1729 1 2016 10 28 1 1729 1111 1111 1729 (1887 1920) (1877 1947) 1729 (, 2016:17) 12 3 1728 9 3 729 1729 = 12 3 + 1 3 = 10 3 + 9 3 (1) 1 1
2 1729 1729 19 13 7 = 1729 = 12 3 + 1 3 = 10 3 + 9 3 13 7 = 91 = 6 3 5 3 = 4 3 + 3 3 7 = 7 = 2 3 1 3 19 = 19 = 3 3 2 3 1729 1 6 4 1729 = 1 7 13 19 6 + 1 (n + 1) 3 n 3 = 3n 2 + 3n + 1 = 6 n(n + 1) 2 + 1 3 2 2 r = a 3 + b 3 = c 3 + d 3 ({a, b} = {c, d}) (2) a b c d 1 1 2
3.1 a + b c + d a + b c + d a 3 + b 3 = c 3 + d 3 = r = (a, b, c, d > 0) a b c d a b f(x) = x 3 x > 0 a + b < c + d 2 a+b a b ( b ) c + d c d b = 0 c = d a+b c+d 1 < c + d r a + b < 2 3 2 1 3 r = 3 4 1.5874 (3) 3 4 [1; 1, 1, 2, 2, 1, 3, 2, 3, 1, 3, 1, 30, 1, 4, 1, 2, 9,... ] 3 2 < 3 4 < 8 5 = 23 5 19 2 2 3 = 19 12 < 3 4 < 27 17 = 33 17 2 2 5 2 3 2 7 = 100 63 < 3 4 < 227 143 = 227 11 13 a b c d ( ) 3.2 Z[ω] a b 1 3 ω = 1 + 3 i 2 a + b ω Z[ω] 2 y = x 3 (x 0) x y 3
Z[ω] ( 1 ) ±1, ± ω, ± ω 2 (= (1 + ω)) 6 6 6 ω 2 = 1 3 i 2 1 ω ω ω 2 a+b ω (a b) b ω a + b ω 2 3 + ω 2 ω 2 2 ω 3 ω 2 ( 1 + ω + ω 2 = 0 ) ω 2 ( ) Z[ω] p Z[ω] 1. 3 = ω 2 λ 2, λ = 1 ω, λ = 1 ω = ω 2 λ, = (1 ω) (1 ω 2 ) = ( 1 + ω) ( 1 + ω 2 ) 2. p 1 (mod 3) p = π π = x 2 x y + y 2, π = x + y ω, π = x + y ω 2 ϵ π(ϵ ), π π ϵ ϵ 6 3. p 2 (mod 3) p Z[ω] 7 (3 + ω)(3 + ω 2 ) 1 ( ) (3 + ω) 6 6 4
1 ω ω 2 2 3 3 1, ω ω, ω 2 ω 2, 1 1 3 + ω 2 ω 3 ω 2 ω 2 + 2 ω 2 2 + 3 ω ω 2 ω 2 3 ω 2 1 ω 1 + 2 ω 3 ω + ω 2 2 ω 2 3 1 3 ω 2 ω + 3 ω 2 ω 2 2 ω 2 2 3 ω ω + 2 ω 2 3 ω 2 + 1 ω 1 2 ω 3 ω ω 2 2 ω 2 + 3 6 + 1 P +1 6 1 P 1 P 1 P +1 2 a+b ω 3 a+b ω 2 P 1 a b 1729 = 7 13 19 = (3+ω) (3+ω 2 ) (3+4 ω) (3+4 ω 2 ) (5+2ω) (5+2ω 2 ) ( ) 1729 = 13 7 19 = 13 (3 + ω) (3 + ω 2 ) (5 + 2 ω) (5 + 2 ω 2 ) = 13 (3 + ω) (5 + 2 ω 2 ) (3 + ω 2 ) (5 + 2 ω) = (12 + 1) (12 + ω 2 ) (12 + ω) = 12 3 + 1 3 = 19 7 13 = 19(3 + ω) (3 + ω 2 ) (3 + 4 ω) (3 + 4 ω 2 ) = 19(3 + ω) (3 + 4 ω 2 ) (3 + ω 2 ) (3 + 4 ω) = (10 + 9) (10 + 9 ω 2 ) (10 + 9 ω) = 10 3 + 9 3 5
r 3.3 a + b c + d (2) r = a 3 +b 3 = (a+b) (a 2 a b+b 2 ) = c 3 +d 3 = (c+d) (c 2 c d+d 2 ) (4) a + b c + d a + b = e, c + d = f r = e f g r e f g 3 a 2 a b + b 2 = f g, c 2 c d + d 2 = e g n n 3 n (mod 6) r a 3 + b 3 a + b c 3 + d 3 c + d (mod 6) r e f g e f g (mod 6) ( 3 ) r e f 6 r 2 3 e f 2 3 e f r 2 3 e f g 6 1 6 1 e f g P +1 e f g P 1 a 3 + b 3 = (a + b) (a 2 a b + b 2 ) = (a + b) (a + b ω) (a + b ω 2 ) 6
(a + b ω) (a + b ω 2 ) P 1 p a b (a + b) p c 3 + d 3 = (c + d) (c 2 c d + d 2 ) = (c + d) (c + d ω) (c + d ω 2 ) p 1 a+b c+d r e f g P +1 = {7, 13, 19, 31, 37,... } e = a + b f = c + d (3) 13 7 > 3 4 1 < 19 13 < 3 4 e f g e = 13 f = 19 g = 7 13 19 7 = 1729 (1) 3.4 a + b c + d a + b c + d h (4) a + b = e = h e, c + d = f = h f r = h e f g r h e f g 4 a 2 a b + b 2 = f g, c 2 c d + d 2 = e g a 3 + b 3 = (a + b) (a 2 a b + b 2 ) = (a + b) (a + b ω) (a + b ω 2 ) c 3 + d 3 = (c + d) (c 2 c d + d 2 ) = (c + d) (c + d ω) (c + d ω 2 ) (a + b ω) (a + b ω 2 ) 2 P 1 (a + b ω) (a + b ω 2 ) (a + b) 7
f g = (a + b ω) (a + b ω 2 ) e g = (c + d ω) (c + d ω 2 ) e f e f g 2 P 1 h 2 e f g 2 6 h 2 3 e f g 5 2 h 5 1 r 3 a+b c+d 3 a 2 a b + b 2 = (a + b) 2 3 a b a 2 a b + b 2 c 2 c d + d 2 3 a 2 a b + b 2 3 a + b 3 r 3 2 r h e f g e f g p (p P +1 ) 3 2 2 q 2 (q P 1 ) e f g {3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28,... } e f g 3 h 3 e f g 2 P 1 h 2 3.1 a + b c + d 3 4 1729 = 12 3 + 1 3 = 10 3 + 9 3 c + d a + b 19 13 1.4615 < 3 2 1729 r 8
c + d a + b 3 2 ( e f 2 : 3 ) (2) r 1729 e f 1 1 < f e < 3 2 (e, f ) = (3, 4) (7, 9) (9, 12) 2 3 h a + b = h e c + d = h f a + b c + d e f h a + b c + d 3 4 6 18 24 7 9 9 63 81 9 12 54 486 648 9 13 9 81 108 12 13 6 72 78 12 16 24 288 384 13 16 4 52 64 c 3 +d 3 c+d c d 10 3 +9 3 = 1729 10 9 19 r 1729 c + d 1729 r a + b c + d 19 (1) 9
4 32 32 10 10 1 12 1729 3 a a n 1 1 (mod n) n 1910 23 3 (6k + 1)(12k + 1)(18k + 1) 3 1939 k 1 7 13 19 = 1729 19 3 3 10 (6k+1) (12k+1) (18k+1) 3 k = 6, 35,45 1729 10
: (2) a, b, c, d r 1000000000 = 10 9 r 10 4 r 10 9 r 10 4 r 10 5 r 10 6 r 10 7 r 10 8 r 10 9 2 10 43 150 487 1570 5 4.3 3.4884 3.2467 3.2238 10 r r 10 5 r 10 4 5 r 10 9 r 10 8 3 r 15 12 3 + 1 3 = 10 3 + 9 3 = 1729 = 7 13 19 16 3 + 2 3 = 15 3 + 9 3 = 4104 = 2 3 3 3 19 24 3 + 2 3 = 20 3 + 18 3 = 13832 = 2 3 7 13 19 27 3 + 10 3 = 24 3 + 19 3 = 20683 = 13 37 43 32 3 + 4 3 = 30 3 + 18 3 = 32832 = 2 6 3 3 19 34 3 + 2 3 = 33 3 + 15 3 = 39312 = 2 4 3 3 7 13 34 3 + 9 3 = 33 3 + 16 3 = 40033 = 7 2 19 43 36 3 + 3 3 = 30 3 + 27 3 = 46683 = 3 3 7 13 19 39 3 + 17 3 = 36 3 + 26 3 = 64232 = 2 3 7 31 37 40 3 + 12 3 = 33 3 + 31 3 = 65728 = 2 6 13 79 48 3 + 4 3 = 40 3 + 36 3 = 110656 = 2 6 7 13 19 48 3 + 6 3 = 45 3 + 27 3 = 110808 = 2 3 3 6 19 51 3 + 12 3 = 43 3 + 38 3 = 134379 = 3 5 7 79 53 3 + 8 3 = 50 3 + 29 3 = 149389 = 31 61 79 54 3 + 20 3 = 48 3 + 38 3 = 165464 = 2 3 13 37 43 11
r 1000000 = 10 6 a + b c + d 12 3 + 1 3 = 10 3 + 9 3 = 1729 = 7 13 19 27 3 + 10 3 = 24 3 + 19 3 = 20683 = 13 37 43 34 3 + 9 3 = 33 3 + 16 3 = 40033 = 7 2 19 43 53 3 + 8 3 = 50 3 + 29 3 = 149389 = 31 61 79 58 3 + 9 3 = 57 3 + 22 3 = 195841 = 37 67 79 67 3 + 30 3 = 58 3 + 51 3 = 327763 = 31 97 109 82 3 + 51 3 = 75 3 + 64 3 = 684019 = 7 19 37 139 89 3 + 2 3 = 86 3 + 41 3 = 704977 = 7 13 61 127 98 3 + 35 3 = 92 3 + 59 3 = 984067 = 7 3 19 151 9 6 1 (mathquest) Z[ω] ( ) G.H., 2016,. 30. 12