ロマンティックな9つの数 #ロマ数ボーイズ
TRANSCRIPT
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)
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
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⇡p19 = 963 + 744- 0.22 . . .
e⇡p43 = 9603 + 744- 0.00022 . . .
e⇡p67 = 52803 + 744- 0.0000013 . . .
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)