, p 1, 7 (mod 8) = p 1, p 3, 5 (mod 8) 17 ( ) p, q ( ) ( ) q p = ( 1) p 1 q p q G G (i) (ii) (iii) a, b, c G (ab)c = a(bc) e G a G ae = ea = a a G aa 1 = a 1 a = e a 1 (ii) e e G a (iii) a 1 a 1 a 19 G H G, H G (i) (ii) a, b H ab H a H a 1 H H G 110 n GL(n, Q) 111 X X X, S X

9 1 7 X X 112 G G f a, b G f(ab) = f(a)f(b) ( ) f f ( ) G G G G 113 G N G g N = g 1 Ng G N G G H G ah = {ah h H} H G a ( ) H G G/H 114 N G G/N an, bn (an)(bn) = abn G/N G N f : G G G G Im(f) = {f(g) g G}, Ker(f) = {x G f(x) = 1} Ker(f) f, Im(f) f 115 ( ) f : G G Ker(f) G, Im(f) G Im(f) G/Ker(f)

10 1 8 SL(Z) ± = SL(Z) + = { r s t u { r s t u SL(Z) = SL(Z) ± SL(Z) + r, s, t, u Z, ru st = ±1 r, s, t, u Z, ru st = 1 } }, SL(Z) ±, SL(Z) +, SL(Z) 116 SL(Z) + SL(Z) ± Proof SL(Z) + SL(Z) ± 110 GL(2, Q) SL(Z) ± GL(2, Q) 19 A, B SL(Z) ± AB = A B = ±1 AB SL(Z) ± r s u s SL(Z) ± A = A 1 = SL(Z) ± t u t r SL(Z) ± GL(2, Q), SL(Z) ± T = r s t u tξ + u 0 ξ C T (ξ) = rξ + s tξ + u ξ T (ξ)

11 1 9 SL(Z) ± T = r s t u r, T = s t u T (ξ), T (T (ξ)) T (T (ξ)) = rt (ξ) + s r ξ + s r tt (ξ) + u = t ξ + u + s t r ξ + s t ξ + u + u = T (T (ξ)) = T T (ξ) (rr + st )ξ + (rs + su ) (tr + ut )ξ + (ts + uu ) 117 SL(Z) ± T, T T (ξ), T (T (ξ)) T (T (ξ)) = T T (ξ)

12 2 21,, 22, 21 ω 0 ( ) {ω 0, ω 1, } ( ) {k 0, k 1, } k 0 = ω 0 1 ω 0 k 0 ω 1 = ω 1 0 < ω 0 k 0 < 1 ω 0 k 0 ω 1 > 1 ω 1 > 1 k 1 = ω 1 ω 1 k 1 ω 2 = ω 2 ω 2 > 1 1 ω 1 k 1 ω j > 1 k j = ω j 1 j n k j N ω j k j ω j = d (c > 0) c k j = q, ω j+1 = c r q, r d c ω j+1 c, n ω n = k n, ( ) {ω 0, ω 1,, ω n } ( ) {k 0, k 1,, k n } 10

13 2 11 {ω 0, ω 1,, ω n } {k 0, k 1,, k n } k 0, ω 1, k 0, k 1, ω 2, k 0, k 1, k 2, ω 3,, k 0,, k n 1, ω n, ω n = k n k 0, ω 1 = k ω 1 k 0, k 1, ω 2 = k k 0, k 1, k 2, ω 3 = k 0 + k ω 2 k k ω 3 k 0, k 1,, k n 1, k n = k 0 + k k n k n ω 0 = k 0, ω 1 = k 0, k 1, ω 2 = = k 0,, k n 1, k n k 0,, k n 1, k n n + 1, ω 0 = k 0,, k n 1, k n ω 0 k 0 k 1,, k n k 0, k 1,, k n 1, k n 21 h 0 h 1,, h n, h 0,, h n 1, h n α h n > 1 α = h 0, h 1,, h n α h n = 1 α = h 0, h 1,, h n α

14 h 0 h 1,, h n α = h 0,, h n α = h 0,, h n α = h 0,, h n α ω n + 1 ω = k 0, k 1,, k n 1, k n k n = = + k n k n k n ω = k 0, k 1,, k n n k n > k n = + 1 k n ω = k 0, k 1,, k n 1, 1 n , = = = = 4, 1, 2, 4 61, = 4, 1, 2, 3, 1 13 ω = ω 0 {k 0, k 1,, k n } T j = kj (j = 0, 1, 2,, n)

15 2 13 ω j (0 j n) 0 p8 117 ω 0 = k ω 1 ω 0 = k 0ω ω 1 = T 0 (ω 1 ), ω 1 = k ω 2 ω 1 = k 1ω ω 2 = T 1 (ω 2 ), ω n 1 = k n ω n ω n 1 = k n 1ω n + 1 ω n = T n 1 (ω n ) ω = T 0 T 1 T j 2 T j 1 (ω j ), j = 1, 2,, n T 0 T 1 T j 2 T j 1 = pj p j q j q j pj p j pj 1 p j 1 kj 1 1 kj 1 p j 1 + p j 1 p j 1 q j q j = q j 1 q j = k j 1 q j 1 + q j 1 q j 1 T 0 T 1 T j 2 T j 1 = pj p j 1 q j q j 1, { pj = k j 1 p j 1 + p j 2 q j = k j 1 q j 1 + q j 2 (21) ω = k 0, k 1,, k j 1, ω j = T 0 T 1 T j 2 T j 1 (ω j ) = p jω j + p j 1 q j ω j + q j 1 (22) j = n ω n = k n ω = k 0, k 1,, k n 1, k n = T 0 T 1 T n 2 T n 1 (k n ) = p nk n + p n 1 q n k n + q n 1 (23) k 0,, x = p nx + p n 1 q n x + q n 1

16 2 14 x x = 1, k n = 1 (23) 23 (23) p j q j p j 1 q j 1 = T 0T 1 T j 2 T j 1 = ( 1) j (24) 22 ω 0, {ω 0, ω 1, } {k 0, k 1, } k 0 = ω 0 ω 0 ω 0 k < ω 0 k 0 < 1 ω 1 = ω 1 ω 1 > 1 ω 0 k 0 k 1 = ω 1 k 1 N 1 0 < ω 1 k 1 < 1 ω 2 = ω 2 ω 2 > 1 ω 1 k 1, ω n > 1 k n = ω n, ω n+1 = 1 > 1 k n+1 = ω n+1 ω n k n, {ω 0, ω 1, } {k 0, k 1, } j 1 k j N {ω 0, ω 1, } {k 0, k 1, }

17 2 15 ω 0 = k 0, ω 1 = k ω 1 = k 0, k 1, ω 2 = k = k 0, k 1, k 2, ω 3 = k 0 + k ω 2 k k ω 3 = k 0, k 1,, k n 1, ω n = k 0 + k k n ω n k 0, k 1,, k n 1, ω n ω 0 {h j } j 0, h j N (j 1), lim h 0, h 1,, h n = h 0, h 1, h 2, n, ω 0 {k j } ω 0 = lim n k 0, k 1,, k n = k 0, k 1, k 2, ( ) 24 h 0 h 1, h 2,, h n 1, β > 1 α = h 0, h 1,, h n 1, β h 0, h 1,, h n 1, β α

18 2 16 ω = ω 0 {ω j } {k j } T n = kn (n = 0, 1, 2, ) T n (cf p8) ω 0 = k ω 1 ω 0 = k 0ω ω 1 = T 0 (ω 1 ) ω 1 = k ω 2 ω 1 = k 1ω ω 2 = T 1 (ω 2 ) ω n = k n + 1 ω n+1 ω n = k nω n ω n+1 = T n (ω n+1 ) ω = T 0 T 1 T n 2 T n 1 (ω n ) T 0 T 1 T n 2 T n 1 = pn p n q n q n, T 0 T 1 T n 2 T n 1 = pn p n 1 q n q n 1, { pj = k j 1 p j 1 + p j 2 q j = k j 1 q j 1 + q j 2 (25) ω = k 0, k 1,, k n 1, ω n = T 0 T 1 T n 2 T n 1 (ω n ) = p nω n + p n 1 q n ω n + q n 1 (26) p n q n p n 1 q n 1 {q j } = T 0T 1 T n 2 T n 1 = ( 1) n (27) q 0 = 0, q 1 = 1, q 0 < q 1 q 2 < q 3 <, lim n q n = (28)

19 2 17 (26) ω p n = q n = < p n ω n + p n 1 p n q n ω n + q n 1 = (p n q n 1 p n 1 q n ) q n (q n ω n + q n 1 ) q n ( 1) n 1 q n (q n ω n + q n 1 ) = 1 q n (q n ω n + q n 1 ) 1 q n (q n k n + q n 1 ) = 1 q n q n+1 25 ω {k j } (25) {p j, q j } ω p n q n < 1 < 1 q n q n+1 qn 2 26 ω {k j } {p j, q j } p n q n = k 0, k 1,, k n 1 (n = 1, 2, ) Proof α = k 0, k 1,, k n 1 22 α, 23 (23) (21) k 0, k 1,, k n 1 = p n 1k n 1 + p n 2 q n 1 k n 1 + q n 2 = p n q n 27 ω {k j } lim k 0, k 1,, k n = ω n Proof ω k 0, k 1,, k n = ω p n+1 < 1 q n+1 q n+1 2 (28) lim n q n = lim n k 0, k 1,, k n = ω

20 2 18 k 0, k 1,, k n,, ω = k 0, k 1,, k n, ω 28 n 1 k n 1 {k n } η = lim n k 0, k 1,, k n, η = k 0, k 1,, k n, η Proof {k n } T n = kn 1 1 0, T 0 T 1 T n 2 T n 1 = q n pn p n 1 q n 1 {p n }, {q n } 26, (27) p n q n = k 0, k 1,, k n 1 p n q n p n 1 q n 1 = p nq n 1 p n 1 q n q n q n 1 = ( 1)n q n q n 1 pn+1 p ( n 1 = ( 1)n 1 1 ) q n+1 q n 1 q n q n 1 q n+1 (28) 1 q n 1 1 q n+1 > 0, p 2k+1 p ( 2k 1 = ( 1)2k 1 1 ) > 0 q 2k+1 q 2k 1 q 2k q 2k 1 q 2k+1 p1 < p 3 < < p 2k 1 < p 2k+1 < q 1 q 3 q 2k 1 q 2k+1 p 2k+2 p ( 2k = ( 1)2k ) < 0 q 2k+2 q 2k q 2k+1 q 2k q 2k+2 p 2 q 2 > p 4 q 4 > > p 2k+2 q 2k+2 > p 2k q 2k >

21 2 19 p 1 q 1 < p 3 q 3 < < p 2k 1 q 2k 1 < < p 2k q 2k < p 4 q 4 < p 2 q 2 { } { } p2k+1 p2k, q 2k+1 q 2k p 2k+1 lim = ω 1, k q 2k+1 p 2k lim = ω 2 k q 2k ω 1 = ω 2 ( pn lim p ) n 1 1 = lim = 0 n q n q n 1 n q n q n 1 lim k p n+1 0, k 1,, k n = lim = η n n q n+1 η lim n k 1,, k n = η 1 η = k η 1, η 1 > k 1 1 η = k 0 η 1 = k 1, η 2 = 1 η 1 k 1 η 2 = k 2,, η n = k n, {k n } η η = k 0, k 1,, k n, η

22 3 ( ),, 31,,, 32,, 33,,, 31 a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 a b/2 b/2 c b/2 c x y 20

23 3 21 f D = b 2 4ac = 4 a b/2 b/2 c f,,, a, b, c f f f 1, f, f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x y f (x, y ) = a x 2 + b x y + c y 2 = x y a b /2 b /2 c x y a b /2 r s T = t u b /2 c = r t s u a b/2 b/2 c r s SL(Z) ± f(x, y) f (x, y ) f(x, y) f (x, y ) T SL(Z) +, T SL(Z) f(x, y) f (x, y ) f (x, y ) f(x, y) x y = r s t u x y t u (31) a = ar 2 + brt + ct 2 b = 2ars + b(ru + st) + 2ctu (32) c = as 2 + bsu + cu 2

24 3 22, f, f, f,, a b/2 b/2 c, a b /2 b /2 c, a b /2 b /2 c, 31 Proof, 1 0 E = 0 1 a b/2 1 0 a b/2 1 0 b/2 c = 0 1 b/2 c 0 1 f f f f a b /2 b /2 c = T t a b/2 b/2 c T (T t T ) T SL(Z) ± a b/2 b/2 c = (T 1 ) t a b /2 b /2 c T 1 f f f f f f a b /2 b /2 c = T t 1 a b/2 b/2 c T 1, a b /2 b /2 c a = T2 t b /2 b /2 c T 2 T 1, T 2 SL(Z) ± a b /2 b /2 c = (T 1 T 2 ) t a b/2 b/2 c T 1 T 2 f f

25 f f n (i) (ii) f(x, y) = n f (x, y) = n Proof a b /2 r t a b/2 r s b /2 c = s u b/2 c t u r s t u SL(Z) ± a b/2 u t a b /2 u s b/2 c = s r b /2 c t r f(x, y) = n (α, β) n = α = α β β a b/2 α b/2 c β u t a b /2 u s s r b /2 c t r α β α u s α β = t r β f (α, β ) = n f (x, y) = n f (x, y) = n 33 f f f f

26 3 24 Proof a, b, c a b /2 r t a b/2 r s b /2 c = s u b/2 c t u r s t u SL(Z) ± a b/2 u t a b /2 u s b/2 c = s r b /2 c t r, a = a u 2 b ut + c t 2 b = 2a su + b (ru + st) 2c rt c = a s 2 b rs + c r 2 a, b, c a, b, c a, b, c f 34 f, f D, D D = D Proof D = 4 a b /2 b /2 c = 4 r s t u a b/2 b/2 c r t s u 2 r s = 4 t u = 4 a b/2 b/2 c a b/2 b/2 c = D 34 D D D h(d) D,

27 3 25 h + (D) D h + (D) h(d) 32 a( 0), b, c f(x) = ax 2 +bx+c Q f(ξ) = 0 ξ ξ a, b, c f(x) = ax 2 + bx + c x = 0 i i 2 = 1 i = 1 f(x) = ax 2 + bx + c D D > 0 D < 0 ξ = b + D, ξ = b D 2a 2a ξ = b + i D, ξ = b i D 2a 2a, ξ, ξ ξ ξ ξ ξ ax 2 + bx + c D = b 2 4ac ξ aξ 2 + bξ + c = 0 ξ tξ + u = 0 t = u = 0 r s p8 T = SL(Z) ± t u η = T (ξ) = rξ + s tξ + u (33) η ξ T SL(Z) ± (33) η ξ η

28 3 26 Proof aξ 2 + bξ + c = 0 a b/2 ξ ξ 1 b/2 c 1 = aξ 2 + bξ + c = 0 a b/2 r t a b /2 r s b/2 c = s u b /2 c t u a, b, c, tξ + u 0 a η 2 + b η + c a b /2 η = η 1 b /2 c 1 ( ) 2 1 r t a b /2 r s = ξ 1 tξ + u s u b /2 c t u ( ) 2 1 a b/2 ξ = ξ 1 tξ + u b/2 c 1 ( ) 2 1 ( = aξ 2 + bξ + c ) = 0 tξ + u ξ 1 a η 2 + b η + c = 0 a b/2 4 b/2 c = 4 a b /2 b /2 c ξ η a x 2 + b x + c Q η p25 ξ T (33) T (ξ) T, T SL(Z) ± T (T (ξ)) = T T (ξ) 36 T SL(Z) ± (33) ( ) Proof η ξ = T 1 (η) 35 T (ξ) = T T 1 (η) = η

29 3 27 T T (ξ) = T (ξ ) ξ = T 1 (T (ξ)) = T 1 (T (ξ )) = ξ T T SL(Z) ± (33) 1 0 T = ±E E = 0 1 r s Proof T = T (i) = i r = u, t = s T (2i) = 2i t u s = 4t s = t = 0 T = r 2 = ±1 r = u = ±1 T = ±E T T = 1, T = 1 35 T SL(Z) ± (33) SL(Z) ±, ±E 115 SL(Z) ± /{±E} M ± SL(Z) + /{±E}, M + ξ η ξ η ξ η, 37 Proof, ξ E(ξ) = ξ ξ ξ ξ η η = T (ξ) T SL(Z) ± ξ = T 1 (η) η ξ ξ η, η ζ η = T (ξ), ζ = T (η) T, T SL(Z) ± ζ = T T (ξ) ξ ζ

30 f(x, y) = ax 2 + bxy + cy 2 f a 0, ax 2 + bx + c Q f ax 2 + bx + c ξ ξ ax 2 bx c f 1 (x, y) = a 1 x 2 + b 1 xy + c 1 y 2, f 2 (x, y) = a 2 x 2 + b 2 xy + c 2 y 2, ξ 1, ξ 2 38 f 1 (x, y) f 2 (x, y) ξ 1 ξ 2 Proof a2 b 2 /2 r t a1 b 1 /2 r s b 2 /2 c 2 = s u b 1 /2 c 1 t u r s t u SL(Z) + a 1 = a 2 u 2 b 2 ut + c 2 t 2 b 1 = 2a 2 us + b 2 (ru + st) 2c 2 rt c 1 = a 2 s 2 b 2 rs + c 2 r 2 D > 0 b 2 + D r + s rξ 2 + s tξ 2 + u = 2a 2 t b 2 + D + u 2a 2 = (2a 2s b 2 r) + r D (2a 2 u b 2 t) + t D

31 3 29 = (4a2 2su 2a 2 b 2 st 2a 2 b 2 ru + 4a 2 c 2 rt) + 2a 2 (ru st) D 4a 2 2u 2 4a 2 b 2 ut + 4a 2 c 2 t 2 = 2a 2b 1 + 2a 2 D 4a 2 a 1 = b 1 + D 2a 1 = ξ 1 ξ 1 = rξ 2 + s tξ 2 + u ξ 1 ξ 2 D < 0 ξ 2 f rξ 2 + s 2 tξ 2 + u f 1 39 ξ 1 ξ 2 f 1 (x, y) f 2 (x, y) Proof ξ 1 ξ 2 r s t u (tξ 2 + u) 2 ξ 1 = rξ 2 + s tξ 2 + u SL(Z) + a 1 ξ1 2 + b 1 ξ 1 + c 1 = 0 ( ) 2 ( ) rξ2 + s rξ2 + s a 1 + b 1 + c 1 = 0 tξ 2 + u tξ 2 + u (a 1 r 2 + b 1 rt + c 1 t 2 )ξ (2a 1 rs + b 1 (ru + st) + 2c 1 tu)ξ 2 + (a 1 s 2 + b 1 su + c 1 u 2 ) = 0 b 2 1 4a 1 c 1 = b 2 2 4a 2 c 2 ξ 2, ±1 a 1 r 2 + b 1 rt + c 1 t 2 = ±a 2 2a 1 rs + b 1 (ru + st) + 2c 1 tu = ±b 2 a 1 s 2 + b 1 su + c 1 u 2 = ±c 2

32 3 30 ( ) r t s u a1 b 1 /2 b 1 /2 c 1 r s t u a2 b 2 /2 = ± b 2 /2 c 2 f 1 f 2 r t s u a1 b 1 /2 b 1 /2 c 1 r s t u = a2 b 2 /2 b 2 /2 c 2, f 1 f 2 ξ 2 f 2 ξ 1 = rξ 2 + s tξ 2 + u f 1 f 1 f f 1 (x, y) f 2 (x, y), ξ 1 ξ 2 311

33 4 D,,, 41, 42,,,, 43 f(x, y) = ax 2 +bxy+cy 2 D = b 2 4ac < 0 ac > 0 a c ( f(x, y) = ax 2 + bxy + cy 2 = a x + b 2a y ) 2 + ( 4ac b 2 D < 0 f(x, y) (x, y) (0, 0) a > 0 f(x, y) > 0, a < 0 f(x, y) < 0 a > 0 f, a < 0 f 32,, f f f f 31 4a ) y 2

34 4 32 D h + (D) h + (D) = 2 h + (D) 41 f(x, y) = ax 2 + bxy + cy 2 c > a b > a c = a b 0 (41) ( ) 41 D f(x, y) = ax 2 + bxy + cy 2 D b 3 Proof b a c b 2 ac D = b 2 4ac < 0 D = b 2 4ac = 4ac b 2 4b 2 b 2 = 3b 2 D b 3 b, 4ac = b 2 D a, c D 42 f(x, y) = ax 2 + bxy + cy (i) f(x, y) = x 2 + y 2 ±, ± (ii) (iii) 1 0 f(x, y) = x 2 + xy + y 2 ± f(x, y) ± 0 1, ± , ±

35 4 33 r s Proof X = f(x, y) t u r t a b/2 r s a b/2 s u b/2 c t u = b/2 c a b/2 r s u t a b/2 b/2 c t u = s r b/2 c a(r u) + bt = 0 as + ct = 0 c(r u) bs = 0 (42) s = 0 t = 0 as + ct = 0 a, c > 0 s = t = ru st = ru = 1 X = ± 0 1 s 0 t 0 as + ct = 0 a, c > 0 st < 0 (i) (ii) r = u st 1 ru st = r 2 st = 1 r = u = 0 st = X = ± (42) b = a = c a, b, c a = c = 1, b = 0 f(x, y) = x 2 + y 2 r u b = 0 a(r u) + bt = 0 r = u b 0 r = 0 bt = au, bs = cu, st = 1 a, b, c a = c = b = 1, u = ±1 X = 0 1 ±, f(x, y) = x 2 + xy + y u = 0 X = ±, f(x, y) = x 2 + xy + y 2 1 0

36 4 34 r 0 u 0 st < 0 ru st = 1 ru < 0 b 0 t = a ( r + u ) r + u, b s = c ( r + u ) r + u b st r r u + u 2 > ur + 2, ru st = 1 ru = st 1 r, s, t, u ξ (i) ξ = i ±, ± (ii) (iii) ξ = 1 + 3i ± ± 0 1, ± , ± f ξ = X + Y i f 1 2 X < 1, ξ > X 0, ξ = 1 Proof f(x, y) = ax 2 + bxy + cy 2 ξ = X Y i 2X = b a, ξ 2 = X 2 + Y 2 = c a

37 4 35 f (41) c > a b > a c = a b 0,, 1 2 X < 1 2, ξ > X 0, ξ = 1 ξ = X +Y i 1 2 X < 1 2, ξ > 1 1 X 2 0, ξ = 1 i 1+ 3i 2, Proof f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x y a > c a b/2 b/2 c = c b/2 b/2 a ( ) a c b > a a b + 2na > a n 1 0 n 1 a b/2 b/2 c 1 n 0 1 = a (b + 2na)/2 (b + 2na)/2 c + nb + n 2 a ( )

38 4 36 a c + nb + n 2 a a > c + nb + n 2 a ( ) a b /2 b /2 c, a = c + nb + n 2 a < c = a, b = (b + 2na), ( ) a b /2 b /2 c, a = a, a b > a,, (1, 1) a 46 f(x, y) = ax 2 + bxy + cy 2 0 < f(x, y) a (x, y) (i) (ii) (iii) c > a (x, y) = (±1, 0) c = a > b (x, y) = (±1, 0), (0, ±1) c = a = b (x, y) = (±1, 0), (0, ±1), ±(1, 1) f(x, y) 0 a Proof 0 < f(x, y) a ( f(x, y) = ax 2 + bxy + cy 2 = a x + b 2a y ) 2 + ( 4ac b 2 4a ) y 2 a 4ac 4a 2, b 2 a 2 y 2 y = 0, ±1 ( ) 4a a 4a2 4ac b 2 3a y = 0 x = ±1 0 < f(x, y) a (±1, 0) y = ±1 f(x, ±1) = ax 2 ± bx + c a b ax 2 ± bx 0 f(x, ±1) c c > a f(x, ±1) > a (±1, 0)

39 4 37 c = a ax 2 ± bx = 0 f(x, ±1) = a ax 2 ± bx = 0 a > b x = 0, (0, ±1) a = b x = ±1 ax 2 ± bx = 0 ±(1, 1) 47 Proof f(x, y) = ax 2 + bxy + cy 2 f (x, y) = a x 2 + b xy + c y 2 f = f 46 f, f 0 a, a a = a r t s u a b/2 b/2 c r s t u = a b /2 b /2 c r s (1, 1) f(r, t) = ar 2 + t u brt + ct 2 = a c > a 46 r = ±1, t = 0 b = ±2sa + b a b, b > a b = b c = c f = f c = a b c > a = a f = f c = a = a b 2 = D + 4ac = (b ) 2 c = a b, b 0 b = b f = f 45, h + (D) D 49 D < 0 h + (D)

40 4 38 ξ, D = 20 f(x, y) = ax 2 + bxy + cy 2 41 b 20 = 2 3 b b = 0, ±1, ±2, 4ac = b 2 D 4ac = 20, 21, 24 ac = 5, 6 (41) (a, b, c) (a, b, c) = (1, 0, 5), (2, 2, 3) D = 20 f 1 (x, y) = x 2 + 5y 2, f 2 (x, y) = 2x 2 + 2xy + 3y 2 h + ( 20) = 2 D = 36 f(x, y) = ax 2 + bxy + cy 2 41 b 36 = 3 3

41 4 39 b b = 0, ±1, ±2, ±3 4ac = b 2 D 4ac = 36, 37, 40, 45 ac = 9, 10 (41) (a, b, c) (a, b, c) = (1, 0, 9), (3, 0, 3), (2, 2, 5) D = 36 f 1 (x, y) = x 2 + 9y 2 f 3 (x, y) = 2x 2 + 2xy + 5y 2 h + ( 36) = 3 f 2 (x, y) = 3x 2 + 3y 2 D = 47 f(x, y) = ax 2 + bxy + cy 2 41 b 47 = 3 3 b b = 0, ±1, ±2, ±3 4ac = b 2 D 4ac = 47, 48, 51, 56 ac = 12, 14 (41) (a, b, c) (a, b, c) = (1, 1, 12), (2, 1, 6), (2, 1, 6), (3, 1, 4), (3, 1, 4) D = 47 f 1 (x, y) = x 2 + xy + 12y 2 f 2 (x, y) = 2x 2 + xy + 6y 2 f 3 (x, y) = 2x 2 xy + 6y 2 f 4 (x, y) = 3x 2 + xy + 4y 2 f 5 (x, y) = 3x 2 xy + 4y 2 h + ( 47) = 5

42 4 40 D < 0 D h + (D) D D h + (D) 3 f(x, y) = x 2 + xy + y f(x, y) = x 2 + y f(x, y) = x 2 + xy + 2y f(x, y) = x 2 + 2y f(x, y) = x 2 + xy + 3y f 1 (x, y) = x 2 + 3y 2, f 2 (x, y) = 2x 2 + 2xy + 2y 2 2 f 1 (x, y) = x 2 + xy + 4y 2, f 2 (x, y) = 2x 2 + xy + 2y 2 2 f 1 (x, y) = x 2 + 4y 2, f 2 (x, y) = 2x 2 + 2y f(x, y) = x 2 + xy + 5y f 1 (x, y) = x 2 + 5y 2, f 2 (x, y) = 2x 2 + 2xy + 3y 2 2 f 1 (x, y) = x 2 + xy + 6y 2, f 2 (x, y) = 2x 2 + xy + 3y 2 3 f 3 (x, y) = 2x 2 xy + 3y 2 f 1 (x, y) = x 2 + 6y 2, f 2 (x, y) = 2x 2 + 3y 2 2 f 1 (x, y) = x 2 + xy + 7y 2, f 2 (x, y) = 3x 2 + 3xy + 3y 2 2 f 1 (x, y) = x 2 + 7y 2, f 2 (x, y) = 2x 2 + 2xy + 4y 2 2 f 1 (x, y) = 2x 2 + xy + 4y 2, f 2 (x, y) = 2x 2 xy + 4y 2 3 f 3 (x, y) = x 2 + xy + 8y 2 f 1 (x, y) = x 2 + 8y 2, f 2 (x, y) = 3x 2 + 2xy + 3y 2 3 f 3 (x, y) = 2x 2 + 4y 2 f 1 (x, y) = x 2 + xy + 9y 2, f 2 (x, y) = 3x 2 + xy + 3y 2 2 f 1 (x, y) = x 2 + 9y 2, f 2 (x, y) = 3x 2 + 3y 2 3 f 3 (x, y) = 2x 2 + 2xy + 5y 2 f 1 (x, y) = x 2 + xy + 10y 2, f 2 (x, y) = 2x 2 + xy + 5y 2 4 f 3 (x, y) = 2x 2 xy + 5y 2, f 4 (x, y) = 3x 2 + 3xy + 4y 2 f 1 (x, y) = x y 2, f 2 (x, y) = 2x 2 + 5y f(x, y) = x 2 + xy + 11y f 1 (x, y) = x y 2, f 2 (x, y) = 2x 2 + 2xy + 6y 2 4 f 3 (x, y) = 3x 2 + 2xy + 4y 2, f 4 (x, y) = 3x 2 2xy + 4y 2 f 1 (x, y) = x 2 + xy + 12y 2, f 2 (x, y) = 2x 2 + xy + 6y 2 5 f 3 (x, y) = 2x 2 xy + 6y 2, f 4 (x, y) = 3x 2 + xy + 4y 2 f 5 (x, y) = 3x 2 xy + 4y 2

43 5 D, D,, 51,, D 52,,, m ξ ξ = k 0, k 1,, k n 1, ξ n (n = 1, 2,, m 1) ξ ξ 0 = ξ, ξ 1, ξ 2,,ξ m 1 51 f(x, y) = ax 2 + bxy + cy 2, a > 0, b < 0, c < 0, a + b + c < 0, a b + c > 0 (51) 41

44 5 42 f(x, y) g(x) = ax 2 + bx + c g(1) = a + b + c, g(0) = c, g( 1) = a b + c (51) a > 0, g( 1) > 0, g(0) < 0, g(1) < 0 (52) (52) a > 0, c = g(0) < 0, a + b + c = g(1) < 0, a b + c = g( 1) > 0 2b = g(1) g( 1) < 0 b < 0 (51) (51) (52) g(x) = ax 2 + bx + c ξ, ξ ξ = b + D, ξ = b D 2a 2a (52) ξ > 1 0 > ξ > 1 ξ > 1 > ξ > 0 (53) D (53) a = ξ ξ > 0 a > 0 g( 1) > 0, g(0) < 0, g(1) < 0 51 f(x, y) = ax 2 + bxy + cy 2 b + D 2a > 1 > b + D 2a g(x) = ax 2 + bx + c 1 ξ, ξ > 0 ξ > 1 > ξ > 0 ξ f(x, y) ξ f(x, y), ξ

45 D Proof f(x, y) = ax 2 + bxy + cy 2 D 51 b + D > 0 D > b = b b, 4ac = b 2 D a, c D (p26), 36 ( ) ω p16, (26), (27) ω = k 0, k 1,, k n 1, ω n = p nω n + p n 1 q n ω n + q n 1, p n q n 1 p n 1 q n = ( 1) n ω n 54 ω ω = k 0, k 1,, k n 1, ω n n ω ω n, n ω ω n

46 ξ η η = T (ξ) = rξ + s r s tξ + u, T = t u SL(Z) ±, t > u > 0, ξ > 1 T n η = k 0, k 1,, k n 1, ξ, r = p n, s = p n 1, t = q n, u = q n 1 T = 1 n, T = 1 n Proof r t r t = k 0, k 1,, k n 1 t > u > 0 t > 1 ru st = ±1 (r, t) = 1 r 0 n 2 21 T j = kj 1 1 0, T 0 T 1 T j 2 T j 1 = pj p j 1 q j q j 1 r t = k 0, k 1,, k n 1 = p n 1k n 1 + p n 2 = p n q n 1 k n 1 + q n 2 q n r (p n, q n ) = 1 q n > 0 t = p n r = p n, t = q n q n, 23 n T n ru st = p n q n 1 p n 1 q n = ( 1) n (54) ru st = p n u q n s = p n q n 1 p n 1 q n p n (u q n 1 ) = q n (s p n 1 ) q n u q n 1 q n = t > u > 0, n 2

47 5 45 q n q n 1 > 0 u q n 1 < q n u = q n 1, s = p n 1 k 0, k 1,, k n 1, ξ = p nξ + p n 1 q n ξ + q n 1 = rξ + s tξ + u = η k 0, k 1,, k n 1, ξ η q j j t = q n n T = 1 n, T = 1 n (54) η = , ξ = η = 1, ξ = 1, 3, 1, 1, ξ η ξ 4 2, 55 n η ξ 56 ξ, η ξ, η ξ = k 0, k 1,, k l 1, ξ l, η = h 0, h 1,, h m 1, η m ξ l = η m ξ η l, m, ξ η l, m, Proof ξ η η = T (ξ) = rξ + s r s tξ + u, T = t u SL(Z) ± T T T tξ + u > 0 T = T T ξ ξ = k 0, k 1,, k n 1, ξ n = p nξ n + p n 1 q n ξ n + q n 1 η = T (ξ) = r s t u pn p n 1 q n q n 1 (ξ n ) An B n C n D n = r s t u pn p n 1 q n q n 1

48 5 46 An B n An B n η = A nξ n + B n C n ξ n + D n = C n D n (ξ n ), C n D n SL(Z) ± 25 ξ p n q n < 1 q n q n+1 ξq n p n < q n q n+1 q n, q n q n+1 < 1 p n = ξq n + δ n, δ n < 1 q n C n = tp n + uq n = t(ξq n + δ n q n ) + uq n = (tξ + u)q n + tδ n q n, D n = tp n 1 + uq n 1 = (tξ + u)q n 1 + tδ n 1 q n 1, tξ + u > 0, lim n q n = + l q l > q l 1 > 0 tδ l 1 tδ l q l 1 q l < tξ + u C l > D l > 0 η = A lξ l + B l C l ξ l + D l, (C l > D l > 0, ξ l > 1) 55 m η = h 0, h 1,, h m 1, ξ l ξ η l ξ ξ l η ξ l 55 m l ξ ξ l η ξ l m ξ η l ξ ξ l, η ξ l m l ξ ξ l, η ξ l m

49 5 47 ω = k 0, k 1,, k n, m, l n > l k n+m = k n, m ω ω = k 0, k 1,, k n 1, ω n n > l ω n+m = ω n m ω = k 0, k 1,, k l, h 1, h 2,, h m, h 1, h 2,, h m, h 1, h 2, m ω = k 0, k 1,, k l, h 1, h 2,, h m ω = k 0, k 1,, k l, 57 ω ω Proof ω, m, l ω = k 0, k 1,, k l, h 1,, h m η = h 1, h 2,, h m 54 ω η η = h 1,, h m, η (26) η = rη + s tη + u, r s t u SL(Z) ±, t > 0

50 5 48 tη 2 + (u r)η s = 0, t > 0 η η ω 58 2 ω Proof ω aω 2 + bω + c = ω, 1 a b/2 ω b/2 c 1 = 0 a 0, b, c ω (26) ω = k 0, k 1,, k n 1, ω n = p nω n + p n 1 q n ω n + q n 1 aω 2 + bω + c = 0, (q n ω n + q n 1 ) 2 a b/2 pn ω n + p n 1 0 = p n ω n + p n 1, q n ω n + q n 1 b/2 c q n ω n + q n 1 p n q n a b/2 pn p n 1 ωn = ω n, 1 p n 1 q n 1 b/2 c q n q n 1 1 A n B n /2 ωn = ω n, 1 B n /2 1 C n A n = ap n 2 + bp n q n + cq n 2, B n = 2ap n p n 1 + b(p n q n 1 + p n 1 q n ) + 2cq n q n 1, C n = ap n bp n 1 q n 1 + cq n 1 2, B n 2 4A n C n = b 2 4ac, 25 ω p n q n < 1 qn 2 p n = ωq n + δ n q n, ( δ n < 1)

51 5 49 ( A n = a ωq n + δ ) 2 ( n + bq n ωq n + δ ) n 2 + cq n q n q n = (aω 2 + bω + c)q n 2 + (2aω + b)δ n + a aω 2 + bω + c = 0 A n = (2aω + b)δ n + a ( δn 2 q n 2 ) ( δn 2 q n 2 ) A n < 2 aω + b + a C n < 2 aω + a + b, B n 2 = 4A n C n + (b 2 4ac) 4(2 aω + a + b ) 2 + b 2 4ac A n, B n, C n n (A p, B p, C p ) = (A q, B q, C q ) = (A r, B r, C r ) p, q, r ω p, ω q, ω r A p x 2 + B p x + C p = 0, ω p = ω q, p < q ω ω = k 0, k 1,, k p 1, ω p = k 0, k 1,, k p 1,, k q 1, ω q ω 59 ω Proof ω 57 ω, n ω = k 0, k 1,, k n 1, ω

52 5 50 (26) ω = p nω + p n 1 q n ω + q n 1 q n ω 2 + (q n 1 p n )ω p n 1 = 0 g(x) = q n x 2 + (q n 1 p n )x p n 1 = 0 p42, (52) n 1 q n > 0, g(0) = p n 1 < 0 q n q n 1 0, p n > p n 1 1 g( 1) = (q n q n 1 ) + (p n p n 1 ) > 0 p 1 = k 0 q 1 = 1 p n+1 = p n k n + p n 1, q n+1 = q n k n + q n 1 n 1 p n > q n > 0 ( g(1) = q n + q n 1 p n p n 1 = q n 1 p ) ( n + q n 1 1 p ) n 1 < 0 q n q n 1 ω 510 ω ω = k 0, k 1,, k n 1, ω n (n = 1, 2, ) ω 1, ω 2, 2 Proof ω g(x) g(x) = ax 2 + bx + c, g(ω) = 0, a > 0, g( 1) > 0, g(0) < 0, g(1) < 0 ω = k g(ω) = 0 ω 2 1 ω 1 (ak bk 0 + c)ω (2ak 0 + b)ω 1 + a = 0 A = (ak bk 0 + c), B = (2ak 0 + b), C = a, h(x) = Ax 2 + Bx + C h(ω 1 ) = 0 1 k 0 = ω 0 k 0 1 < k 0 < ω < k 0 + 1

53 5 51 g(k 0 1) < 0, g(k 0 ) < 0, g(k 0 + 1) > 0 A = g(k 0 ) > 0 h(1) = g(k 0 + 1) < 0, h( 1) = g(k 0 1) > 0, h(0) = a < 0 h ω 1 ω 2, ω 3, 511 ω Proof ω ω > 1 k 0 1 ω ω = k 0, k 1,, k n 1, ω n ω 58, m, n(m > n) ω m = ω n n = 0 ω n > 0 ω n 1 = k n ω n, ω m 1 = k m ω m ω n 1 ω m 1 = k n 1 k m 1 Z ω j ω j ω n 1 = b n 1 + D, ω m 1 = b m 1 + D 2a n 1 2a m 1, ω n 1 ω m 1 Z ω n 1 ω m 1 D 0 D 2a n 1 = ω n 1, ω m 1 D 2a m 1 ω n 1 = b n 1 D, ω m 1 = b m 1 D 2a n 1 2a m 1 ω n 1 ω m 1 = ω n 1 ω m 1 = k n 1 k m 1 Z

54 ω n 1, ω m 1 1 < ω n 1 < 0, 1 < ω m 1 < 0 ω n 1 ω m 1 < 1, ω n 1 ω m 1 ω n 1 ω m 1 = 0 k n 1 = k m 1, ω n 1 = ω m 1 ω m = ω n ω n 1 = ω m 1 ω 0 = ω m n ω 53 D h(d), 512 Proof ξ ξ ξ = k 0, k 1,, k n 1, ξ n 58, m, n(m > n) ξ n = ξ m ξ = k 0, k 1,, k n 1, ξ n = k 0, k 1,, k m 1, ξ m = k 0, k 1,, k m 1, ξ n ξ n = k n,, k m 1, ξ n ξ n 59 ξ n n 510 ξ n+1 n n + 1, n ξ = p nξ n + p n 1 q n ξ n + q n 1, p n q n 1 p n 1 q n = ( 1) n = 1 ξ ξ n

55 D h(d), h + (D) 514 ξ, ξ = k 0, k 1,, k n 1, ξ n ξ m ξ 0 = ξ, ξ 1,, ξ m 1, m ξ 0, ξ 1,, ξ m 1 m ξ 0, ξ 2,, ξ m 2, ξ ξ 1, ξ 3,, ξ m 1 Proof ξ 0,, ξ m 1 54, 510 m ξ ξ 0, ξ 1,, ξ m 1 54 ξ ξ 2n m ξ n = ξ m+n n m + n ξ ξ m+n = ξ n m ξ 0, ξ 1,, ξ m 1 m 54 ξ 0, ξ 2,, ξ m 2, ξ ξ 1, ξ 3,, ξ m 1 n ξ ξ n 56, r s ξ r = ξ n+s n + s r, m n ξ ξ n 515 ξ, ξ = k 0, k 1,, k n 1, ξ n, m ξ ξ 0 = ξ, ξ 1,, ξ m 1 Proof η ξ, η = u 0, u 1,, u n 1, η n, 56 ξ r = η s r, s η η s = u j, u j+1,, η = ξ r η s η ξ 0 = ξ, ξ 1,, ξ m 1 ξ ξ 0 = ξ, ξ 1,, ξ m 1 η s

56 D = 60 f(x, y) = ax 2 + bxy + cy = 7 53 b = 1, 2, 3,, 7 4ac = b 2 D b, 4ac = 56, 44, 24 ac = 14, 11, 6 p41 (51) (a, b, c) (6, 6, 1), (3, 6, 2), (2, 6, 3), (1, 6, 6) 4 f(x) f 1 (x, y) = 6x 2 6xy y 2, f 2 (x, y) = 3x 2 6xy 2y 2, f 3 (x, y) = 2x 2 6xy 3y 2, f 4 (x, y) = x 2 6xy 6y 2 ξ 1 = , ξ 2 = ,, ξ 3 = , ξ 4 = ξ 1 = 1, 6, ξ 2 = 2, 3, ξ 3 = 3, 2, ξ 4 = 6, ξ 1 ξ 4, ξ 2 ξ 3, ξ 1 ξ 2 h(d) = h + (D) = 4 D = 136 f(x, y) = ax 2 +bxy +cy = 11 b = 1, 2, 3,, 11 4ac = b 2 D 4ac = 132, 120, 100, 72, 36 ac = 33, 30, 25, 18, 9 (51) (a, b, c) 10 (5, 6, 5), (5, 4, 6), (6, 4, 5), (2, 8, 9), (3, 8, 6), (6, 8, 3), (9, 8, 2), (1, 10, 9), (3, 10, 3), (9, 10, 1)

57 5 55 f(x, y) = ax 2 + bxy + cy 2 f 1 (x, y) = 5x 2 6xy 5y 2, f 2 (x, y) = 5x 2 4xy 6y 2, f 3 (x, y) = 6x 2 4xy 5y 2, f 4 (x, y) = 2x 2 8xy 9y 2, f 5 (x, y) = 3x 2 8xy 6y 2, f 6 (x, y) = 6x 2 8xy 3y 2, f 7 (x, y) = 9x 2 8xy 2y 2, f 8 (x, y) = x 2 10xy 9y 2, f 9 (x, y) = 3x 2 10xy 3y 2, f 10 (x, y) = 9x 2 10xy y 2 ξ 1 = ξ 5 = , ξ 2 = , ξ 6 = ξ 9 = , ξ 10 = ,, ξ 3 = , ξ 4 = , 2, ξ 7 = , ξ 8 = , 9 ξ 1 = 1, 1, 3, 3, 1, 1, ξ 2 = 1, 1, 1, 3, 3, 1, ξ 3 = 1, 3, 3, 1, 1, 1, ξ 4 = 4, 1, 10, 1, ξ 5 = 3, 3, 1, 1, 1, 1, ξ 6 = 1, 1, 1, 1, 3, 3, ξ 7 = 1, 10, 1, 4, ξ 8 = 10, 1, 4, 1, ξ 9 = 3, 1, 1, 1, 1, 3, ξ 10 = 1, 4, 1, 10 ξ 1 ξ 2 ξ 3 ξ 5 ξ 6 ξ 9, ξ 4 ξ 7 ξ 8 ξ 10, ξ 1 ξ 4 h(d) = 2 h + (D) = 4

58 5 56 D, D, D D h(d) h + h(d) h + (D) 5 f(x, y) = x 2 xy y 2 ξ = ξ = f(x, y) = x 2 2xy y 2 ξ = ξ = f 1 (x, y) = 2x 2 2xy y 2 ξ 1 = ξ 1 = 1, f 2 (x, y) = x 2 2xy 2y 2 ξ 2 = ξ 2 = 2, 1 13 f(x, y) = x 2 3xy y 2 ξ = ξ = f 1 (x, y) = x 2 3xy 2y 2 ξ 1 = ξ 1 = 3, 1, f 2 (x, y) = 2x 2 xy 2y 2 ξ 2 = ξ 2 = 1, 3, 1 f 3 (x, y) = 2x 2 3xy y 2 ξ 3 = ξ 3 = 1, 1, 3 20 f 1 (x, y) = 2x 2 2xy 2y 2 ξ 1 = ξ 1 = f 2 (x, y) = x 2 4xy y 2 ξ 2 = ξ 2 = 4 21 f 1 (x, y) = x 2 3xy 3y 2 ξ 1 = ξ 1 = 3, f 2 (x, y) = 3x 2 3xy y 2 ξ 2 = ξ 2 = 1, 3 24 f 1 (x, y) = x 2 4xy 2y 2 ξ 1 = ξ 1 = 4, f 2 (x, y) = 2x 2 4xy y 2 ξ 2 = ξ 2 = 2, 4 28 f 1 (x, y) = x 2 4xy 3y 2 ξ 1 = ξ 1 = 4, 1, 1, f 2 (x, y) = 2x 2 2xy 3y 2 ξ 2 = ξ 2 = 1, 1, 4, 1 f 3 (x, y) = 3x 2 2xy 2y 2 ξ 3 = ξ 3 = 1, 4, 1, 1 f 4 (x, y) = 3x 2 4xy y 2 ξ 4 = ξ 4 = 1, 1, 1, 4 29 f(x, y) = x 2 5xy y 2 ξ = ξ = f 1 (x, y) = x 2 4xy 4y 2 ξ 1 = ξ 1 = 4, f 2 (x, y) = 2x 2 4xy 2y 2 ξ 2 = ξ 2 = 2 f 3 (x, y) = 4x 2 4xy y 2 ξ 3 = ξ 3 = 1, 4 33 f 1 (x, y) = 2x 2 3xy 3y 2 ξ 1 = ξ 1 = 2, 5, 2, f 2 (x, y) = 3x 2 3xy 2y 2 ξ 2 = ξ 2 = 1, 2, 5, 2 f 3 (x, y) = x 2 5xy 2y 2 ξ 3 = ξ 3 = 5, 2, 1, 2 f 4 (x, y) = 2x 2 5xy y 2 ξ 4 = ξ 4 = 2, 1, 2, 5 37 f 1 (x, y) = 3x 2 xy 3y 2 ξ 1 = ξ 1 = 1, 5, f 2 (x, y) = x 2 5xy 3y 2 ξ 2 = ξ 2 = 5, 1, 1 f 3 (x, y) = 3x 2 5xy y 2 ξ 3 = ξ 3 = 1, 1, 5 40 f 1 (x, y) = 3x 2 2xy 3y 2 ξ 1 = ξ 1 = 1, 2, f 2 (x, y) = 2x 2 4xy 3y 2 ξ 2 = ξ 2 = 2, 1, 1 f 3 (x, y) = 3x 2 4xy 2y 2 ξ 3 = ξ 3 = 1, 1, 2 f 4 (x, y) = x 2 6xy y 2 ξ 4 = ξ 4 = 6

59 6 f(x, y) = ax 2 + bxy + cy 2 n f(x, y) = n x, y n f n,, n f 0, f n f n, 61 f(x, y) = ax 2 + bxy + cy 2 D, n (r, t) f(x, y) = n f(r, t) = n r, t n f n f f(x, y) = x 2 + y = 8, 8 f, (α, β) f(x, y) = n, d, d 2 n α = α d, β = β d, n = n d 2 (α, β ) f(x, y) = n n, 1 f(x, y) = n 61 m 2 D (mod 4n), 0 m < 2n m, n D 57

60 6 58 Proof l = m2 D 4n, g(x, y) = nx 2 + mxy + ly 2 g m 2 4nl = D g(1, 0) = n, n D g 62 n, D m 2 D (mod 4n), 0 m < 2n m Proof 61 n D f(x, y) = ax 2 + bxy + cy 2 f(x, y) = n (r, t) r, t 1 r s x = s, y = u T = t u n m/2 m/2 l = T t a b/2 b/2 c ry tx = 1 (61) T, T x f(x, y) g(x, y ) = nx 2 + mx y + ly 2 y = r s t u x y n = ar 2 + brt + ct 2 m = 2ars + b(ru + st) + 2ctu l = as 2 + bsu + cu 2 (61) (s 0, u 0 ) s = s 0 + rk, u = u 0 + tk (k Z)

61 6 59 m = 2ars + b(ru + st) + 2ctu m = 2ar(s 0 + rk) + b(ru 0 + rtk + s 0 t + rtk) + 2ct(u 0 + tk) = 2ars 0 + b(ru 0 + s 0 t) + 2ctu 0 + 2nk m 2ars 0 + b(ru 0 + s 0 t) + 2ctu 0 (mod 2n) k m 0 m < 2n g f D = b 2 4ac = m 2 4nl m 2 D (mod 4n), 0 m < 2n m m 2 D (mod 4n) 0 m < 2n m, f(x, y) = ax 2 + bxy + cy 2 n 62 f n f(x, y) g(x, y ) = nx 2 + mx y + ly 2 D = b 2 4ac = m 2 4nl m 2 D (mod 4n), 0 m < 2n (62) (62) m, l, n m/2 m/2 l = r t s u a b/2 b/2 c r s t u

62 6 60 r s T = t u m, l T n f (r, t) f(x, y) = n D h + (D), f g m 2 D (mod 4n), (0 m < 2n) m n f, n f, 42 T, f(x, y) = n f(x, y) = n, 62 p ( ) { 1 1, p 1 (mod 4) = p 1, p 3 (mod 4) ( ) { 2 1, p 1, 3 (mod 8) = p 1, p 5, 7 (mod 8) ( ) { 3 1, p 1, 11 (mod 12) = p 1, p 5, 7 (mod 12) ( ) { 3 1, p 1 (mod 3) = p 1, p 2 (mod 3) ( ) { 5 1, p 1, 4 (mod 5) = p 1, p 2, 3 (mod 5) ( ) { 5 1, p 1, 3, 7, 9 (mod 20) = p 1, p 11, 13, 17, 19 (mod 20)

63 6 61 D = 3 p40 h + ( 3) = 1 3 f(x, y) = x 2 + xy + y 2 p 3 62 m 2 3 (mod 4p), 0 m < 2p m x 2 3 (mod p) x 2 3 (mod p) α (0 α < p), m α α + p m 2 3 (mod 4p), 0 m < 2p x 2 3 (mod p), p = 3 p 1 (mod 3) 100 3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97 f(x, y) = x 2 + xy + y 2 3 = = f(1, 1) 7 = = f(2, 1) 13 = = f(3, 1) 19 = = f(3, 2) 31 = = f(5, 1) 37 = = f(4, 3) 43 = = f(6, 1) 61 = = f(5, 4) 67 = = f(7, 2) 73 = = f(8, 1) 79 = = f(7, 3) 97 = = f(8, 3) D = 4 p40 h + ( 4) = 1 4 f(x, y) = x 2 + y 2

64 6 62 p 4 62 m 2 4 (mod 4p), 0 m < 2p m m m = 2m 0 m (mod p), 0 m 0 < p m 0 m = 2m 0 m 2 4 (mod 4p), 0 m < 2p 4 p x 2 1 (mod p) ( ) 1, = 1 p 1 (mod 4) p 100 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97 f(x, y) = x 2 + y 2 5 = = f(1, 2) 13 = = f(2, 3) 17 = = f(1, 4) 29 = = f(2, 5) 37 = = f(1, 6) 41 = = f(4, 5) 53 = = f(2, 7) 61 = = f(5, 6) 73 = = f(3, 8) 89 = = f(5, 8) 97 = = f(4, 9) D = 20 p40 h + ( 20) = 2 20 f 1 (x, y) = x 2 + 5y 2, f 2 (x, y) = 2x 2 + 2xy + 3y 2

65 6 63, 1 f 1, f 2 f 1 f 2 p 20 m 2 20 (mod 4p), 0 m < 2p m m m = 2m 0 m (mod p), 0 m 0 < p m 0 m = 2m 0 m 2 20 (mod 4p), 0 m < 2p 20 p p = 5 p 1, 3, 7, 9 (mod 20) 100 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89 f 1, f 2 ( p f 1 (x, y) = x 2 + 5y 2 = p x 2 p (mod 5) p = 5 = 1 5) p = 5 p 1, 9 (mod 20) f 2 (x, y) = 2x 2 + 2xy + 3y 2 = p f 2 (x, y) 5 p 5 4x 2 + 4xy + ( 6y 2 ) = 2p (2x ( + y) ) 2 + 5y 2 = 2p, (2x + y) 2 2p (mod 5) 2p 2 ( p = 1 = 1 = ) p 3, 7 (mod 20)

66 f 1 (x, y) f 2 (x, y) f 1 (x, y) = x 2 + 5y 2 f 2 (x, y) = 2x 2 + 2xy + 3y 2 5 = = f 1 (0, 1) 3 = ( 1) + 3 ( 1) 2 = f 2 (1, 1) 29 = = f 1 (3, 2) 7 = = f 2 (1, 1) 41 = = f 1 (6, 1) 23 = ( 3) + 3 ( 3) 2 = f 2 (2, 3) 61 = = f 1 (4, 3) 43 = ( 1) + 3 ( 1) 2 = f 2 (5, 1) 89 = = f 1 (3, 4) 47 = ( 3) + 3 ( 3) 2 = f 2 (5, 3) 67 = ( 5) + 3 ( 5) 2 = f 2 (1, 5) 83 = ( 3) + 3 ( 3) 2 = f 2 (7, 3) D = 24 p40 h + ( 24) = 2 20 f 1 (x, y) = x 2 + 6y 2, f 2 (x, y) = 2x 2 + 3y 2, 1 f 1, f 2 f 1 f 2 p 24 f(x, y) m 2 24 (mod 4p), 0 m < 2p m m m = 2m 0 m (mod p), 0 m 0 < p m 0 m = 2m 0 m 2 24 (mod 4p), 0 m < 2p 24 p p = 3 ( ) 6 = 1 p

67 6 65, p = 3 p 1, 5, 7, 11 (mod 24) , 5, 7, 11, 29, 31, 53, 59, 73, 79, 83, 97 f 1, f 2 f 1 (x, y) = x 2 + 6y 2 = p p 3 x 2 p (mod 3), ( p = 1 p 1, 7 (mod 24) 3) f 2 (x, y) = 2x 2 + 3y 2 = p 2x 2 p (mod 3) p = 3 ( ( ) p 2x 2 = = 3) 3 ( ) 2 = 1 3 p = 3 p 5, 11 (mod 24) 100 f 1 (x, y) f 2 (x, y) f 1 (x, y) = x 2 + 6y 2 f 2 (x, y) = 2x 2 + 3y 2 7 = = f 1 (1, 1) 3 = = f 2 (0, 1) 31 = = f 1 (5, 1) 5 = = f 2 (1, 1) 73 = = f 1 (7, 2) 11 = = f 2 (2, 1) 79 = = f 1 (5, 3) 29 = = f 2 (1, 3) 97 = = f 1 (1, 4) 53 = = f 2 (5, 1) 59 = = f 2 (4, 3) 83 = = f 2 (2, 5) D = 5 p56 h(5) = h + (5) = 1, f(x, y) = x 2 xy y 2 p 5 m 2 5 (mod 4p), 0 m < 2p

68 6 66 m x 2 5 (mod p) 0 x < p α 2 5 (mod p), 0 α < p α α α + p m m 2 5 (mod 4p), 0 m < 2p 5 p = 5 p 1, 4 (mod 5) 100 5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89 f(x, y) = x 2 xy y 2 5 = ( 1) ( 1) 2 = f(2, 1) 11 = ( 1) ( 1) 2 = f(3, 1) 19 = = f(5, 1) 29 = ( 1) ( 1) 2 = f(5, 1) 31 = ( 2) ( 2) 2 = f(5, 2) 41 = ( 1) ( 1) 2 = f(6, 1) 59 = ( 2) ( 2) 2 = f(7, 2) 61 = ( 3) ( 3) 2 = f(7, 3) 71 = = f(9, 1) 79 = ( 3) 3 2 = f(8, 3) 89 = = f(10, 1) D = 12 p56 h(12) = 1, h + (12) = 2, f 1 (x, y) = 2x 2 2xy y 2, f 2 (x, y) = x 2 2xy 2y 2 p 12 m 2 12 (mod 4p), 0 m < 2p

69 6 67 m m, m = 2m 0 m 0 m (mod p), 0 m 0 < p m 0 m = 2m 0 m 2 12 (mod 4p), 0 m < 2p 12 p x 2 3 (mod p), p = 3 p 1, 11 (mod 12) 100 3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97 f 1, f 2 f 1 (x, y) = 2x 2 2xy y 2 = p 4x 2 4xy 2y 2 = 2p (2x y) ( ) 2 3y 2 = 2p 2p (2x y) 2 2p (mod 3) p = 3 = 1 ( ) ( ) 3 2p 2 ( p = 1 = 1 = ) p = 3 p 2 (mod 3) f 2 (x, y) = x 2 2xy 2y 2 = p 3 f 2 ( ) p 3 (x y) 2 3y 2 = p (x y) 2 p (mod 3) p 1 (mod 3) 100 f 1 (x, y) f 2 (x, y) f 1 (x, y) = 2x 2 2xy y 2 f 2 (x, y) = x 2 2xy 2y 2 3 = = f 1 (2, 1) 13 = = f 2 (5, 1) 11 = ( 1) ( 1) 2 = f 1 (2, 1) 37 = ( 3) 2( 3) 2 = f 2 (5, 3) 23 = ( 1) ( 1) 2 = f 1 (3, 1) 61 = ( 1) 2( 1) 2 = f 2 (7, 1) 47 = ( 3) ( 3) 2 = f 1 (4, 3) 73 = ( 3) 2( 3) 2 = f 2 (7, 3) 59 = ( 1) ( 1) 2 = f 1 (5, 1) 97 = ( 1) 2( 1) 2 = f 2 (9, 1) 71 = ( 3) ( 3) 2 = f 1 (5, 3) 83 = = f 1 (7, 1)

70 6 68 D = 20 p56 h(20) = h + (20) = 2, f 1 (x, y) = 2x 2 2xy 2y 2, f 2 (x, y) = x 2 4xy y 2 p 20 m 2 20 (mod 4p), 0 m < 2p m m m = 2m 0 m (mod p), 0 m < p m 0 m = 2m 0 m 2 20 (mod 4p), 0 m < 2p 20 p x 2 5 (mod p), p = 5 p 1, 4 (mod 5) 100 5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89 f 1 (x, y) = 2x 2 2xy 2y 2, f 2 (x, y) = x 2 4xy y 2

71 f 2 (x, y) 5 = = f 2 (9, 2) 11 = = f 2 (6, 1) 19 = ( 3) ( 3) 2 = f 2 (2, 3) 29 = ( 2) ( 2) 2 = f 2 (3, 2) 31 = ( 1) ( 1) 2 = f 2 (4, 1) 41 = ( 4) ( 4) 2 = f 2 (3, 4) 59 = ( 1) ( 1) 2 = f 2 (6, 1) 61 = ( 2) ( 2) 2 = f 2 (5, 2) 71 = ( 5) ( 5) 2 = f 2 (4, 5) 79 = ( 7) ( 7) 2 = f 2 (4, 7) 89 = ( 4) ( 4) 2 = f 2 (5, 4)

72 References 1,, ,,, , 2,, ,, H MStark, (, ),,, W J LeVeque, Fundamentals of Number Theory, Dover,

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