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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), · · ...
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, ...
e⇡
p
163
= 262537412640768743.99999999999925...
12 digits
Google
 
e⇡
p
d
e⇡
p
d
= ( )3 + 744 – ( )
e⇡
p
19
= 963
+ 744 - 0.22 . . .
e⇡
p
43
= 9603
+ 744 - 0.00022 . . .
e⇡
p
67
= 52803
+ 744 - 0.0000013 . . .
e⇡
p
163
163
= 6403203
+ 744 - 0.00000000000075 . . .
12 digits
 
e⇡
p
d
 
() X = 1 q – 1
Q(
p
-d)
()
(=))
((=)
(1912 )
(1913 )
X = 0, · · · , q - 2
X2
+ X + q
tsujimotter
http://tsujimotter.hatenablog.com/entry/...
j(⌧) =
1
q
+ 744 + 196884q + 21493760q3
+ · · ·
j- q-
⌧ =
1 +
p
-d
2
q = -
1
e⇡
p
d
e⇡
p
d
= -j
✓
1 +
p
-d
2
◆
+ 744 - 196...
Romantic formula
1
⇡
=
12
6403203/2
1X
k=0
(6k)!(163 · 3344418k + 13591409)
(3k)!(k!)3(-640320)3k
j
✓
1 +
p
-163
2
◆
= -64...
Heegner Number
Q(
p
-d) Heegner Number h log " - 32
21 ⇡
p
d < e
⇡
p
d
100
h0
log "0
- 80
33 ⇡
p
d < e
⇡
p
d
100
---- (1)
...
Heegner Number
Heilbronn, Linfoot
10 Heegner Number d > e1000000
(B)
d
(B)(A)
d > e1000000) d < 10500
d (QED)
tsujimotter
...
ロマンティックな9つの数 #ロマ数ボーイズ
ロマンティックな9つの数 #ロマ数ボーイズ
ロマンティックな9つの数 #ロマ数ボーイズ
ロマンティックな9つの数 #ロマ数ボーイズ
ロマンティックな9つの数 #ロマ数ボーイズ
ロマンティックな9つの数 #ロマ数ボーイズ
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ロマンティックな9つの数 #ロマ数ボーイズ

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#ロマ数ボーイズ で tsujimotter が発表したスライドです。
https://connpass.com/event/50636/

tsujimotter のノートブック
http://tsujimotter.hatenablog.com

Publié dans : Formation
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ロマンティックな9つの数 #ロマ数ボーイズ

  1. 1. @tsujimotter 2017/04/01 #
  2. 2. Q( p -d)
  3. 3. p -1 13 = ⇣ 2 + 3 p -1 ⌘ ⇣ 2 - 3 p -1 ⌘ 13 = ⇣ 2 + 3 p -1 ⌘ ⇣ 2 - 3 p -1 ⌘
  4. 4. 1471 = ⇣ 2 + 3 p -163 ⌘ ⇣ 2 - 3 p -163 ⌘ p -163
  5. 5. Q( p -d) p -d Q( p -1) Q( p -163) Q( p -5)
  6. 6. 6 = 2 · 3 = ⇣ 1 + p -5 ⌘ ⇣ 1 - p -5 ⌘ Q( p -5)
  7. 7. 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 Q( p -7), Q( p -11), Q( p -19), Q( p -43), Q( p -67), Q( p -163)
  8. 8. d = -1, -2, -3, -7, -11, -19, -43, -67, -163
  9. 9. X2 - X + 41 4 1
  10. 10. 12 – 1 + 41 = 22 – 2 + 41 = 32 – 3 + 41 = 42 – 4 + 41 = 52 – 5 + 41 = 62 – 6 + 41 = 72 – 7 + 41 =
  11. 11. 82 – 8 + 41 = 92 – 9 + 41 = 102 – 10 + 41 = 112 – 11 + 41 = 122 – 12 + 41 = 132 – 13 + 41 = 142 – 14 + 41 =
  12. 12. 152 – 15 + 41 = 162 – 16 + 41 = 172 – 17 + 41 = 182 – 18 + 41 = 192 – 19 + 41 = 202 – 20 + 41 = 212 – 21 + 41 =
  13. 13. 222 – 22 + 41 = 232 – 23 + 41 = 242 – 24 + 41 = 252 – 25 + 41 = 262 – 26 + 41 = 272 – 27 + 41 = 282 – 28 + 41 =
  14. 14. 292 – 29 + 41 = 302 – 30 + 41 = 312 – 31 + 41 = 322 – 32 + 41 = 332 – 33 + 41 = 342 – 34 + 41 = 352 – 35 + 41 =
  15. 15. 362 – 36 + 41 = 372 – 37 + 41 = 382 – 38 + 41 = 392 – 39 + 41 = 402 – 40 + 41 = 412 – 41 + 41 = = 412
  16. 16.   () X = 1 q – 1 q = 1 + d 4
  17. 17. 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
  18. 18. e⇡ p 163 = 262537412640768743.99999999999925... 12 digits
  19. 19. Google
  20. 20.   e⇡ p d e⇡ p d = ( )3 + 744 – ( )
  21. 21. e⇡ p 19 = 963 + 744 - 0.22 . . . e⇡ p 43 = 9603 + 744 - 0.00022 . . . e⇡ p 67 = 52803 + 744 - 0.0000013 . . .
  22. 22. e⇡ p 163 163 = 6403203 + 744 - 0.00000000000075 . . . 12 digits
  23. 23.   e⇡ p d   () X = 1 q – 1
  24. 24. Q( p -d) () (=)) ((=) (1912 ) (1913 ) X = 0, · · · , q - 2 X2 + X + q tsujimotter http://tsujimotter.hatenablog.com/entry/prime-generating-polynomials
  25. 25. j(⌧) = 1 q + 744 + 196884q + 21493760q3 + · · · j- q- ⌧ = 1 + p -d 2 q = - 1 e⇡ p d e⇡ p d = -j ✓ 1 + p -d 2 ◆ + 744 - 196884 ✓ 1 e⇡ p d ◆ - 21493760 ✓ 1 e⇡ p d ◆3 - · · ·
  26. 26. 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]
  27. 27. Heegner Number Q( p -d) Heegner Number h log " - 32 21 ⇡ p d < e ⇡ p d 100 h0 log "0 - 80 33 ⇡ p d < e ⇡ p d 100 ---- (1) ---- (2) (1), (2) |b log " + b0 log "0 | < e- B B < ⇣ 4n2 -1 l2n log A ⌘(2n+1)2 < 10250 ⇣ B = 140 p d ⌘ ) d < 10500 (A)
  28. 28. 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

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