( ) ( ) 1729 (, 2016:17) = = (1) 1 1
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1 ( ) ( ) 1729 (, 2016:17) = = (1) 1 1
2 = 1729 = = = 91 = = = 7 = = 19 = = (n + 1) 3 n 3 = 3n 2 + 3n + 1 = 6 n(n + 1) r = a 3 + b 3 = c 3 + d 3 ({a, b} = {c, d}) (2) a b c d 1 1 2
3 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 < r = (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 = = < 3 4 < = = < 3 4 < = a b c d ( ) 3.2 Z[ω] a b 1 3 ω = i 2 a + b ω Z[ω] 2 y = x 3 (x 0) x y 3
4 Z[ω] ( 1 ) ±1, ± ω, ± ω 2 (= (1 + ω)) ω 2 = 1 3 i 2 1 ω ω ω 2 a+b ω (a b) b ω a + b ω 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
5 1 ω ω , ω ω, ω 2 ω 2, ω 2 ω 3 ω 2 ω ω ω ω 2 ω 2 3 ω 2 1 ω ω 3 ω + ω 2 2 ω ω 2 ω + 3 ω 2 ω 2 2 ω ω ω + 2 ω 2 3 ω ω 1 2 ω 3 ω ω 2 2 ω P P 1 P 1 P +1 2 a+b ω 3 a+b ω 2 P 1 a b 1729 = = (3+ω) (3+ω 2 ) (3+4 ω) (3+4 ω 2 ) (5+2ω) (5+2ω 2 ) ( ) 1729 = = 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) = = = 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) ( ω 2 ) ( ω) =
6 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 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
7 (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 e f g e = 13 f = 19 g = = 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
8 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 ) 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 a + b c + d = = c + d a + b < r 8
9 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 c 3 +d 3 c+d c d = r 1729 c + d 1729 r a + b c + d 19 (1) 9
10 a a n 1 1 (mod n) n (6k + 1)(12k + 1)(18k + 1) k = (6k+1) (12k+1) (18k+1) 3 k = 6, 35,
11 : (2) a, b, c, d r = 10 9 r 10 4 r 10 9 r 10 4 r 10 5 r 10 6 r 10 7 r 10 8 r r r 10 5 r r 10 9 r r = = 1729 = = = 4104 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
12 r = 10 6 a + b c + d = = 1729 = = = = = = = = = = = = = = = = = = = = = = = = = (mathquest) Z[ω] ( ) G.H., 2016,
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福岡大学人文論叢47-3
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2000年度『数学展望 I』講義録
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1 (1) X = AB + AB, Y = C D + C D, Z = AD + AD P A, B, C, D P = (XY + X Y + X Y )(Y Z + Y Z + Y Z )(ZX + Z X + Z X ) (2) Q A, B, C, D Q = AB C D + AB C
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SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
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