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@tsujimotter 2017/04/01 #

ロマンティックな9つの数 #ロマ数ボーイズ

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@tsujimotter

2017/04/01 #

Q(p-d)

p-1

13 =⇣2+ 3

p-1

⌘⇣2- 3

p-1

⌘13 =

⇣2+ 3

p-1

⌘⇣2- 3

p-1

1471 =⇣2+ 3

p-163

⌘⇣2- 3

p-163

p-163

Q(p-d)

p-d

Q(p-1) Q(

p-163) Q(

p-5)

6 = 2 · 3 =⇣1+

p-5

⌘⇣1-

p-5

⌘Q(

p-5)

Q(p-5),Q(

p-6),Q(

p-10),Q(

p-13),Q(

p-14),Q(

p-15), · · ·

Q(p-15),Q(

p-17),Q(

p-21),Q(

p-22),Q(

p-23), · · ·

Q(p-1),Q(

p-2),Q(

p-3),Q(

p-7),Q(

p-11),Q(

p-19),Q(

p-43),Q(

p-67),Q(

p-163)

Q(p-1),Q(

p-2),Q(

p-3),Q(

p-7),Q(

p-11),Q(

p-19),Q(

p-43),Q(

p-67),Q(

p-163)

d = -1, -2, -3, -7, -11, -19, -43, -67, -163

X2 - X+ 41 4 1

12 – 1 + 41 = 22 – 2 + 41 = 32 – 3 + 41 = 42 – 4 + 41 = 52 – 5 + 41 = 62 – 6 + 41 = 72 – 7 + 41 =

82 – 8 + 41 = 92 – 9 + 41 = 102 – 10 + 41 = 112 – 11 + 41 = 122 – 12 + 41 = 132 – 13 + 41 = 142 – 14 + 41 =

152 – 15 + 41 = 162 – 16 + 41 = 172 – 17 + 41 = 182 – 18 + 41 = 192 – 19 + 41 = 202 – 20 + 41 = 212 – 21 + 41 =

222 – 22 + 41 = 232 – 23 + 41 = 242 – 24 + 41 = 252 – 25 + 41 = 262 – 26 + 41 = 272 – 27 + 41 = 282 – 28 + 41 =

292 – 29 + 41 = 302 – 30 + 41 = 312 – 31 + 41 = 322 – 32 + 41 = 332 – 33 + 41 = 342 – 34 + 41 = 352 – 35 + 41 =

362 – 36 + 41 = 372 – 37 + 41 = 382 – 38 + 41 = 392 – 39 + 41 = 402 – 40 + 41 =

412 – 41 + 41 = = 412

 

() X = 1 q – 1

q =1+ d

4

X2 – X + X = 1

X2 – X + X = 1, 2

X2 – X + X = 1, 2, 3, 4, 5

X2 – X + X = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

X2 – X + X = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

X2 – X + X = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …, 40

e⇡p163

= 262537412640768743.99999999999925...12 digits

Google

 

e⇡pd

e⇡pd = ( )3 + 744 – ( )

e⇡p19 = 963 + 744- 0.22 . . .

e⇡p43 = 9603 + 744- 0.00022 . . .

e⇡p67 = 52803 + 744- 0.0000013 . . .

e⇡p163

e⇡p163 = 6403203 + 744- 0.00000000000075 . . .

12 digits

 

e⇡pd

 

() X = 1 q – 1

Q(p-d) ()

(=))

((=)

(1912 )

(1913 )

X = 0, · · · , q- 2

X2 + X+ q

tsujimotter http://tsujimotter.hatenablog.com/entry/prime-generating-polynomials

j(⌧) =1

q+ 744+ 196884q+ 21493760q3 + · · ·

j- q-

⌧ =1+

p-d

2q = -

1

e⇡pd

e⇡pd = -j

✓1+

p-d

2

◆+ 744- 196884

✓1

e⇡pd

◆- 21493760

✓1

e⇡pd

◆3

- · · ·

Romantic formula

1

⇡=

12

6403203/2

1X

k=0

(6k)!(163 · 3344418k+ 13591409)

(3k)!(k!)3(-640320)3k

j

✓1+

p-163

2

◆= -6403203

[Chudnovsky brothers 1989]

Heegner Number

Q(p-d) Heegner Number

h log "- 32

21⇡pd < e

⇡p

d100

h 0log " 0 - 80

33⇡pd < e

⇡p

d100

---- (1)

---- (2)

(1), (2) |b log "+ b 0log " 0| < e-�B

B <⇣4n

2

�-1l2n logA⌘(2n+1)2

< 10250

⇣B = 140

pd⌘

) d < 10500(A)

Heegner Number

Heilbronn, Linfoot

10 Heegner Number d > e1000000

(B)

d

(B)(A)

d > e1000000) d < 10500

d (QED)

tsujimotter http://tsujimotter.hatenablog.com/entry/class-numbers-of-imaginary-quadratic-fields