[1]border < p(n) < 0.584 のp(n)生成ルールを考える。
---
[2][break : prime again (3)],[5]の一覧を再掲。
n,p,?,result,(up/down),(border/or/0.584),comments,
--
2,3,1,O,up,border,
2,5,1,O,up,0.584,@before (0.584), except border.
--
3,7,1,O,up,border,
3,11,1,X,up,0.584,@before (0.584), except border.
3,11,((1,2)),O,down,
3,13,2,O,up,
--
4,15,,,border,@!p(15),
4,17,2,O,up,
4,19,4,O,up,,
4,23,3,O,up,0.584,@before (0.584), p(17,19)=?(2,4),
4,29,3,O,up,
--
5,31,6,O,up,border,
5,37,4,O,up,
5,41,6,O,up,
5,43,8,O,up,,
5,47,7,O,up,0.584,@before (0.584), p(37,41,43)=?(4,6,8),
5,53,7,O,up,
5,59,(7,8),X,(down,up),,@@@(redo)@@@
5,59,((9,10)),O,up,
5,61,11,O,down,
--
6,63,,,border,@!p(63),
6,67,8,O,down,
6,71,10,O,up,
6,73,(13,14),O,(down,up),
6,79,11,O,down,
6,83,(13,14),(X,△),(down,up),@@@(redo)@@@
6,83,((15,16)),O,up,
6,89,13,O,up,,
6,95,,,,0.584,@!p(95),@before (0.584), p(67,71,73,79,83,89)=?(8,10,(13,14),11,(14?),13),
6,95,,,,0.584,@!p(95),@?()=(set(p*3)?,set(p*3)?)=?((67,71,73),(79,83,89)),
6,97,13,O,up,
6,101,(15,16),(△,△),(down,up),@@@(redo)@@@
6,101,((17,18)),O,down,
6,103,18,O,down,
6,107,18,O,up,
6,109,(20,21),O,(down,up),
6.113,(18,19),O,(down,up),
--
7,127,14,O,down,border,
7,131,(21,22),(O,△),(up,up),
7,137,(20,21),(O,△),(down,up),
7,139,(24,25,26),(X,O,△),(down,up,up),
7,149,19,O,up,
7,151,(26,27,28),(△,O,△),(down,up,up),
7,157,(23,24),O,(up.up),
7,163,(23,24,25),(△,O,△),(down,up,up),
7,167,(26,27),(X,O),(down,down),
7,173,(25,26),O,(down,up),
7,179,(26,27),(O,△),(down,up),
7,181,(32,33,(33,34)),O,(down,up,down),
7,191,(25,26),(O,△),(down,up),0.584,@before (0.584), p(131,137,139)=?(21,20,25), p(149,151,157)=?(19,27,(23,24)), p(163,167,173)=?(24,27,(25,26)), p(179,181)=?(26,(32,33,(33,34))),
---
[3][2]で、borderから0.584までに限定し、result='O'のみとする。
n,p,?,result,(up/down),(border/or/0.584),comments,
--
2,3,1,O,up,border,
2,5,1,O,up,0.584,
--
3,7,1,O,up,border,
3,11,((1,2)),O,down,0.584.
--
4,15,,,border,@!p(15),
4,17,2,O,up,
4,19,4,O,up,,
4,23,3,O,up,0.584,
--
5,31,6,O,up,border,
5,37,4,O,up,
5,41,6,O,up,
5,43,8,O,up,,
5,47,7,O,up,0.584,
--
6,63,,,border,@!p(63),
6,67,8,O,down,
6,71,10,O,up,
6,73,(13,14),O,(down,up),
6,79,11,O,down,
6,83,((15,16)),O,up,
6,89,13,O,up,,
6,95,,,,0.584,@!p(95),
--
7,127,14,O,down,border,
7,131,21,O,up,
7,137,20,O,down,
7,139,25,O,up,
7,149,19,O,up,
7,151,27,O,up,
7,157,(23,24),O,(up.up),
7,163,24,O,up,
7,167,27,O,down,
7,173,(25,26),O,(down,up),
7,179,26,O,down,
7,181,(32,33,(33,34)),O,(down,up,down),
7,191,25,O,down,0.584,
---
[4][3]から、各nの border < p(n) < 0.584 に対して、max(diff(p(n),p(?))を示すと、
n,max(diff(p(n),p(?)),comments,
--
2,na,
3,na,
4,4,2^(4-2),
5,8,2^(5-2),
6,16,2^(6-2),
7,34,2^(7-2)=32?
--
n=7をよく見ると、
7,181,(32,33,(33,34)),O,(down,up,down),
があり、同一pで、複数のp(?)がある場合、min(diff(p(n),p(?))を選択すると、n=7は、(33,34)ではなく、単独の32となる。
---
一覧を再考。
n,min(diff(p(n),p(?))),max(diff(p(n),p(?)),comments,
--
2,na,na,
3,na,na,
4,2,4,2^(4-2)=4,
5,4,8,2^(5-2)=8,
6,8,16,2^(n-2)=16,@(15,16)はセットもの。
7,19,32,2^(n-2)=32,
8,?,?,?<2^(n-2)=64,@@@(todo), @n=8,32<?=c(38,39,...,56,)<64, @see comments for detail.
9,?,?,?<2^(n-2)=128,@@@(todo), @n=9,64<?=c(68,69,,113,114,)<128,
10,?,?,?<2^(n-2)=256,@@@(todo), @n=10,128<?=c(138△?,139△?,(146,147),,,205,206,)<256,
...
---[5][4]から、生成ルールは、以下になるか？
m=c(1,2,3,,,), ex) 0.584962501..=log2(5+1),
(2^m.0)-1 < p(n) < (2^m.584962501)-1,
p(n)^2=p(n-1)^2+p(?)^2,
2^(m-3) <= diff(p(n),p(?)) <= 2^(m-2),
diff(p(n),p(n-1))=1,
---
[6]上記は、前半pに関するものだが、後半pは、前半pとペアリングルールを期待している。が、既に、[break : prime again (4)]で示したように、(0.584)の位置が、中央ではなく、count(前半p) > count(後半p)の関係があることから、これをどのように収拾するか、前途多難？/or/前半pは、ペアなしとなるか？
end.
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3 件のコメント:
How to use [The Nth Prime Page]. @備忘録,
[0]1000近傍の素数を求める。
[1][Pi function]に、1000を入力。pi(1000)=168.
[2][Nth prime]に、168を入力。168th=997.
169を入力。169th=1009.
[3]結果を一覧する。
nth,p,memos
168,997,1000=!p.
169,1009,
---end.
[4]@@@(todo)を処理する。
---
n=(8,9,10)の、min(diff(p(n),p(?))),max(diff(p(n),p(?)))を求める。
全部ではなく、ピンポイント(先頭と、0.584の手前の２箇所)で求めるので、
正確ではないかもしれない。
===
n=8,255<p<383(=0.584),
---
[1]先頭,255<near(p),
54th=251,
55th=257, @sel=this.
p=257,sqrt(257^2-251^2)=55.208,
16th=53, @(n-1,?)=(1,(55-16)=39), 2^(8-3)=32<39<2^(8-2)=64,O*all,up,result=O,
17th=59, @(n-1,?)=(1,(55-17)=38), 32<38<64,O*3,X*1,down,result=O,
---@s(251,53)=256.534,O, up={@s(251.4,53)=256.925,O, @s(251,53.4)=256.617,O, @s(251.4,53.4)=257.008,O,},
---@s(251,59)=257.841,X, down={@s(250.5,59)=257.354,O, @s(251,58.5)=257.727,X, @s(250.5,59)=257.354,O, @s(250.5,58.5)=257.240,O,},
---
[2]0.584の手前,near(p)<383,
74th=373,
75th=379, @sel=this.
76th=383,
p=379,sqrt(379^2-373^2)=67.171,
19th=67, @(n-1,?)=(1,(75-19)=56), 32<56<64, O*all,up,result=O,
20th=71, @(n-1,?)=(1,(75-20)=55), 32<55<64, O*2,X*2,down,result=△,
---@s(373,67)=378.969,O, up={@s(373.4,67)=379.363,O, @s(373,67.4)=379.040,O, @s(373.4,67.4)=379.434,O},
---@s(373,71)=379.697,X, down={@s(372.5,71)=379.206,O, @s(373,70.5)=379.604,X, @s(372.5,70.5)=379.112,O,},
---
n=8,32<?=c(38,39,...,56,)<65,
===
n=9,511<p<767(=0.583=!p),
---
[1]先頭,511<near(p),
97th=509,
98th=521, @sel=this.
p=521,sqrt(521^2-509^2)=111.175,
29th=109, @(n-1,?)=(1,(98-29)=69), 64<69<128, O*all,up,result=O,
30th=113, @(n-1,?)=(1,(98-30)=68), 64<68<128, O*all,up,result=O,
---@s(509,109)=520.540,O, up={@s(509.4,109)=520.931,O, @s(509,109.4)=520.624,O, @s(509.4,109.4)=521.015,O,},
---@s(509,113)=521.392,O, down={@s(508.5,113)=520.904,O, @s(509,112.5)=521.284,O, @s(508.5,112.5)=520.796,O,},
---
[2]0.584の手前,near(p)<767,
134th=757,
135th=761, @sel=this.
136th=769,
p=761,sqrt(761^2-757^2)=77.923,
21th=73, @(n-1,?)=(1,(135-21)=114), 64<114<128, O*all,up,result=O,
22th=79, @(n-1,?)=(1,(135-22)=113), 64<113<128, O*all,down,result=O,
---@s(757,73)=760.511,O, up={@s(757.4,73)=760.909,O, @s(757,74.4)=760.550,O, @s(757.4,74.4)=760.948,O,},
---@s(757,79)=761.111,O, down={@s(756.5,79)=760.613,O, @s(757,78.5)=761.059,O, @s(756.5,78.5)=760.561,O,},
---
n=9,64<?=c(68,69,,113,114,)<128,
===
n=10,1023<p<1535(=0.584=!p),
---
[1]先頭,1023<near(p),
172th=1021,
173th=1031, @sel=this.
p=1031,sqrt(1031^2-1021^2)=143.248,
34th=139, @(n-1,?)=(1,(173-34)=139), 128<139<256, O*2,X*2,up,result=△,
35th=149, @(n-1,?)=(1,(173-35)=138), 128<138<256, O*2,X*2,down,result=△,
---@s(1021,139)=1030.418,X, up={@s(1021.4,139)=1030.814,O, @s(1021,139.4)=1030.472,X, @s(1021.4,139.4)=1030.868,O,},
---@s(1021,149)=1031.814,X, down={@s(1020.5,149)=1031.320,O, @s(1021,148.5)=1031.742,X, @s(1020.5,148.5)=1031.248,O,},
^^^:anther one...
p(n)^2=p(n-1)^2+p(?)^2+p(?-1)^2,
26th=101,
27th=103,
p=1031,sqrt(103^2+101^2)=144.256, @(n-1,?)=(1,(173-27,173,26)=(146,147)), 128<(146,147)<256, O*all,down,
---@s(1021,(103,101))=1031.140,O, down={@s(1021,(102.5,101))=1031.090,O, @s(1021,(103,100.5))=1031.091,O, @s(1021,(102.5,100.5))=1031.041,O,},
---@s(1020.5,(103,101))=1030.645,O, down={@s(1020.5,(102.5,101))=1030.595, @s(1020.5,(103,100.5))=1030.596,O, @s(1020.5,(102.5,100.5))=1030.546,O,},
---
[2]0.584の手前,near(p)<1535,
241th=1523,
242th=1531, @sel=this.
p=1531,sqrt(1531^2-1523^2)=156.307,
36th=151, @(n-1,?)=(1,(242-36)=206), 128<206<256, O*3,X*1,up,result=O,
37th=157, @(n-1,?)=(1,(242-37)=205), 128<205<256, O*all,down,result=O,
---@s(1523,151)=1530.467,X, up={@s(1523.4,151)=1530.865,O, @s(1523,151.4)=1530.506,O, @s(1523.4,151.4)=1530.904,O,},
---@s(1523,157)=1531.070,O, down={@s(1522.5,157)=1530.573,O, @s(1523,156.5)=1531.019,O, @s(1522.5,156.5)=1530.522,O,},
---
n=10,128<?=c(138△?,139△?,(146,147),,,205,206,)<256,
===
ミスプリが多すぎた。４箇所訂正。
(1)n=8,[1],ダブりで削除。
NG:---@s(251,59)=257.841,X, down={@s(250.5,59)=257.354,O, @s(251,58.5)=257.727,X, @s(250.5,59)=257.354,O, @s(250.5,58.5)=257.240,O,},
^^^^^^^^^^^^^^^^^^^^^^^
OK:---@s(251,59)=257.841,X, down={@s(250.5,59)=257.354,O, @s(251,58.5)=257.727,X, @s(250.5,58.5)=257.240,O,},
---
(2)n=8,end,2^6=64,input mistak...,
NG:n=8,32<?=c(38,39,...,56,)<65,
^^
OK:n=8,32<?=c(38,39,...,56,)<64,
---
(3)n=9,[2],input mistak...,cal=OK,
NG:---@s(757,73)=760.511,O, up={@s(757.4,73)=760.909,O, @s(757,74.4)=760.550,O, @s(757.4,74.4)=760.948,O,},
^^^^ ^^^^
OK:---@s(757,73)=760.511,O, up={@s(757.4,73)=760.909,O, @s(757,73.4)=760.550,O, @s(757.4,73.4)=760.948,O,},
---
(4)n=10,[1],input mistak...,
NG:p=1031,sqrt(103^2+101^2)=144.256, @(n-1,?)=(1,(173-27,173,26)=(146,147)), 128<(146,147)<256, O*all,down,
^^^^^^
OK:p=1031,sqrt(103^2+101^2)=144.256, @(n-1,?)=(1,(173-27,173-26)=(146,147)), 128<(146,147)<256, O*all,down,
---
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