1 n =3, 2 n 3 x n + y n = z n x, y, z 3 a, b b = aq q a b a b b a b a a b a, b a 0 b 0 a, b 2

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1 n =3,

2 1 n =3, 2 n 3 x n + y n = z n x, y, z 3 a, b b = aq q a b a b b a b a a b a, b a 0 b 0 a, b 2

3 a, b (a, b) =1a b 1 x 2 + y 2 = z 2, (x, y) =1, x 0 (mod 2) (1.1) x =2ab, y = a 2 b 2, z = a 2 + b 2 (1.2) a, b (a, b) =1, a > b > 0, a+ b 1 (mod 2) (1.3) (1.1) (x, y) =1, x 0 (mod 2) y x y z y z z y z+y (1.1) 2 2 )( z y 2 ) ( x 2 )2 =( z+y 2 z + y 2 = a 2, z y 2 a, b = b 2, a > b > 0, (a, b) =1 a + b a 2 + b 2 = z 1 (mod 2). (1.3) a, b x, y, z (1.2) x 2 + y 2 =(2ab) 2 +(a 2 b 2 ) 2 =(a 2 + b 2 ) 2 = z 2, x>0, y > 0, z > 0, x 0 (mod 2). (x, y) =d d z d y = a 2 b 2, d z = a 2 + b 2, d 2a 2, d 2b 2 (a, b) =1 d y d 2 d =1.(x, y) =1. 3

4 5 x + y = z 1 x + y = z, x > 0, y > 0, z > 0 n = x + y = u 2, (x, y) =1 (2.1) (2.1) u (x, y) =1 x, y u 2 = x + y 1 2 (mod ) u 2 2 (mod ) u 2 = x + y 1 (mod ). u x, y x y a, b x 2 =2ab, y 2 = a 2 b 2, u = a 2 + b 2, a>0, b > 0, (a, b) =1, a+ b 1 (mod 2). a b y 2 1 (mod ) a b b = 2c ( 1 2 x)2 = ac, (a, c) =1. a = d 2, c = f 2, d > 0, f > 0, (d, f) =1, d y 2 = a 2 b 2 = d f f + y 2 = d (2f 2 ) 2 + y 2 =(d 2 ) 2, (2f 2,y)=1.

5 l, m 2f 2 =2lm, d 2 = l 2 + m 2, l > 0, m > 0, (l, m) =1. f 2 = lm (l, m) =1 (r, s) =1 l = r 2, m = s 2, r > 0, s > 0 d 2 = r + s, d d 2 = a a 2 <a 2 + b 2 = u. (2.2) (2.1) (2.2) d u u 6 n =3 n =3 a, b a 2 +3b 2 = s 3 u, v a = u(u 2 9v 2 ), b =3v(u 2 v 2 ) (a, b) =1 (a 2 +3b 2 : ) u, v a + b 3=(1± 3)(u + v 3). ( ) (a, b) =1 a 2 +3b 2 a, b a + b a b (1) a + b a 3b (a 2 +3b 2 )=( )(a 2 +3b 2 ) 5

6 =(a 3b) 2 +3(a + b) 2 (a 2 +3b 2 ) 2 a 2 +3b 2 ( ) 2 ( ) 2 a 3b a + b = +3. u = a 3b, v = a+b u, v a2 +3b 2 = u 2 +3v 2 u + v 3= a 3b + a + b 3 = (a + b 3)(1 + 3). a + b 3= (u + v 3) 1+ 3 =(1 3)(u + v 3), (u, v) =1. (2) a b a +3b (a 2 +3b 2 )=( )(a 2 +3b 2 ) =(a +3b) 2 +3(a b) 2 (a 2 +3b 2 ) 2 a 2 +3b 2 ( ) 2 ( ) 2 a +3b a b = +3. u = a+3b, v = a b u, v a2 +3b 2 = u 2 +3v 2 u + v 3= a +3b + a b 3 = (a + b 3)(1 3). a + b 3= (u + v 3) 1 3 =(1+ 3)(u + v 3), (u, v) =1. 6

7 (a, b) =1, q, r a 2 +3b 2 = q 2 +3r 2 u, v a + b 3=(q ± r 3)(u + v 3) ( ) = q 2 +3r 2 (qb + ar)(qb ar) =b 2 (q 2 +3r 2 ) r 2 (a 2 +3b 2 ). qb + ar qb ar (1) qb + ar 2 (a 2 +3b 2 )=(q 2 +3r 2 )(a 2 +3b 2 ) =(qa 3rb) 2 +3(qb + ar) 2 a 2 +3b 2 ( ) 2 ( ) 2 qa 3rb qb + ar = +3. u = qa 3rb, v = qb+ar u, v a2 +3b 2 = u 2 +3v 2 u + v 3= qa 3rb + qb + ar 3 = (q + r 3)(a + b 3) a + b 3= (u + v 3) q + r 3 =(q r 3)(u + r 3), (u, v) =1. 7

8 (2) qb ar (a 2 +3b 2 )=(q 2 +3r 2 )(a 2 +3b 2 ) =(qa +3rb) 2 +3(qb ar) 2 2 a 2 +3b 2 ( ) 2 ( ) 2 qa +3rb qb ar = +3. u = qa+3rb, v = qb ar u, v a2 +3b 2 = u 2 +3v 2 u + v 3= a + b 3= (u + v 3) q r 3 qa +3rb + qb ar 3 = (q r 3)(a + b 3) =(q + r 3)(u + r 3), (u, v) =1. x a 2 +3b 2 α 2 +3β 2 (α, β Z) a2 +3b 2 γ 2 +3δ 2 (γ,δ Z) x ( ) a 2 +3b 2 = xy (x : ) y 2 a 2 +3b 2 a 2 +3b 2 ( ) y x = c 2 +3d 2 (c, d Z). y k y = 1 2 n ( i ) y γ 2 +3δ 2 (γ,δ Z) a 2 +3b 2 = 1 (u 2 +3v 2 ) a2 +3b = u 2 +3v 2

9 u 2 +3v 2 = m 2 +3n 2,, k2 +3l 2 = α 2 +3β 2 2 n x α 2 +3β 2 (α, β Z) ( ) x a 2 +3b 2 q, r x = q 2 +3r 2 a = mx ± c, c < 1 2 x b = nx ± d, d < 1 2 x c 2 +3d 2 =(a mx) 2 +3(b nx) 2 =(a 2 +3b 2 ) 2(am +3bn)x +(m 2 +3n 2 )x 2 x a 2 +3b 2 y c 2 +3d 2 = xy xy = c 2 +3 d 2 ( 1 2 x ) 2 +3( 1 2 x ) 2 = x 2, x>0 y x x = y c 2 +3d 2 = x 2 x x 2 1 (mod ) x 2 = c 2 +3d 2 c 2 +3d 2 c d c 2 0, d 2 1 (mod ). c 2 +3d 2 3 (mod ) 9

10 x 2 c 2 +3d 2 (mod ) x 2 c 2 +3d 2 x y y <x (c, d) =e e 1 e x e a, e b (a, b) =1 e x y f 2 +3g 2 = xz, (f,g) =1, z < y e =1 y = z x α 2 +3β 2 (α, β Z) z γ 2 +3δ 2 (γ,δ Z) w a 2 +3b 2 q 2 +3r 2 (q, r Z) x (f,g) =1 f 2 +3g 2 γ 2 +3δ 2 (γ,δ Z) w w z y<x (a, b) =1 a + b 3=±(q 1 ± r 1 3)(q2 ± r 2 3) (qn ± r n 3) q i,r i q 2 i +3r 2 i (i =1, 2,,n). ( ) 2 a 2 +3b 2 a 2 +3b 2 a 2 +3b 2 1 a 2 +3b 2 a + 3=(q ± r 3)(u + v 3) a 2 +3b 2 = q 2 +3r 2 (q, r Z) (u, v) =1u + v 3 a + b 3=(q 1 ± r 1 3) (qn ± r n 3)(u + v 3), 10

11 u 2 +3v 2 =1 u 2 +3v 2 =1 v =0, u = ±1, a + b 3=±(q 1 ± r 1 3)(q2 ± r 2 3) (qn ± r n 3) (a, b) =1 a 2 +3b 2 =(q r 2 1) (q 2 n +3r 2 n) a 2 +3b 2 a + b 3 a 2 +3b 2 q + r 3 q r 3 ( ) = q 2 +3r 2 q, r = = q 2 +3r 2 q + r 3=(q ± r 3)(u + v 3). = (u 2 +3v 2 ) u 2 +3v 2 =1 u = ±1, v =0. q + r 3=±(q + r 3). q + r 3 q r 3 q 2 +3r 2 a, b 11

12 ( ) (a + b 3) = (u + v 3) 3 u, v a 2 +3b 2 = 1 n 2 2k a 2 +3b 2 3 k i 3 n a + b 3=±(u + v 3) 3 (u, v Z). (u + v 3) 3 = ±( u v 3) 3 a + b 3=(u + v 3) 3 = u 3 +3u 2 v 3 9uv 2 3v 3 3 =(u 3 9uv 2 )+(3u 2 v 3v 3 ) 3 a = u(u 2 9v 2 ), b =3v(u 2 v 2 ). 7 x 3 + y 3 = z 3 x 3 + y 3 = z 3, xyz 0 ( ) x 3 + y 3 = z 3 x 3 + y 3 +( z) 3 =0 x 3 + y 3 + z 3 =0, xyz 0 x, y, z x, y, z 12

13 x, y z z z x + y =2a, x y =2b ( z) 3 = x 3 + y 3 =(a + b) 3 +(a b) 3 =2a(a 2 +3b 2 ) (3.1) a, b a 2 +3b 2 2a b 2a a 2 +3b 2 a a 2 +3b 2 (a, b) =1 (2a, a 2 +3b 2 )=1 3 (1) (2a, a 2 +3b 2 )=1 (3.1) 2a a 2 +3b 2 r, s 2a = r 3, a 2 +3b 2 = s 3. a = u(u 2 9v 2 ), b =3v(u 2 v 2 ) u, v s = u 2 +3v 2 v u 0 u (u, v) =1 r 3 =2a =2u(u 3v)(u +3v) 2u, u 3v, u +3v u v u 3v, u +3v 2u u ± 3v u u ± 3v u ±3v (u, v) =1 u (2u, u ± 3v) =1(u 3v, u +3v) =1 2u, u 3v, u +3v 2u = l 3, u 3v = m 3, u+3v = n 3 13

14 l, m, n l 3 + m 3 + n 3 =0 u b 0 z 3 = 2a(a 2 +3b 2 ) = l 3 (u 2 9v 2 )(a 2 +3b 2 ) 3 l 3 > l 3. z (2) (2a, a 2 +3b 2 )=3a =3c b ( z) 3 =2a(a 2 +3b 2 )=18c(3c 2 + b 2 ). 18c 3c 2 + b 2 r, s 18c = r 3, 3c 2 + b 2 = s 3. (1) b = u(u 2 9v 2 ), c =3v(u 2 v 2 ) u, v s = u 2 +3v 2 ( ) r 3 2 = c =2v(u + v)(u v). 3 3 (1) 2v, u + v, u v 2v = l 3, u+ v = m 3, u v = n 3 l, m, n l 3 + m 3 + n 3 =0 z 3 =18 c (3c 2 + b 2 )=5 v(u 2 v 2 ) (3c 2 + b 2 ) =27 l 3 u 2 v 2 (3c 2 + b 2 ) 27 l 3 > l 3. z (1) (2) 1

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