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平成 29年度 東邦大学理学部情報科学科 卒業論文

ゴールドバッハ予想における

組み合わせ数についての考察

学籍番号 5514028 氏名 小野澤純一

要旨

「4より大きいすべての偶数は 2つの奇素数の和であらわせるであろう」という

ゴールドバッハ予想とその類似問題について、数式処理システム Mapleにおける

計算機実験により、ゴールドバッハ予想に関する現象を観測した。

本研究では 2016年度卒業生である髙嶋一毅先輩の実験よりも、さらに大きな数に

おける現象について観測した。具体的に、偶数を２つの奇素数の和で表すことがで

きる組み合わせ数が何通りあるのかということをグラフに表し規則や法則を見つ

けようと試みた。またそれに関連して「７より大きい奇数は三つの奇素数の和で表

せるであろう」という弱いゴールドバッハ予想についても同様に検証した。その結

果前者については、新しい法則性を見つけることができた。

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目次

第一章 序論

第二章 二つの予想の検証

2.1 実験 1 ゴールドバッハ予想について

2.1.1 ゴールドバッハ予想とは

2.1.2 研究目的

2.1.3 研究方法

2.1.4 研究結果とグラフ

2.1.5 研究成果

2.2 実験 2 弱いゴールドバッハ予想について

2.2.1 弱いゴールドバッハ予想とは

2.1.2 研究目的

2.1.3 研究方法

2.1.4 研究結果とグラフ

2.1.5 研究成果

2.3 ゴールドバッハ予想と弱いゴールドバッハ予想との相関関係

第三章 まとめと考察・今後の課題

謝辞

参考文献

付録

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第一章 序論

本研究で扱っているゴールドバッハ予想とは、「４より大きい全ての偶数は、２

個の奇素数の和で表すことができるであろう」というものである。一見するととて

も簡単な内容に感じるだろう。しかし、数学界においてとても有名な未解決問題と

なっている。なぜ解決されないのか。そこには、無限という莫大な値の取り扱いの

難しさに関係している。小さな値の予想であれば容易に計算することが可能である。

しかしながら、偶数も素数も無限大に存在する。そうなれば奇素数の和も無限大に

存在するであろうことが容易に予想することができる。今回の研究では、ある偶数

χに対してこの予想が適応されているのかを Maple を用いて確かめた。しかし、

今回の作業において次の偶数χ+2 では、一からの検証が必要となってしまい終わ

りがない。よって、Mapleの検証は有力な検証方法の一つとはなるが証明にはなら

ないことが分かる。

Maple の検証の終わりとしては反例を見つけゴールドバッハ予想は間違ってい

るという形で結論付ける必要があるが、反例を見つけるのも至難の業だ。

ゴールドバッハ予想の解決には確率論（所定の素数の組み合わせパターンが存在

する確率）、古典的な背理法、リーマン予想方面からの解析論などがある。また、

一般化されたリーマン予想が正しいならば、弱いゴールドバッハ予想「すべての 5

より大きな奇数は、3つの素数の和であらわすことができるであろう」が導かれる

ことが知られている。

今回の研究では、主にゴールドバッハ予想について、計算機ソフト Maple を用いて

行った。内容は、昨年度卒業生である髙嶋一毅先輩の研究を拡張して検証するという

ものになっている。本稿の構成は下記の通りである。第二章でゴールドバッハ予想、

弱いゴールドバッハ予想についての研究を行い、実行結果やグラフ、二つの予想の

関係性を示す。そして第三章では、研究に対してのまとめと考察、高嶋先輩の予想

の真偽に関する知見を述べる。

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第二章 二つの予想の検証

2.1 実験 1 ゴールドバッハ予想について

2.1.1 ゴールドバッハ予想とは

すべての 4よりも大きな偶数は、2つの奇素数の和であらわすことができるで

あろう。

2.1.2 研究目的

実際にゴールドバッハ予想に関する現象を観測する。またそれに伴いある偶数

が 2つの奇素数の和で表す方法が何通りあるのかを調べ、グラフで表し、規

則・法則を見つける。

2.1.3 研究方法

数式処理システムMapleを用いてゴールドバッハ予想における計算機実験を

行い、その実行結果を Excelにまとめ、グラフに表す。

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2.1.4 研究結果とグラフ

以下、5002から 30000までの組み合わせ数を出力した①プログラム

②データとそれを表した③グラフである。

① プログラム

a := proc (n)

local i, c;

c := 0;

for i from 3 to prevprime(n) do

if isprime(i) = true then

if n-i <> 2 then

if isprime(n-i) = true then

c := c+1

end if

end if

end if

end do;

if 0 < c then

printf("%d %d¥n", n, ceil((1/2)*c))

end if

end proc

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② データ（5002から 30000までの出力結果）

付録 1に示す。

③グラフ（0から 30000までの出力結果）

0

100

200

300

400

500

600

700

800

0 5000 10000 15000 20000 25000 30000

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2.1.5 研究成果

今回の研究結果とグラフから以下のことが分かった。

① 得られたデータをもとにグラフを作成し検証した結果、大まかに 3つのグラフ

に分かれているのではないかと考えられる。以下その特徴を示す。

Ⅰ 3の倍数で組み合わせ数が増加しているグラフ

Ⅱ 30の倍数で組み合わせ数が増加しているグラフ

Ⅲ Ⅰ・Ⅱにあてはまらない組み合わせ数のグラフ

2016年度卒業生である髙嶋一毅先輩の研究では、Ⅰのグラフの特徴

についての記載はされていたが、Ⅱについては記載されていなかった。

今回の研究からⅡについて新しく見つかった特徴であるといえる。

② 得られたデータをもとにグラフを作成し検証したところ、グラフの概形から 2

つの仮説を立てることができた。

・グラフは単調増加になっているのが見て取れる。このことから偶数を

より大きくしても組み合わせ数は大きくなっていくと容易に予想する

ことができる。

・前述の通り、単調増加のグラフになっているが、組み合わせ数の増加

量に着目すると、一定になっていくのではないかと予想できる。このこ

とから偶数をより大きくしていくと、組み合わせ数の増加量は、何らか

の直線になるのではないかと予想される。

この仮説について、30000 までのデータでは傾きが一定になるとい

う確実な結果を得ることができなかった。この仮説を検証するには非常

に大きな偶数について検証する必要がある。その検証を行うには、研究

室内のパソコンでは膨大な時間かかかるため、現段階での検証は難しか

った。

実際に 300000までの偶数の組み合わせ数を出力するのにおよそ 34

時間かかった。このことからも、より大きな偶数について検証するには

膨大な時間がかかることが考えられる。

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2.2 実験２ 弱いゴールドバッハ予想について

2.2.1 弱いゴールドバッハ予想とは

すべての 5より大きな奇数は、3つの素数の和であらわすことができるであろう。

2.2.2 研究目的

実際に弱いゴールドバッハ予想に関する現象を観測する。またそれに伴いある奇

数が 3つの素数の和で表す方法が何通りあるのかを調べ、グラフで表し、規則・

法則を見つける。

2.2.3 研究方法

数式処理システム Mapleを用いて弱いゴールドバッハ予想における計算機実験

を行い、その実行結果を Excelにまとめ、グラフに表す。

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2.2.4 研究結果とグラフ

以下、10001から 15001までの組み合わせ数を出力した①プログラム

②データとそれを表した③グラフである。

① プログラム

gb := proc (n)

local i, c, j;

c := 0;

for i from 3 to prevprime(n) do

if isprime(i) = true then

for j from i to prevprime(n) do

if isprime(j) = true then

if isprime(n-i-j) = true then

if n-i-j <= i then

c := c+1

end if

end if

end if

end do

end if

end do;

printf("%d,", c)

end proc

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② データ（10001から 15001までの出力結果）

付録２に示す。

③ グラフ（5から 15001までの出力結果）

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 2000 4000 6000 8000 10000 12000 14000 16000

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2.2.5 研究成果

今回の研究から以下のことが分かった。

① 得られたデータをもとにグラフを作成し検証した結果、大まかに 2つのグラ

フに分かれているのではないかと考えられる。以下その特徴を示す。

Ⅰ 3の倍数で組み合わせ数が減少しているグラフ

Ⅱ Ⅰにあてはまらない組み合わせ数のグラフ

2016年度卒業生である髙嶋一毅先輩の研究において、組み合わせ数

が減少している奇数が３の倍数であると記載されている。この検証にお

いて全く同じ検証結果になった。

② 得られたデータをもとにグラフを作成し検証したところ、グラフの概形か

ら１つの仮説を立てることができる。

・グラフは単調増加になっているのが見て取れる。このことから奇数を

より大きくしても組み合わせ数は大きくなっていくと容易に予想する

ことができる。

実際に弱いゴールドバッハ予想は、2013年にペルーの数学者である

ハラルド・ヘルフゴッド氏によって証明され、すべての５よりも大きな奇

数が、３つの素数の和で表すことができることが分かっている。今回の

Mapleを用いた検証から得られたグラフからも、証明を裏付けるような結

果になったであろう。

2.3 ゴールドバッハ予想と弱いゴールドバッハ予想との相関関係

今回の検証から、それぞれの予想の間に何らかの相関関係があるのではないか

と考えた。以下、検証した結果である。

・どちらの予想も単調増加関数となったが、組み合わせ数に大きな隔たりが

あるため関係性については検証できなかった。

・どちらの予想も 3の倍数で大きく組み合わせ数が変化することが分かる。

しかし、ゴールドバッハ予想は３の倍数で組み合わせ数が増加するのに対し

て、弱いゴールドバッハ予想では、減少するため関連性は薄いのではないか

と考えられる。

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第三章 まとめと考察・今後の課題

今回の研究から以下のことが分かった。

ゴールドバッハ予想

・3の倍数で組み合わせ数が増加するが、その中でも 30の倍数で組み合わせ数が

さらに増加する。

弱いゴールドバッハ予想

・データ及びグラフの概形から、奇数を十分に大きくすると、組み合わせもそれに

伴って増加していくことが分かる。この結果は、2013年に H.A.HELFGOTTが発

表した弱いゴールドバッハ予想の証明が正しいことの裏付けとなっている。

また今回の研究から、以下の仮説を立てることができた。

ゴールドバッハ予想

・6から 30000の検証した結果、すべての偶数において組み合わせ数が存在するこ

とが分かった。この結果から、偶数を十分に大きくしても奇素数の和で表すことが

できる組み合わせが存在するのではないかと考えられる。

・グラフの概形から、グラフに接する接線の傾きは一定になっているのではないか

と分かった。この結果から、偶数を十分に大きくしていったときに何らかの値で傾

きが収束するのではないかと考えられる。

高嶋先輩の実験より実行範囲を拡張したことで、ゴールドバッハ予想の組み合わ

せ数についての新しい法則性を見つけることができた。しかし、ゴールドバッハ予

想についての証明に直接的につながるような法則性を見つけることができなかっ

た。

今後の課題は以下の通りである。

・ゴールドバッハ予想の新たな法則性を探るため、さらに大きな偶数で検証する必

要がある。また仮説の検証についても同様である。

・ゴールドバッハ予想の証明をするための手掛かりを探るため、新たな視点からゴ

ールドバッハ予想について検証を進めていく必要がある。

・ゴールドバッハ予想と弱いゴールドバッハ予想との関連性について、様々な視点

から検証を行い、その関連性から証明に結び付けられないか検証する。

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謝辞

本研究を行うにあたって、ご指導いただいた白柳潔教授に厚く御礼申し上げます。

また、東邦大学在学期間にお世話になりました多くの先生方にも、改めて御礼申し

上げます。

参考文献

THE TERNARY GOLDBACH CONJECTURE IS TRUE

著 H.A.HELFGOTT

数論〈未解決問題〉の辞典

著 リチャード・K・ガイ

プライムナンバーズ 魅惑的で楽しい素数の事典

著 David Wells

東邦大学理学部情報科学科 2016年度卒業研究論文

「ゴールドバッハ予想とその類似における組み合わせ数」

著 髙嶋 一毅

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付録１（ゴールドバッハ予想の組み合わせ数）

76,56,121,63,56,154,68,67,145,63,75,119,56,79,121,82,64,116

,63,70,184,59,66,115,54,80,117,76,64,122,87,61,123,61,77,17

6,57,58,118,51,83,159,64,61,118,82,66,120,77,68,167,56,72,1

30,55,101,127,65,61,122,76,65,148,64,53,167,61,65,121,69,84

,123,59,61,140,76,73,121,59,60,167,59,59,144,65,90,110,69,6

2,121,89,69,126,61,59,159,67,71,118,64,90,130,58,61,144,75,

63,132,63,60,160,73,61,131,56,85,115,63,86,119,84,64,139,57

,64,198,68,62,124,61,77,116,70,59,118,85,62,120,57,84,175,6

0,67,115,53,87,142,63,63,112,82,72,140,77,64,163,55,62,128,

58,107,122,69,64,126,86,64,138,69,69,166,65,63,138,70,83,11

4,59,73,122,82,77,119,61,70,166,60,64,143,57,81,132,59,68,1

24,102,68,132,64,68,156,63,75,128,63,86,141,58,61,139,75,67

,126,63,65,169,70,87,123,54,93,121,63,75,127,85,68,135,74,6

4,218,55,67,129,52,82,127,84,63,144,80,65,118,65,80,164,61,

67,126,60,98,149,62,68,127,88,70,130,71,73,172,66,65,134,54

,103,128,65,65,138,79,70,169,72,70,173,63,67,129,70,89,126,

68,75,127,81,79,133,69,71,177,60,66,163,65,93,124,67,63,124

,96,64,132,64,67,198,64,79,136,56,90,120,64,67,151,83,74,12

5,67,64,169,79,73,130,60,86,125,73,81,140,86,70,131,65,74,2

04,63,67,147,64,89,128,78,64,127,84,65,145,63,89,186,62,72,

132,64,87,168,66,61,127,110,66,130,77,69,174,58,73,127,61,1

03,143,71,79,132,84,76,162,71,69,172,64,76,136,73,87,144,61

,63,128,87,78,132,75,65,167,61,64,166,70,98,130,68,63,141,9

7,67,147,63,65,171,69,87,129,70,101,135,65,67,150,87,70,131

,76,74,191,91,68,131,66,89,128,62,80,131,87,73,157,74,69,21

6,67,74,140,74,98,127,79,74,132,89,76,134,71,87,182,64,74,1

53,71,96,161,66,65,146,87,70,139,79,73,202,67,73,133,55,107

,133,73,75,138,81,77,167,75,69,177,63,73,129,75,99,132,78,7

2,126,87,81,140,63,72,178,62,73,195,61,92,137,69,69,146,111

,66,144,68,75,173,74,84,126,65,91,151,61,65,161,99,72,140,7

1,77,177,76,68,132,59,98,147,68,84,138,93,70,153,72,67,222,

64,79,140,61,100,132,84,68,142,99,68,132,73,95,192,64,72,14

1,61,91,162,71,76,152,85,69,145,81,75,190,60,76,142,66,123,

154,77,71,138,81,69,163,69,71,180,72,73,132,84,91,136,69,66

15

,119,87,86,161,75,68,195,61,83,163,72,90,148,74,79,144,109,

77,135,77,74,198,66,89,140,75,95,146,68,77,159,98,73,147,72

,73,218,78,71,138,69,101,138,68,80,145,108,82,143,69,77,216

,69,82,137,70,101,138,96,69,150,86,77,156,65,93,185,70,75,1

50,61,98,161,76,81,142,95,72,140,80,88,194,71,80,144,61,130

,140,73,72,138,102,78,190,72,75,191,77,78,164,80,98,156,77,

69,140,96,92,146,71,71,184,72,76,187,76,94,138,74,74,150,11

1,67,148,74,79,183,69,91,138,73,107,145,66,76,188,95,69,163

,72,72,193,82,72,149,72,109,138,76,94,151,105,78,140,64,67,

241,74,75,148,66,94,145,84,79,152,97,82,166,77,83,190,69,75

,141,69,101,184,71,81,137,99,78,137,85,70,202,70,78,149,89,

117,142,75,73,145,91,75,177,73,76,211,70,89,138,85,97,149,7

9,70,142,91,105,149,73,71,218,66,72,176,69,92,142,86,69,138

,114,70,151,80,78,194,69,89,165,68,108,147,76,75,177,99,81,

148,77,83,187,83,69,153,75,104,145,77,89,169,105,78,149,72,

79,239,71,71,151,76,95,175,101,72,144,94,82,144,79,91,188,7

3,77,151,71,112,172,81,78,147,95,71,141,98,74,191,71,80,159

,70,122,151,81,73,162,100,81,183,79,82,195,79,78,146,88,107

,152,76,76,136,93,95,166,73,77,205,78,80,170,75,99,146,77,7

4,144,119,77,178,73,81,192,77,97,154,80,103,162,77,83,178,1

17,83,151,68,77,206,100,76,148,76,99,146,77,101,154,94,78,1

51,67,82,268,73,73,162,70,103,160,102,79,147,100,90,162,76,

94,205,73,77,157,77,115,186,92,73,145,96,79,151,95,78,206,7

8,82,161,72,119,150,83,84,155,96,83,180,76,87,231,72,77,157

,84,121,158,79,75,168,104,93,153,75,78,210,75,75,186,72,103

,171,77,75,168,113,89,162,81,79,198,71,108,157,75,108,152,7

9,75,195,105,83,159,93,82,204,91,82,150,67,100,154,79,94,16

1,107,84,166,71,89,258,72,82,154,79,123,147,87,76,145,99,82

,162,76,94,200,82,87,175,77,106,193,75,74,160,107,85,163,93

,76,198,78,75,147,77,126,152,78,95,150,105,80,189,79,89,206

,74,83,157,98,108,167,76,80,150,104,95,176,76,86,229,73,80,

183,80,103,152,78,85,153,135,88,152,77,77,203,67,102,160,81

,112,164,86,83,188,112,89,172,85,80,212,82,82,174,76,101,16

2,85,99,158,104,81,158,80,92,254,76,83,157,77,116,157,95,82

,155,106,87,155,74,106,217,75,81,156,79,108,205,81,79,147,1

04,84,151,98,79,236,76,86,154,77,131,150,78,80,155,99,81,19

16

1,98,82,210,74,73,167,98,98,157,79,77,176,107,110,157,77,74

,214,74,81,183,78,127,163,81,86,167,116,77,152,79,86,206,84

,98,160,68,103,155,91,74,191,107,83,194,84,79,209,81,81,158

,73,126,157,84,106,176,104,84,161,74,76,251,82,87,182,88,10

7,157,93,83,161,103,97,168,77,98,250,90,85,151,74,117,190,7

6,86,161,105,99,159,105,94,218,79,83,170,77,132,159,84,82,1

68,102,83,213,81,86,224,81,81,189,97,105,163,80,88,161,114,

102,166,87,86,211,83,91,204,78,110,165,77,99,158,142,81,169

,83,89,223,72,107,159,86,115,192,73,79,197,116,88,169,80,86

,217,87,94,162,87,119,183,78,99,168,110,86,166,86,86,267,80

,95,161,82,118,164,101,85,189,119,85,163,88,98,233,80,85,16

0,79,126,199,84,84,164,118,84,169,108,90,225,89,84,165,77,1

34,158,84,90,162,105,94,233,88,82,210,71,83,183,96,112,171,

96,95,178,106,104,159,79,94,220,77,86,208,87,113,159,94,84,

163,121,92,166,86,86,243,78,106,167,72,133,164,80,93,197,10

8,100,165,87,86,216,98,93,200,85,113,170,89,99,174,114,83,1

67,76,86,274,85,83,185,77,122,169,99,88,170,106,80,178,77,1

21,219,80,90,171,75,112,200,94,85,157,125,83,186,98,79,224,

70,80,169,72,140,182,82,92,175,110,89,198,91,83,212,82,101,

170,98,111,167,86,91,167,112,110,172,96,90,230,76,85,200,81

,116,173,82,87,184,138,91,176,91,90,227,87,108,172,82,134,1

85,78,89,208,113,90,157,83,87,238,104,93,163,83,125,157,91,

106,182,106,92,194,79,93,292,76,97,163,84,123,160,102,95,18

0,104,91,169,91,100,225,86,88,167,96,112,193,86,95,171,111,

97,171,105,81,251,88,82,177,81,144,175,86,94,190,100,101,20

7,94,93,238,83,103,174,98,114,165,106,98,166,119,107,180,81

,89,219,84,85,228,82,119,188,78,96,172,145,82,183,85,101,22

6,88,109,195,80,117,174,83,95,202,134,90,182,92,88,228,109,

83,172,76,117,184,86,100,163,118,90,169,86,108,271,80,97,17

7,85,132,160,102,86,182,111,88,186,97,107,228,91,86,188,87,

114,214,82,93,195,130,85,171,102,84,240,80,89,176,92,158,17

3,94,97,182,111,87,207,90,89,232,93,98,166,105,124,195,90,8

3,167,114,108,195,89,84,231,78,86,215,96,125,185,89,105,169

,139,85,175,92,87,247,75,114,187,100,126,171,81,96,226,113,

80,180,86,87,272,94,90,185,89,116,172,89,104,166,116,109,18

5,94,85,282,88,92,180,93,117,166,108,95,170,106,93,176,92,1

17

08,239,95,90,198,84,117,214,87,87,191,118,88,182,107,95,251

,86,92,174,83,142,171,94,91,173,125,100,229,89,90,242,96,97

,181,106,132,195,88,92,168,115,112,176,95,88,239,88,101,238

,99,136,182,93,96,173,138,86,180,104,95,244,88,111,175,90,1

14,195,91,96,244,115,91,182,95,98,236,105,92,177,91,125,179

,92,125,181,124,91,195,95,87,281,102,90,181,87,126,190,93,9

1,175,141,93,204,83,103,244,91,102,188,90,116,211,93,118,17

7,121,93,194,102,93,256,89,93,182,95,148,199,91,94,180,117,

88,218,93,95,262,96,93,184,114,125,178,93,95,182,120,125,18

2,95,92,246,87,101,226,90,129,178,104,95,173,141,90,191,96,

96,273,84,104,220,89,117,178,88,91,214,127,94,191,96,101,24

2,110,95,202,90,133,185,96,119,183,136,99,184,89,93,303,89,

93,191,92,126,203,122,96,210,120,102,177,90,115,247,98,105,

182,92,128,209,101,90,193,124,96,175,123,92,241,86,93,186,9

0,164,194,100,98,211,123,104,227,96,89,259,98,87,199,104,13

6,184,94,90,190,119,110,182,96,90,245,109,96,224,96,121,188

,101,98,176,150,95,211,100,102,252,87,116,189,97,122,183,93

,108,215,138,97,215,88,96,249,118,99,197,102,120,184,87,112

,179,133,96,189,89,94,329,96,95,204,98,129,181,116,107,186,

124,103,188,96,109,252,93,97,179,91,121,254,104,95,186,116,

97,183,118,93,261,86,87,209,107,155,189,91,105,189,121,97,2

33,107,107,249,95,100,189,123,129,182,94,94,203,148,113,194

,93,95,264,95,90,229,89,120,215,96,92,182,152,99,214,108,10

7,243,92,123,203,100,138,183,85,95,223,125,99,199,105,96,25

5,115,111,188,94,126,185,104,114,225,130,98,193,96,97,298,9

7,108,195,102,142,199,120,91,186,128,102,197,99,113,257,106

,93,213,88,145,224,94,93,187,135,100,209,117,97,253,97,105,

211,86,170,199,101,109,188,123,94,232,97,104,257,104,103,20

2,120,130,199,104,102,193,123,121,191,94,110,295,97,98,245,

90,132,194,93,94,199,159,109,221,96,97,261,77,127,197,91,13

7,190,117,93,234,135,100,194,93,106,264,112,97,212,91,126,1

91,93,124,202,123,105,192,99,114,324,101,110,194,97,140,220

,119,98,191,140,101,197,96,118,284,93,101,193,106,121,254,9

8,104,184,134,99,192,117,102,254,96,128,195,103,161,193,103

,101,196,133,101,235,113,93,286,108,99,194,124,135,185,98,9

0,219,132,117,191,108,99,260,103,99,230,98,151,205,102,110,

18

200,147,118,205,95,101,258,101,118,206,94,135,206,98,100,25

9,124,98,219,96,91,264,129,104,196,100,130,195,103,130,202,

144,104,204,86,99,316,102,104,208,110,156,200,117,101,196,1

46,103,214,96,118,301,98,102,211,94,134,233,100,109,223,141

,112,200,122,108,266,105,103,202,103,162,200,113,113,196,12

6,95,248,98,105,269,113,99,217,120,139,194,93,104,195,136,1

35,211,103,110,269,102,98,255,99,127,197,99,92,192,191,99,2

09,104,99,263,93,121,194,106,139,238,104,109,235,132,97,203

,105,102,256,113,109,190,99,131,218,112,119,198,137,92,198,

110,96,315,99,101,208,109,136,200,120,102,241,131,100,201,1

09,124,254,97,116,194,96,157,244,106,100,204,124,101,194,11

4,109,292,111,101,214,96,162,212,107,101,204,123,98,264,124

,112,271,102,112,194,122,138,199,107,118,200,130,136,196,10

6,105,294,98,101,234,111,132,216,100,100,213,161,118,206,10

3,96,304,93,135,203,102,141,203,113,104,247,136,117,200,103

,104,277,113,102,216,98,156,199,109,130,215,129,103,200,98,

101,330,110,108,245,107,135,218,130,115,214,133,109,202,102

,138,277,108,117,209,98,141,252,97,113,194,152,113,209,116,

114,281,103,106,206,94,170,231,98,98,204,141,111,291,97,103

,269,114,121,212,119,139,197,106,111,196,142,125,215,118,10

3,279,104,100,248,98,138,201,98,112,219,166,112,218,109,110

,279,98,127,205,107,164,230,100,99,245,129,112,206,103,106,

274,142,108,217,113,135,194,92,134,226,137,115,237,99,105,3

23,102,122,216,107,139,207,119,122,210,134,109,210,110,126,

302,104,116,205,110,153,255,92,98,199,135,104,205,135,106,3

04,100,111,211,96,172,212,108,105,208,141,130,249,100,98,26

3,103,110,214,121,145,225,107,109,241,137,119,218,96,103,28

0,95,102,290,102,144,192,108,106,200,169,113,216,103,116,28

0,99,129,216,105,155,212,106,105,254,155,113,204,111,118,27

1,121,106,216,97,138,228,103,128,203,138,113,212,105,101,34

0,114,115,229,106,149,200,120,116,210,148,113,205,119,139,2

81,105,117,208,99,141,251,107,102,239,145,107,232,125,112,2

85,96,110,198,102,192,205,111,113,217,142,112,256,100,102,2

87,117,112,201,123,156,237,102,112,214,133,126,231,111,102,

289,95,111,262,110,145,231,101,130,205,173,113,205,105,101,

277,104,127,211,135,140,217,107,105,255,145,111,208,99,105,

19

311,131,115,210,104,141,231,112,143,210,139,123,232,105,111

,362,111,108,216,109,138,205,139,108,208,137,109,225,118,14

4,288,106,107,230,97,140,266,112,107,210,133,123,226,125,11

8,282,115,104,219,104,170,200,108,111,242,153,103,240,108,1

12,287,106,107,231,120,151,231,115,109,203,139,135,210,109,

114,301,104,118,251,99,162,209,103,110,222,172,109,217,121,

103,291,106,135,241,105,136,220,111,118,275,141,120,224,113

,108,291,134,110,213,104,159,212,98,137,229,147,112,218,106

,120,350,121,122,220,106,147,209,122,104,216,145,120,263,99

,123,291,106,106,216,108,132,268,107,125,226,157,107,209,13

8,109,279,102,104,215,115,179,215,108,120,213,155,108,257,1

06,106,340,109,109,218,129,148,234,108,105,218,143,144,226,

102,119,286,102,117,266,118,142,218,124,120,214,174,113,216

,112,101,290,108,152,246,120,146,212,106,113,252,139,112,21

8,110,121,321,129,109,227,105,146,238,115,127,212,163,110,2

16,119,113,348,106,113,231,108,148,247,130,115,220,142,137,

217,109,133,292,109,125,227,114,141,256,112,111,233,155,111

,206,152,112,301,108,109,222,108,174,221,115,107,237,137,10

9,272,115,109,289,113,114,215,133,173,214,112,122,213,140,1

38,234,100,112,296,110,104,279,106,148,219,116,122,220,184,

112,256,116,109,288,122,131,229,113,157,223,105,126,262,148

,110,225,107,116,284,133,120,242,117,147,224,115,133,214,13

6,117,239,110,111,393,111,117,223,110,155,225,136,115,218,1

61,123,211,109,131,301,108,108,218,103,158,277,130,122,217,

140,113,218,127,115,300,103,104,237,113,182,240,117,118,234

,147,112,264,113,123,293,105,113,218,140,158,213,110,115,22

6,158,143,220,111,110,285,98,116,263,108,150,247,117,116,22

2,164,116,222,122,108,316,105,153,236,112,147,221,105,118,2

77,151,122,215,128,110,319,125,115,222,107,148,235,110,135,

254,152,131,220,111,115,350,112,122,230,105,160,223,135,113

,250,163,113,228,114,129,295,124,120,216,119,154,258,121,10

9,231,138,113,246,133,117,292,113,109,224,109,191,239,117,1

37,215,148,116,261,103,114,294,115,112,264,142,148,221,107,

116,241,145,138,218,111,114,327,121,108,270,112,168,225,121

,115,218,201,122,226,109,115,291,104,153,238,116,147,228,12

3,116,262,153,122,253,107,119,302,131,116,251,117,159,233,1

20

08,144,229,170,116,213,113,133,379,114,113,223,107,155,225,

135,123,232,166,113,236,108,144,302,115,112,246,116,152,328

,115,111,221,145,115,227,133,116,297,109,123,242,120,184,22

7,109,121,216,148,122,280,128,115,310,111,129,229,129,167,2

20,114,115,255,153,137,243,117,113,341,101,111,278,118,172,

215,118,112,231,178,111,227,129,117,302,129,134,250,120,157

,224,110,125,292,142,129,268,113,119,301,144,123,225,116,17

0,224,117,165,244,155,128,231,113,123,376,112,114,229,122,1

63,223,141,119,237,152,124,226,125,139,348,114,122,230,114,

180,278,113,121,229,146,135,222,150,118,324,108,121,242,106

,179,223,128,121,220,148,115,281,116,114,308,122,135,250,12

3,166,242,115,119,226,155,140,235,118,145,313,106,115,274,1

22,164,248,117,113,229,202,121,251,106,111,300,109,140,237,

115,163,268,115,124,278,167,123,247,115,123,306,135,135,231

,104,158,221,122,170,235,148,122,234,133,124,363,112,116,23

0,113,169,255,135,117,264,163,115,219,119,136,308,108,117,2

47,124,176,283,124,121,240,151,120,246,141,112,312,134,129,

232,112,192,240,124,129,232,158,114,302,121,123,344,121,116

,232,132,158,236,119,139,233,151,159,229,119,128,317,122,11

7,291,132,174,246,118,125,242,176,134,235,116,116,368,116,1

47,259,118,157,234,121,125,320,153,134,239,118,118,311,136,

122,240,112,175,229,144,162,236,149,124,238,118,122,365,112

,125,279,120,164,239,158,128,229,163,118,244,115,164,308,11

6,118,257,111,165,296,125,119,248,186,124,254,145,115,322,1

25,123,248,116,197,255,116,116,229,159,128,300,115,132,328,

124,140,253,142,158,251,112,115,238,158,147,254,140,124,334

,105,114,302,127,173,238,119,115,261,199,128,241,126,127,34

6,109,138,228,121,184,232,119,140,302,153,130,240,126,118,3

11,152,134,233,129,153,254,114,140,241,159,130,268,124,118,

392,107,124,230,145,157,223,144,143,232,152,128,258,122,148

,322,116,143,234,127,156,305,126,118,241,160,124,244,142,12

1,386,125,120,249,118,205,251,119,131,234,158,127,285,144,1

20,325,118,127,243,130,165,241,135,122,237,170,162,244,129,

122,328,119,118,316,116,160,237,118,128,256,188,126,262,126

,151,316,117,142,243,113,167,259,137,126,298,177,132,246,12

6,134,324,149,126,267,117,177,259,128,151,249,156,124,242,1

21

20,124,394,125,141,269,112,156,244,153,124,259,167,133,236,

138,152,344,131,127,258,112,168,286,126,129,266,172,135,232

,141,130,331,119,121,247,118,237,248,119,122,251,158,123,31

3,118,127,359,137,126,247,140,165,245,128,123,235,179,155,2

61,121,125,330,116,128,288,126,168,250,114,157,255,204,128,

251,126,125,322,129,149,257,133,172,264,121,125,292,173,138

,245,123,125,349,152,120,266,124,174,245,114,151,229,161,14

1,271,132,121,391,110,141,241,121,168,246,163,131,255,163,1

27,249,125,149,377,131,124,275,119,168,296,126,123,248,166,

126,253,164,132,321,116,136,252,127,218,236,122,120,231,171

,139,296,118,127,347,119,127,263,156,172,270,129,124,274,17

9,154,255,126,128,340,119,133,312,120,165,262,132,127,242,1

93,122,242,147,128,332,120,153,264,129,188,257,121,139,335,

165,127,256,127,125,334,139,135,270,124,179,261,124,149,240

,180,139,246,121,117,376,155,122,242,130,164,258,147,125,25

4,161,129,295,124,168,321,121,131,255,120,173,299,122,138,2

38,172,125,269,144,135,336,124,133,247,132,207,235,128,129,

242,169,133,318,129,136,370,128,127,258,139,162,263,127,141

,255,164,166,261,128,129,333,109,125,335,129,181,281,132,12

0,255,187,123,242,134,128,330,116,152,288,128,182,249,132,1

24,297,169,129,240,139,142,335,145,142,248,123,175,257,133,

153,244,182,124,251,124,122,433,109,125,254,123,166,279,150

,127,261,156,128,250,143,147,366,114,144,263,124,176,298,13

3,134,256,166,156,251,169,140,344,130,130,253,124,196,237,1

37,125,310,160,129,313,132,131,338,127,124,252,150,184,246,

143,151,249,158,165,271,124,128,363,135,129,312,120,187,255

,119,134,251,206,127,302,131,128,357,123,168,283,122,170,25

3,122,140,305,169,139,253,127,140,348,172,133,256,157,170,2

40,126,162,250,165,130,253,121,136,446,129,130,258,130,187,

268,150,129,248,163,141,272,137,161,343,115,132,264,125,162

,292,138,146,268,182,141,252,155,133,371,122,131,280,117,20

8,255,117,156,270,181,133,315,136,145,356,122,134,263,156,1

74,282,122,133,256,190,159,289,126,139,333,121,136,311,142,

173,281,119,131,248,196,133,246,131,130,333,134,189,266,131

,180,277,127,132,312,176,126,261,147,132,366,155,126,267,11

7,178,235,129,152,282,181,133,251,140,136,408,122,137,288,1

22

21,193,253,148,127,264,164,141,271,135,169,350,143,143,258,

123,186,306,129,131,277,165,135,283,157,126,339,121,134,283

,117,217,249,145,146,260,167,136,308,136,136,356,128,129,25

9,169,190,261,132,140,270,163,154,259,134,130,379,121,141,3

35,121,178,256,137,133,248,207,168,278,122,130,336,132,165,

257,127,175,254,140,131,316,168,129,258,142,151,355,144,130

,286,123,188,256,131,155,267,175,131,273,122,149,433,122,13

2,260,140,166,257,149,126,251,202,126,255,136,164,346,119,1

37,259,121,165,336,135,157,255,168,134,266,152,138,344,132,

149,262,126,206,293,130,126,247,171,138,325,135,133,342,132

,132,254,161,177,279,144,131,285,172,161,258,135,138,344,12

9,137,311,132,200,256,130,147,263,203,138,261,131,134,382,1

42,155,266,118,188,280,141,130,310,183,131,295,139,135,334,

154,130,266,125,174,264,133,177,277,175,145,282,123,133,425

,122,133,257,143,181,253,158,142,299,162,135,270,134,168,39

5,142,134,262,128,174,317,143,147,261,165,152,266,162,125,3

52,139,153,261,128,231,253,151,141,260,182,126,315,131,128,

347,139,133,343,153,179,265,132,136,262,169,161,259,140,146

,340,141,126,332,142,196,254,119,141,276,240,134,270,129,14

3,345,135,161,267,120,186,309,129,134,317,180,135,306,133,1

41,347,150,153,268,122,182,253,125,161,286,193,135,262,136,

134,420,125,135,287,127,194,256,162,150,297,179,142,257,141

,169,347,127,138,267,124,205,348,144,142,256,170,131,268,18

1,136,360,148,135,271,141,212,265,127,141,266,178,141,349,1

40,136,402,136,151,269,162,186,281,127,157,250,170,172,268,

137,144,399,121,143,317,151,184,258,134,144,251,213,138,279

,146,145,408,122,166,268,151,184,263,130,134,319,176,167,27

6,124,146,357,160,149,272,138,180,294,165,180,293,173,133,2

85,132,143,439,126,138,305,129,174,266,180,142,257,168,138,

295,136,187,356,137,138,279,124,213,325,128,139,264,201,131

,286,159,137,353,134,134,298,128,221,299,132,147,266,179,14

3,340,145,133,355,141,155,290,164,196,266,131,145,265,174,1

79,265,142,161,366,123,132,323,128,181,276,129,137,306,216,

141,295,130,144,361,141,172,279,132,194,269,131,145,311,190

,147,277,141,160,358,170,133,305,130,176,261,142,184,258,17

7,140,310,128,143,431,126,159,276,137,195,291,168,160,280,1

23

74,145,275,130,164,359,127,137,268,157,188,351,144,146,269,

182,134,273,167,134,424,124,156,266,132,238,277,137,137,296

,186,154,325,140,143,383,133,139,278,149,181,280,147,142,27

7,179,174,274,144,148,362,128,142,371,128,188,285,139,145,2

76,216,152,274,139,149,366,140,171,299,142,189,266,146,133,

361,200,140,284,135,144,395,161,140,274,133,190,309,161,172

,286,173,154,279,136,135,447,129,172,277,138,210,265,169,15

4,279,184,149,264,148,169,368,132,150,315,135,196,331,141,1

44,304,185,139,274,164,140,396,148,145,272,134,244,297,134,

146,280,200,136,341,137,156,362,154,145,278,167,195,283,140

,146,283,181,157,333,131,139,367,144,135,354,149,191,263,13

1,162,262,250,143,289,134,145,374,136,175,287,153,181,287,1

33,160,330,192,149,309,135,142,417,164,143,283,141,191,295,

138,179,267,188,167,270,129,142,466,134,147,277,149,198,277

,183,140,271,181,146,278,143,174,374,145,153,317,140,206,33

7,134,149,272,177,137,290,166,163,405,141,148,297,130,230,2

83,141,142,281,202,147,350,143,145,380,139,148,281,155,199,

321,134,144,286,184,189,278,154,137,384,145,157,342,137,192

,265,149,148,304,207,137,282,165,154,384,136,178,288,143,19

0,289,151,144,398,183,138,284,139,151,374,166,148,279,132,2

33,301,144,175,293,185,148,289,150,145,421,153,158,321,137,

196,287,165,142,295,197,149,313,148,182,385,145,145,283,125

,199,340,144,158,284,186,148,289,166,168,410,144,149,273,14

5,242,284,141,152,287,180,148,384,143,143,417,142,156,305,1

66,191,278,138,144,293,199,195,280,140,139,396,136,148,339,

149,192,283,159,152,281,231,153,292,141,142,375,130,171,305

,138,200,329,142,156,348,194,152,275,138,161,373,165,148,29

6,150,202,292,147,188,293,219,135,288,141,142,477,148,162,2

90,146,194,306,168,149,275,202,145,301,134,167,418,135,165,

303,149,191,334,143,149,306,190,146,311,193,152,384,137,138

,283,145,232,289,148,158,319,182,160,340,146,153,377,143,15

4,276,175,216,280,152,145,303,192,174,322,141,152,379,167,1

45,335,137,186,282,150,141,278,228,146,331,151,153,383,147,

179,310,147,221,290,144,159,344,186,151,289,140,153,376,157

,160,337,163,219,290,150,173,286,188,145,287,148,148,517,15

7,156,287,133,199,306,183,152,306,177,170,288,154,190,384,1

24

38,150,290,138,189,343,158,148,302,188,133,311,176,146,379,

134,151,326,139,253,280,142,152,290,198,155,357,147,165,381

,142,153,300,165,212,286,135,158,288,214,169,316,155,145,38

4,137,141,351,145,191,351,148,154,291,235,151,288,146,147,3

95,139,203,276,147,216,275,146,154,349,207,149,284,165,152,

386,164,173,300,140,202,286,141,182,330,184,145,294,155,156

,507,141,151,285,148,223,284,175,144,286,191,150,303,155,16

3,386,153,164,305,144,201,377,143,144,301,198,151,318,181,1

51,385,135,154,288,134,244,286,156,181,332,182,144,332,151,

157,394,151,151,282,186,195,296,157,156,296,204,197,296,150

,146,431,147,144,356,138,222,287,150,153,289,231,162,313,14

3,141,419,141,172,333,133,199,289,161,150,351,198,148,309,1

38,165,407,180,156,324,142,196,275,140,185,293,194,141,299,

136,189,459,161,151,291,141,198,284,163,153,285,221,142,322

,149,179,397,132,158,323,141,193,381,139,148,296,213,153,30

5,181,147,396,149,175,294,133,246,285,148,162,291,190,145,3

44,163,157,385,155,158,305,173,196,319,149,147,341,189,184,

300,150,147,401,149,149,352,155,238,279,145,148,319,236,159

,304,146,145,397,150,210,294,153,195,296,151,153,373,193,16

2,334,150,155,428,166,148,286,141,203,280,140,203,293,190,1

48,302,177,160,469,152,148,297,152,196,286,171,148,293,189,

158,306,141,176,460,149,161,316,147,193,353,145,150,314,191

,162,300,182,141,394,133,166,296,141,242,313,174,146,305,19

1,156,353,148,155,392,143,154,328,181,219,293,136,158,297,2

07,182,318,144,161,393,144,150,397,143,208,316,150,161,284,

264,161,307,148,164,391,151,186,288,151,206,329,145,157,349

,185,149,326,147,168,393,177,180,311,139,199,302,152,176,29

4,215,149,319,153,144,477,134,160,298,148,214,301,189,153,3

18,210,150,314,153,185,408,142,152,292,140,237,361,162,168,

303,207,158,298,186,165,386,162,159,291,144,248,322,142,150

,292,191,151,413,150,149,415,153,155,292,196,215,325,154,16

7,300,200,179,304,153,152,404,140,169,372,180,206,281,146,1

47,284,231,150,296,148,157,480,138,186,304,153,221,301,150,

168,348,194,169,302,162,159,394,174,158,309,142,209,295,189

,181,305,195,161,321,148,152,497,149,150,346,140,196,284,17

7,151,348,191,152,314,144,200,426,151,164,313,141,200,361,1

25

66,155,303,216,166,305,170,152,391,151,160,308,142,276,317,

147,162,302,199,157,360,162,149,395,143,171,333,170,194,287

,161,155,322,197,187,303,171,159,428,173,155,361,146,204,29

8,146,156,326,241,167,302,166,174,404,153,199,299,149,229,2

99,159,155,368,218,162,323,145,156,406,195,166,315,151,207,

301,140,182,302,230,156,350,147,153,490,142,165,297,151,200

,316,179,191,313,201,143,306,161,191,415,156,156,304,168,20

2,392,149,159,299,213,153,318,194,152,457,142,161,310,154,2

47,304,154,158,300,218,164,386,172,147,403,152,174,321,181,

202,303,169,157,307,203,190,335,145,162,445,143,141,433,157

,206,295,160,148,305,238,162,319,165,186,399,154,198,300,14

7,200,301,160,158,355,218,165,312,153,166,404,198,166,309,1

61,201,345,160,187,347,199,165,305,143,167,489,158,174,332,

155,204,295,203,154,299,196,152,321,178,188,452,161,160,312

,146,231,357,153,165,337,219,158,309,175,155,417,148,151,33

4,152,277,311,164,166,293,199,153,358,155,156,411,184,163,3

12,185,214,298,145,161,343,205,179,346,154,173,422,156,152,

375,152,214,305,143,162,308,249,168,346,148,156,414,152,199

,307,169,201,327,148,161,379,219,163,296,162,152,453,195,15

5,303,146,202,312,161,198,291,195,177,311,156,166,530,154,1

70,322,158,219,323,211,149,297,188,146,311,157,181,413,144,

163,335,174,194,365,168,153,322,206,144,317,176,178,415,151

,188,322,163,256,317,156,154,312,226,149,376,163,164,467,15

5,149,310,186,207,360,154,150,321,209,185,321,176,163,409,1

53,193,381,165,231,320,154,172,321,241,171,313,170,157,419,

153,186,315,153,209,331,150,159,444,219,145,311,163,151,399

,197,162,304,165,236,318,176,197,315,211,154,336,165,154,49

6,177,160,318,155,232,301,190,165,328,212,151,351,157,207,4

19,152,160,345,152,210,376,168,173,308,209,154,324,185,159,

424,184,162,330,173,255,314,152,163,318,219,169,375,162,177

,449,157,167,345,178,221,313,157,165,308,232,207,346,159,16

3,423,149,164,380,155,228,314,173,165,323,270,166,339,159,1

64,414,158,187,363,143,209,320,160,165,377,205,157,310,170,

177,449,184,152,311,168,210,342,158,182,334,234,164,308,152

,155,516,154,162,315,192,214,346,184,163,312,213,165,314,16

7,192,416,156,186,321,175,203,384,147,179,338,207,160,315,2

26

29,163,471,154,162,319,161,251,322,160,152,348,214,164,421,

178,170,458,157,153,310,188,229,311,167,161,318,211,209,325

,165,159,418,178,165,373,163,224,322,162,166,357,243,159,37

1,156,162,413,160,206,332,167,231,313,171,177,373,210,161,3

46,179,160,432,191,163,313,171,226,332,167,190,352,214,157,

318,165,155,571,165,179,349,152,214,327,192,170,327,214,187

,324,182,190,411,170,161,321,159,206,382,178,169,323,227,16

3,319,188,182,427,163,172,363,159,256,314,157,161,324,223,1

52,447,150,175,433,163,158,337,183,216,323,170,162,321,259,

187,321,161,168,450,152,159,387,161,217,349,166,166,331,272

,164,325,161,164,442,168,208,320,155,216,351,160,155,404,21

6,158,328,186,169,422,200,153,344,173,208,317,163,195,366,2

18,152,325,160,160,520,158,185,331,155,255,330,205,160,306,

215,153,322,163,192,460,181,165,311,152,214,377,184,158,310

,208,152,359,216,167,431,159,166,302,157,272,334,170,176,35

6,199,186,388,161,172,416,160,160,344,215,219,328,169,165,3

64,216,214,329,161,162,492,164,160,396,161,215,324,178,158,

321,250,189,337,176,158,423,154,198,324,170,241,321,176,164

,380,222,151,321,154,160,460,191,155,385,159,210,317,154,20

6,317,220,166,329,159,184,516,167,186,337,161,224,316,192,1

60,316,241,156,338,159,214,422,162,162,306,167,218,429,169,

159,300,210,179,359,184,167,440,166,182,334,158,265,353,159

,168,326,229,165,375,182,173,434,159,160,329,208,230,318,16

3,190,339,208,186,326,163,174,428,151,168,395,161,250,355,1

63,156,331,267,172,334,160,157,421,175,190,323,176,216,307,

169,164,384,212,167,372,173,158,438,196,173,327,154,231,331

,159,226,311,210,168,320,173,174,559,157,173,332,175,212,31

6,197,162,319,214,167,317,178,196,486,161,166,321,156,203,3

97,164,161,342,213,204,316,201,162,424,164,170,347,165,261,

343,170,161,362,225,160,379,164,172,429,157,168,364,185,228

,319,169,170,324,223,195,336,169,193,421,163,165,392,155,24

4,305,176,150,324,278,166,326,163,155,424,162,213,375,155,2

25,368,162,164,425,231,170,328,164,166,429,224,187,322,154,

211,323,165,200,328,215,177,343,178,176,539,155,163,324,155

,223,320,196,156,366,206,176,366,168,210,447,171,162,331,16

3,242,399,163,167,324,253,169,327,195,153,491,175,165,311,1

27

59,265,323,158,174,330,214,171,437,171,167,434,162,174,316,

216,218,364,157,177,323,236,202,348,172,166,435,164,157,381

,207,228,335,164,174,346,258,181,323,161,164,489,154,221,33

1,166,222,353,163,166,401,225,190,362,161,167,480,188,178,3

27,170,209,327,193,207,326,234,168,333,180,169,537,176,164,

383,166,221,320,203,169,331,239,188,335,169,226,451,164,176

,325,161,237,408,168,162,358,245,165,361,198,173,438,168,17

5,350,168,264,363,181,162,357,229,168,396,172,173,449,159,1

85,339,192,252,320,164,165,335,213,202,326,189,167,434,177,

170,421,156,222,346,167,166,371,256,174,340,159,170,443,176

,212,347,171,245,333,170,157,397,213,168,332,172,179,441,21

9,172,347,163,242,350,169,197,336,225,171,418,162,166,551,1

68,174,320,166,223,319,207,189,340,248,172,336,170,200,445,

159,168,328,194,241,389,170,184,360,224,157,327,214,180,489

,167,167,329,154,271,365,165,171,328,220,207,418,178,167,43

0,173,168,367,215,221,337,183,165,344,218,197,331,177,160,4

47,161,177,446,159,229,340,181,165,330,279,165,326,169,187,

526,157,216,333,166,225,342,161,168,389,247,184,321,185,178

,458,192,183,339,175,210,371,173,194,339,212,189,339,170,16

7,558,165,183,338,175,217,329,214,174,357,225,165,330,187,2

13,453,164,163,335,180,219,434,191,167,365,231,176,337,201,

180,477,166,174,335,164,329,337,173,203,336,216,178,408,166

,169,443,196,181,371,192,236,333,172,175,330,239,203,405,17

3,176,458,179,167,405,168,223,335,167,182,345,266,174,356,1

72,177,460,186,203,341,185,226,335,173,159,411,225,181,373,

170,182,512,209,169,345,163,237,338,174,209,366,257,188,335

,167,176,527,181,175,334,155,220,352,231,188,368,219,168,36

1,178,206,455,169,172,392,167,240,440,173,172,332,239,179,3

39,218,192,455,169,177,337,175,305,340,170,167,333,259,177,

412,181,165,466,171,201,338,208,221,375,187,162,339,226,200

,364,176,177,497,163,181,427,177,224,365,169,169,351,283,17

4,342,212,166,450,163,204,329,165,232,345,161,166,459,232,2

00,339,180,176,467,194,181,344,155,266,340,184,199,389,220,

178,352,175,187,541,184,178,345,171,239,349,219,171,338,214

,161,391,184,211,477,172,174,366,164,244,404,174,200,343,22

5,168,362,208,171,454,168,195,367,181,274,345,175,176,341,2

28

29,178,411,167,180,524,183,172,343,223,248,343,171,162,344,

234,239,348,176,187,447,167,180,416,177,233,357,184,169,328

,279,177,378,183,176,483,172,217,386,165,218,363,174,186,41

5,255,166,343,175,193,464,204,178,340,178,227,364,177,227,3

48,261,179,346,166,178,554,166,184,349,175,233,425,205,183,

331,228,168,374,181,216,471,169,186,338,195,232,403,170,172

,339,227,178,372,226,192,464,157,188,353,177,273,347,185,17

2,376,219,165,422,176,177,495,202,179,350,206,249,353,171,1

74,344,220,211,353,187,182,480,188,166,467,184,245,336,167,

170,338,273,189,393,190,167,451,171,201,348,165,247,344,179

,211,457,238,180,347,177,179,460,206,183,359,182,248,337,21

1,209,357,231,178,343,182,168,615,182,187,357,167,260,333,1

95,175,371,230,193,341,181,209,453,168,183,391,172,224,426,

195,176,360,232,186,342,204,175,454,186,164,381,174,276,368

,169,179,363,235,172,419,184,209,457,182,191,345,198,242,34

0,172,166,347,272,221,390,171,175,497,175,171,410,177,233,3

75,170,183,353,314,182,363,173,190,456,177,250,339,163,241,

343,163,194,420,231,169,352,199,177,466,204,171,387,182,235

,391,181,206,396,227,177,344,186,173,547,171,191,344,190,26

7,369,223,182,360,234,184,357,190,221,467,187,194,343,172,2

49,440,174,187,352,233,179,391,216,173,490,168,179,344,182,

281,348,174,182,339,254,175,425,191,171,502,179,178,352,234

,233,351,173,180,347,229,242,391,172,174,521,179,171,432,17

9,226,346,183,182,368,279,201,351,170,191,476,181,216,369,1

77,255,349,217,191,416,248,186,349,177,179,481,219,174,388,

164,248,346,178,214,363,229,174,385,186,198,573,189,186,414

,176,223,356,215,178,353,254,177,346,174,225,491,204,176,35

4,174,245,482,175,170,348,237,170,357,214,193,472,181,199,3

48,160,297,342,195,180,353,240,170,458,199,191,475,179,179,

365,198,232,356,175,175,435,248,209,353,177,194,488,165,175

,429,180,266,354,195,199,353,280,196,349,191,180,450,195,21

1,369,178,240,383,164,170,420,232,182,391,200,182,503,208,1

77,348,191,240,378,175,234,355,227,191,366,180,183,605,179,

200,349,194,236,378,203,190,361,241,181,355,178,214,573,176

,193,357,181,255,422,173,177,337,264,199,350,234,179,476,18

2,189,357,190,281,348,186,182,355,239,195,430,184,195,471,1

29

73,176,391,218,226,379,174,181,403,226,219,375,172,198,478,

190,173,454,174,243,358,200,193,354,318,188,361,180,173,474

,175,227,394,172,266,412,188,189,420,254,180,371,191,190,47

0,221,209,388,181,245,362,185,224,356,254,200,389,198,191,5

78,196,187,364,181,244,357,217,177,405,237,174,362,189,244,

502,182,185,375,180,261,436,179,182,349,228,180,394,221,183

,475,206,185,353,182,289,356,178,184,361,252,188,476,194,18

4,467,173,186,381,209,250,349,176,224,349,249,218,362,182,1

82,506,195,187,458,217,256,398,180,177,360,296,185,392,178,

185,527,183,222,366,184,237,369,187,171,424,239,195,369,191

,192,473,232,193,349,169,248,350,202,215,390,241,178,355,18

2,202,635,188,193,413,177,277,360,220,177,362,232,188,364,2

10,250,482,171,191,371,177,237,426,185,179,391,261,192,374,

242,175,473,179,184,347,178,293,399,180,186,384,232,190,445

,198,177,541,177,209,355,223,237,383,195,181,384,236,229,36

1,202,179,487,175,207,433,178,272,363,194,188,415,300,193,3

73,185,186,489,173,221,396,193,272,395,181,190,441,249,173,

365,187,184,481,263,182,351,174,239,380,184,213,374,267,181

,418,183,201,568,187,188,370,179,254,350,217,210,358,249,18

7,415,172,235,507,169,181,366,208,248,433,178,178,384,278,1

93,363,216,174,553,188,192,392,176,303,373,183,217,358,248,

215,494,201,172,495,187,187,385,219,249,396,221,186,355,257

,229,372,192,191,479,180,189,500,191,251,366,191,172,375,28

4,191,360,189,203,506,200,252,362,183,241,384,183,178,454,2

79,185,364,191,176,565,213,176,389,178,250,403,185,222,356,

240,174,376,202,193,591,187,226,366,187,254,368,235,197,380

,224,210,364,198,232,498,177,193,377,188,245,435,207,186,44

7,253,180,364,212,190,515,183,195,370,181,333,370,204,185,3

89,249,189,469,186,186,486,208,187,392,206,286,353,173,185,

366,242,239,406,182,189,493,198,198,497,193,250,357,181,210

,362,290,198,373,189,190,501,199,225,372,198,266,380,171,20

1,450,253,194,374,192,199,573,219,193,389,186,247,378,184,2

15,395,241,203,402,187,198,601,189,203,361,179,246,360,252,

199,359,260,184,383,199,223,502,183,199,424,193,241,450,186

,195,358,245,187,364,221,206,490,182,196,366,183,311,430,17

8,191,372,274,180,449,186,179,481,188,187,363,240,258,440,1

30

80,197,393,235,232,370,197,187,506,185,226,435,185,243,365,

187,193,363,334,186,381,203,184,553,175,223,376,182,261,398

,177,192,474,251,188,410,212,194,491,210,189,373,181,283,36

5,192,210,377,237,208,368,177,222,596,216,185,388,178,247,3

86,227,186,403,243,184,417,178,231,493,199,207,376,191,252,

456,228,201,372,253,186,371,222,191,490,185,194,393,202,338

,386,184,189,388,246,205,444,191,184,580,201,200,410,226,26

9,370,186,191,384,261,261,394,199,189,523,196,186,452,181,2

53,364,228,202,372,285,183,377,188,213,497,180,240,423,186,

255,379,189,179,484,250,195,402,184,212,479,234,190,392,178

,258,380,193,245,391,295,201,380,181,192,611,188,196,385,18

4,252,413,209,198,389,251,193,407,195,231,494,180,217,372,1

83,260,478,186,197,367,265,190,384,240,186,535,179,204,353,

196,308,387,189,199,433,229,186,437,196,200,496,190,221,369

,218,303,372,177,188,372,258,240,387,180,198,564,193,197,47

4,204,255,372,186,194,386,300,189,437,199,190,496,190,228,4

02,198,258,414,192,214,442,259,201,393,193,198,503,237,212,

374,207,257,377,197,247,427,242,204,383,183,193,671,192,205

,362,192,266,385,233,184,379,261,204,399,193,231,510,191,20

5,380,195,276,493,219,196,387,246,191,392,232,188,514,186,1

93,466,175,302,360,187,202,370,266,185,484,188,241,507,211,

195,379,218,274,374,176,205,384,273,231,406,186,233,514,198

,190,452,193,275,428,188,202,383,300,189,416,188,196,511,18

7,263,435,190,249,375,184,186,444,275,201,385,218,198,506,2

28,198,374,185,270,378,201,247,415,254,197,383,199,199,625,

200,198,397,185,301,432,235,190,370,252,205,384,193,246,514

,196,202,378,199,252,461,193,202,406,259,181,421,220,206,55

0,179,225,378,190,317,387,192,214,375,248,206,468,189,201,5

40,205,200,396,260,267,375,189,189,373,245,241,396,215,198,

565,190,189,470,189,254,374,184,200,393,302,234,371,179,198

,571,181,235,379,187,256,390,211,198,504,243,201,413,190,20

4,513,229,204,424,174,262,384,207,230,388,242,191,381,186,2

10,603,188,202,380,197,283,390,225,188,370,277,187,384,189,

243,517,203,197,416,182,288,530,199,194,383,252,197,376,233

,197,522,199,222,376,177,316,382,210,209,439,254,194,462,22

9,213,506,203,202,380,222,269,400,190,195,427,261,227,413,1

31

97,210,501,193,211,459,188,296,397,193,192,381,330,192,435,

197,206,519,207,250,386,170,269,390,196,211,476,272,202,426

,190,216,518,230,197,401,188,264,432,192,254,397,250,201,38

4,179,197,625,207,196,409,217,263,385,228,204,375,274,196,4

08,185,240,571,194,215,411,191,265,457,198,199,394,287,226,

376,225,208,561,192,197,385,188,333,388,210,195,398,244,197

,463,210,206,506,190,199,462,229,262,381,199,210,411,255,25

5,386,190,218,518,195,200,468,184,262,409,197,185,453,354,1

89,412,198,215,515,189,238,378,187,273,437,215,205,471,263,

226,393,186,205,528,238,214,389,194,293,373,185,237,393,260

,193,401,228,211,670,181,211,417,193,263,388,237,203,435,25

1,203,387,200,246,569,212,193,396,215,300,453,194,195,387,2

50,209,412,231,219,518,215,211,395,177,311,425,190,212,401,

267,196,560,206,190,510,194,212,398,243,263,375,198,222,378

,292,238,428,201,205,521,207,206,479,214,275,421,199,210,38

4,313,202,393,189,200,571,212,242,399,187,263,425,205,217,4

83,265,223,385,199,209,520,231,209,399,218,284,384,204,241,

398,279,197,404,186,201,641,192,222,455,190,260,377,244,200

,423,253,207,428,204,281,604,204,200,410,198,283,457,196,21

1,405,290,194,404,262,215,537,193,205,387,195,316,452,200,1

99,387,260,210,480,204,202,535,235,242,388,235,267,416,201,

206,436,258,241,403,224,214,520,205,197,492,196,262,415,223

,195,459,304,192,382,195,201,520,196,243,388,200,351,404,20

2,187,471,286,195,382,203,201,503,269,223,437,189,273,391,2

09,256,397,261,192,459,191,211,642,210,200,408,189,270,384,

234,217,423,273,203,392,196,274,532,192,202,429,209,268,480

,196,199,404,253,200,440,235,206,588,196,208,408,200,317,39

1,211,214,393,283,216,471,221,202,535,196,195,385,222,273,4

10,223,223,386,244,261,384,205,207,575,199,200,534,197,283,

449,193,204,397,311,201,408,207,222,572,187,249,416,224,262

,416,182,203,485,297,181,418,200,205,543,224,223,386,206,27

0,443,201,242,402,263,183,420,194,207,690,184,226,397,202,2

75,412,234,212,406,258,216,392,232,251,543,202,196,393,200,

283,476,208,217,442,264,214,413,234,214,527,193,202,408,187

,349,412,200,198,435,286,193,494,214,206,531,239,199,383,23

3,270,384,215,202,390,245,254,466,199,192,532,198,196,485,2

32

00,303,406,202,226,401,320,205,394,191,208,538,203,251,430,

207,274,398,200,203,524,251,202,389,198,204,575,278,202,384

,185,270,407,196,259,426,257,216,399,204,222,635,198,217,42

2,203,277,400,281,198,394,262,201,439,204,245,533,195,216,4

59,191,294,463,205,198,392,289,194,397,241,221,556,203,190,

399,188,331,428,207,228,404,285,200,501,203,192,531,202,208

,397,241,266,490,181,208,423,277,251,398,202,205,533,197,21

9,469,225,275,401,203,208,394,320,229,436,224,215,537,196,2

65,386,212,288,405,191,203,538,256,201,385,217,204,581,234,

209,430,190,288,390,193,243,408,268,199,401,216,199,673,220

,217,399,204,264,399,258,211,401,275,232,430,205,238,523,19

5,229,414,202,271,482,201,229,427,264,221,398,236,226,543,2

05,216,424,229,324,411,200,203,409,288,201,496,198,194,624,

210,198,406,236,294,430,196,199,423,264,262,381,199,202,554

,202,220,551,217,280,415,228,208,396,310,208,416,189,202,53

2,219,234,470,188,272,400,194,213,491,300,215,416,203,260,5

12,259,214,396,187,273,406,193,239,393,303,207,504,196,213,

652,193,204,396,215,262,429,230,203,390,291,203,407,198,263

,555,223,244,407,190,268,469,199,220,404,268,215,426,280,20

3,524,206,212,401,204,351,441,200,216,450,272,195,503,210,2

05,513,200,206,447,262,314,392,207,203,438,269,233,407,204,

210,535,218,215,485,200,291,398,209,204,417,324,191,451,234

,205,579,195,239,416,204,279,401,209,220,485,275,212,397,21

0,200,550,248,223,413,207,275,425,195,243,447,266,225,433,1

94,208,719,198,218,400,207,294,420,245,208,465,261,243,399,

206,257,551,198,205,417,206,257,484,244,217,405,284,205,409

,255,206,543,195,217,450,191,353,422,202,213,405,262,201,49

3,202,238,571,213,210,444,239,272,403,201,209,412,308,233,4

03,201,205,581,217,204,509,201,263,463,215,206,395,333,203,

412,200,233,556,195,280,413,212,268,407,205,209,508,273,206

,457,219,212,515,234,225,400,199,281,443,207,267,447,304,21

7,403,207,212,665,216,214,404,201,309,411,247,251,405,264,2

04,408,194,246,584,239,211,414,200,271,535,194,213,434,295,

200,446,247,207,542,201,203,412,238,340,416,207,247,405,276

,214,513,212,200,546,207,250,408,276,288,399,195,222,412,28

1,253,415,200,226,665,207,200,491,212,270,408,214,207,405,3

33

38,221,451,226,210,547,202,253,416,205,273,418,229,197,478,

271,232,406,206,208,565,254,208,505,212,283,410,211,259,437

,255,205,418,207,224,677,212,218,400,209,311,412,274,206,40

4,298,226,434,203,243,551,206,224,412,204,298,538,220,209,4

18,268,201,396,253,200,562,213,235,442,211,340,413,199,211,

443,265,212,497,253,215,538,228,207,431,245,278,399,200,204

,453,262,259,441,205,227,566,207,210,487,204,307,415,204,20

7,406,336,216,469,213,221,559,228,264,448,205,283,437,209,2

07,499,301,207,462,204,202,578,255,206,415,205,293,423,208,

307,421,295,211,408,209,216,673,220,233,398,232,276,454,219

,214,414,268,220,416,209,252,625,199,205,419,251,270,518,20

1,210,408,280,213,390,249,191,570,211,231,448,205,332,442,2

33,206,403,273,206,499,211,217,614,207,214,468,243,276,405,

228,220,411,270,247,415,235,253,551,220,217,498,217,271,414

,215,214,409,385,240,412,199,202,549,202,253,450,211,280,46

4,208,205,522,268,207,446,211,212,604,260,230,443,207,294,4

08,227,247,403,292,216,425,233,223,667,204,218,418,212,304,

424,248,238,465,291,216,420,206,255,564,220,210,461,216,309

,499,215,229,420,269,204,412,248,205,565,254,211,444,197,32

6,427,214,209,413,281,218,582,201,227,546,203,222,420,236,3

15,432,210,234,443,276,253,463,209,213,558,228,205,514,235,

283,403,225,216,423,373,203,433,200,215,627,212,263,434,196

,305,436,209,235,544,275,243,418,216,215,558,246,221,425,21

2,291,442,237,251,460,285,208,432,203,214,667,208,208,458,2

20,267,420,254,213,427,269,205,422,240,289,563,222,227,434,

205,294,495,209,221,424,306,207,405,250,210,651,205,236,418

,199,348,481,205,220,415,275,221,517,233,207,569,211,257,41

3,252,290,438,208,222,426,280,280,424,243,208,582,194,211,5

13,226,283,421,211,214,509,342,205,424,207,214,555,199,248,

427,206,321,473,230,215,495,286,213,439,206,222,559,267,213

,413,209,317,404,208,279,417,301,220,470,214,211,675,209,22

6,481,201,292,411,245,241,421,286,213,432,206,255,568,229,2

21,427,230,279,505,229,215,438,277,223,457,275,246,614,214,

215,412,216,337,416,210,228,411,296,247,553,210,232,570,219

,219,427,247,295,451,228,216,415,283,250,431,220,227,620,22

5,216,586,206,290,424,220,236,411,331,219,446,211,265,559,2

34

09,269,462,216,287,481,204,214,504,313,211,417,219,218,549,

251,233,425,224,313,481,213,278,425,275,213,429,201,216,672

,209,270,416,215,294,421,268,224,426,265,222,466,237,272,64

8,208,216,430,216,293,499,216,209,493,272,205,432,284,231,5

72,218,222,416,208,378,424,210,244,424,267,263,517,204,218,

620,240,224,424,258,301,415,214,226,466,264,273,485,220,216

,568,216,213,512,201,300,421,225,253,439,332,230,431,209,22

7,618,211,257,427,227,325,413,200,225,551,283,215,425,216,2

18,636,260,235,472,217,293,442,212,282,432,293,245,448,223,

216,694,221,261,420,225,298,422,280,214,405,275,221,439,209

,287,577,221,213,471,209,292,587,212,217,415,303,217,480,25

8,244,573,204,215,446,210,363,422,214,242,430,357,202,512,2

21,208,573,224,220,444,241,281,483,199,245,429,297,273,434,

204,247,569,209,241,509,221,295,476,226,229,416,343,235,468

,232,223,575,203,266,459,223,288,428,212,213,572,294,224,43

6,205,234,570,243,248,433,219,340,417,234,257,434,278,210,4

37,198,230,738,255,225,448,212,285,419,250,211,455,288,222,

506,238,269,556,205,228,456,208,284,562,212,234,451,282,222

,427,262,221,607,223,231,424,228,390,423,215,222,469,292,21

8,514,214,215,631,212,215,405,257,288,434,249,220,428,279,2

94,417,225,225,587,210,202,537,207,316,419,241,225,456,338,

221,441,205,231,592,208,277,519,202,295,442,222,219,523,300

,227,459,212,236,590,265,227,422,211,290,456,226,252,430,31

0,219,434,208,250,682,220,221,426,212,288,459,250,214,447,2

75,219,485,212,262,616,216,251,437,221,325,523,220,232,431,

320,228,422,281,222,572,205,229,420,213,349,436,218,254,517

,302,201,531,214,221,589,215,220,430,283,329,466,202,219,43

5,282,257,434,211,229,579,232,212,514,231,290,464,209,227,4

19,357,236,485,202,221,568,207,301,459,214,306,435,206,261,

536,281,224,436,215,223,637,289,231,433,230,300,450,213,261

,435,298,226,467,233,223,768,218,225,417,220,295,425,262,22

1,460,298,271,457,217,265,587,216,223,441,213,305,510,239,2

30,462,287,223,444,259,265,573,217,230,480,199,353,436,235,

230,433,300,214,531,216,259,576,229,224,465,245,327,434,218

,234,448,320,263,431,213,222,597,217,218,610,217,302,471,24

5,212,432,335,218,456,210,240,580,237,302,465,227,328,440,2

35

20,223,526,283,237,439,242,263,590,243,219,430,196,298,434,

241,263,511,292,234,476,207,219,740,212,224,449,218,321,423

,247,223,460,312,217,436,218,268,596,242,223,455,228,298,53

9,219,240,436,294,239,506,246,226,580,209,224,427,210,370,4

89,231,274,436,283,224,572,231,230,570,222,232,438,307,293,

439,214,228,451,282,272,470,212,223,694,209,236,526,204,308

,434,225,217,416,333,266,442,223,225,636,214,292,446,218,31

2,437,252,223,512,291,212,434,251,231,594,260,221,489,224,3

17,443,230,277,443,312,241,452,231,242,708,218,214,441,230,

290,469,273,231,498,318,223,446,209,268,585,217,221,444,224

,294,599,226,237,452,294,217,453,286,235,583,219,251,437,21

9,403,421,222,225,444,307,226,539,241,226,582,222,241,478,2

57,310,445,219,229,495,285,271,464,204,225,604,232,229,515,

230,326,462,221,220,430,338,241,445,209,263,636,239,285,445

,210,303,454,235,215,526,308,227,532,210,218,584,270,227,44

2,205,305,451,229,315,444,325,232,436,210,239,698,220,228,4

67,236,292,455,271,247,434,277,229,439,221,273,663,221,228,

462,222,318,565,222,232,435,283,239,441,272,229,579,212,225

,457,226,372,436,281,234,476,287,224,524,225,231,604,218,23

9,493,247,303,433,217,228,492,279,279,459,226,244,655,215,2

27,529,233,309,448,224,219,435,392,253,445,228,235,606,211,

266,448,231,291,496,213,212,530,313,242,441,223,231,578,249

,243,479,226,293,455,214,277,502,291,240,452,239,236,748,21

3,241,454,221,308,437,293,255,496,286,237,442,213,273,596,2

16,256,446,229,361,523,217,232,441,307,243,458,262,230,605,

245,236,497,232,371,438,220,223,480,291,227,629,221,218,599

,258,233,438,267,291,481,213,253,437,297,276,462,224,245,60

8,230,222,527,237,316,432,228,250,445,363,234,523,230,237,6

83,233,276,456,228,302,442,225,238,573,338,249,447,223,227,

594,260,224,451,211,322,441,248,296,443,279,227,500,246,242

,718,217,237,501,210,295,481,272,226,443,296,246,457,235,31

6,593,221,240,449,246,312,537,221,222,459,346,218,467,284,2

35,607,237,253,452,216,360,559,220,228,455,313,224,538,235,

230,684,222,254,469,265,304,436,230,229,440,311,273,454,286

,244,596,222,224,542,221,328,454,227,225,517,353,247,466,21

6,235,605,219,286,447,226,343,432,262,218,585,279,229,438,2

36

29,233,602,303,231,467,209,302,443,243,277,499,303,233,511,

225,230,756,221,229,454,225,325,451,295,244,449,303,252,451

,220,285,575,220,237,512,251,302,526,227,232,481,295,249,45

0,296,237,669,227,240,458,211,377,452,237,223,467,300,246,5

45,231,254,685,230,223,457,275,303,473,250,239,457,300,264,

491,227,229,612,221,244,601,231,318,442,231,237,466,384,223

,467,216,258,580,211,278,461,242,328,463,226,265,544,328,21

9,461,239,233,596,259,238,436,233,318,580,240,269,444,299,2

30,483,220,229,734,230,258,459,237,309,457,273,248,444,317,

241,453,253,286,604,225,252,453,229,303,577,237,230,494,313

,225,439,271,225,658,230,239,489,217,424,448,241,225,450,29

4,231,591,250,223,615,251,219,458,275,322,447,225,225,490,2

99,309,515,223,234,617,226,227,557,225,304,453,230,274,502,

363,235,453,232,237,618,243,271,475,271,303,443,249,233,550

,300,233,462,228,225,664,276,248,486,221,342,464,230,280,48

6,299,251,498,221,226,710,231,251,519,228,331,498,314,230,4

48,285,232,464,236,276,611,258,233,499,226,320,547,233,231,

449,328,230,459,260,280,625,220,249,446,221,374,470,226,235

,493,348,227,600,236,248,614,232,235,458,268,311,483,227,23

4,456,331,301,457,232,232,640,232,250,579,232,303,479,213,2

54,463,368,232,467,268,242,610,218,270,447,217,315,496,248,

227,605,298,219,476,230,244,670,268,243,451,241,360,455,231

,286,448,317,266,459,229,233,755,242,276,466,232,315,442,28

1,232,465,296,224,520,234,287,698,214,248,466,248,315,541,2

26,258,437,297,222,466,299,240,670,224,243,470,267,373,498,

232,239,450,299,255,567,240,231,661,232,231,444,277,301,450

,231,282,498,295,322,457,231,233,630,224,228,552,224,294,44

8,283,234,464,389,233,491,227,238,633,235,285,528,239,332,4

43,229,228,562,310,227,481,231,257,592,284,227,498,237,321,

455,225,282,481,338,244,458,215,251,728,254,237,462,223,330

,518,265,233,458,309,237,472,227,311,617,247,268,477,229,30

3,553,222,233,495,311,238,497,287,237,617,227,252,497,212,3

84,484,240,252,523,342,222,554,236,230,601,237,236,457,265,

378,464,229,251,478,307,283,473,221,228,607,246,244,605,232

,315,501,243,233,462,362,234,512,232,225,626,230,271,445,24

3,306,485,236,282,583,310,241,438,231,233,602

37

付録２（弱いゴールドバッハ予想の組み合わせ数）

42615,40431,26099,42695,41509,28138,42611,38624,27633,42519

,41086,28364,39543,42153,28352,41396,41891,26303,42820,4162

5,28466,42702,37469,28472,42501,41653,28626,39800,42253,275

21,42957,41949,26215,43163,41975,28551,41821,38614,28688,42

887,41671,28879,39895,40545,28455,43153,42537,26537,43206,4

2224,27584,43333,39005,28848,42657,42244,28987,38699,42508,

28713,43251,42443,26734,43224,40955,28494,43588,39293,28923

,43441,42439,28166,40327,42633,28792,43452,42194,26894,4241

5,42405,28979,43648,39235,29219,43720,41316,29225,40044,430

47,28597,43762,42968,26223,43931,42415,29069,43970,39799,29

067,42408,42647,29290,40389,43193,29126,44079,41933,27121,4

3901,42287,29170,44187,40083,28595,44051,42665,29467,40808,

43593,29267,42212,43317,27152,44319,42926,29402,44523,38829

,29299,44299,43152,29365,41166,43590,28506,44356,43644,2733

8,44590,43338,29564,42951,39530,29599,44416,43363,29842,413

31,42449,29458,44569,44013,27497,44312,43350,28756,44884,40

428,29836,44502,43712,29855,40263,43875,29427,44877,43903,2

7630,45073,42517,29719,45100,40619,30007,44993,43529,29023,

41583,44072,29894,44629,44499,27898,43826,43879,29861,44694

,40636,30266,45215,42768,30176,41806,43937,29896,45413,4448

2,26806,45567,43977,30134,45550,41163,30319,43884,44170,300

95,42209,44113,30170,45391,43359,28001,45485,44447,30012,45

883,41369,29552,44734,44433,30453,42273,44914,30368,44142,4

4751,28011,45904,44376,30099,46064,39743,30590,45553,44698,

30605,42615,45159,29546,45979,44644,28305,46104,44786,30464

,44708,41521,30664,46025,44826,30897,42293,43847,30521,4608

1,45211,28352,46103,45061,29762,46304,41819,30372,46134,451

08,31104,41640,45569,30675,46366,45075,28646,46699,43356,30

843,46577,42032,30715,46459,45388,30229,43063,45724,30616,4

6026,45852,28728,45339,45260,30887,46673,42163,31219,46759,

43880,30837,42871,45825,30885,47010,46052,28009,46866,45533

,31128,47048,42158,31323,45349,45192,31433,43364,46053,3124

8,47042,44780,28836,46657,45867,31131,47342,42887,30203,469

30,45673,31450,43732,46380,30791,45724,46288,29037,47492,45

38

958,31464,47249,41488,31534,47200,45688,31549,43893,46756,3

0518,47331,46362,29015,47417,46080,31506,45675,42955,31695,

47442,46270,31726,44183,45171,31486,47644,46907,28764,47724

,46426,30693,47938,42996,31853,47622,46519,31988,42959,4639

3,31700,47432,47030,29604,48118,45186,31661,47972,43345,319

29,47468,46962,31057,44449,46698,31862,48015,47306,29641,46

653,46677,31612,48269,43237,32227,48270,45550,31991,44738,4

7474,31935,48223,47012,28908,48291,46978,32164,48469,43875,

32346,46424,47096,32323,44454,47454,32215,48652,46204,29806

,48692,47144,32117,48928,43713,31150,48497,47148,32318,4500

5,47956,32277,47271,47929,30003,48921,46805,32199,49096,427

90,32521,48615,47668,32679,45390,48114,31359,48371,47745,30

122,48795,48041,32620,47682,44233,32659,49101,47657,32525,4

5508,46695,32438,49002,47959,30259,49362,48093,31680,49435,

43949,32807,49198,48347,33118,44274,48336,32478,49403,48301

,30411,49235,46537,32682,49595,44856,32985,49506,48441,3205

0,45697,48552,32499,49446,48967,30645,48150,48078,32828,498

18,44840,33158,49792,46215,33203,46036,48878,33002,49671,49

012,29777,49964,48524,33102,49506,45336,33053,48142,48685,3

3427,46368,48878,33166,50087,47701,30447,50221,48646,33188,

49959,45444,32173,49993,48702,33485,46415,48879,33504,48714

,49349,30889,50374,48490,33441,50579,44244,33550,49798,4898

0,33604,46877,49598,32397,50434,49487,30764,50725,49376,332

83,49153,45690,33577,50469,49225,33988,46819,48208,33601,50

299,49332,31171,50979,49655,32671,50790,45799,33874,50728,4

9689,34092,45140,49603,33571,50935,49894,31398,51190,48116,

33701,51360,46132,33641,51097,49747,32844,47317,50153,33891

,50966,50152,31561,49567,49277,33822,51492,46296,34204,5086

2,48496,34225,47454,50491,33792,50940,50499,30666,51280,500

96,34102,51630,46346,34304,49865,50127,33910,47686,50586,34

171,51758,49104,31752,51637,50363,33840,51822,46308,33330,5

1609,50334,34565,47771,50942,34425,50221,50766,31802,50989,

50566,34465,52206,45186,34502,51776,50567,34688,48310,51286

,32979,51950,50768,32105,52317,50916,34563,50621,47156,3473

0,51694,50224,34821,48463,49822,34335,52209,51452,32121,524

89,51058,33598,51927,47298,34853,52316,51193,34927,46692,51

39

387,34665,52424,51543,32040,52744,49510,34860,52821,47701,3

4899,52519,51065,34236,48871,51059,34882,52602,51919,32539,

51174,51205,34972,52890,47652,34975,52286,49870,35114,48921

,52004,34905,52783,52078,31638,53121,51574,34815,52815,4794

2,35275,51443,51739,35366,49165,51958,35409,53234,50102,325

82,53117,51584,35161,53415,48354,34511,53231,51845,35576,48

695,52529,35439,51565,52498,32609,53417,51676,35470,53860,4

6773,35220,53317,52240,35546,49756,52773,34463,53066,52466,

33083,53819,51918,35675,51998,48484,35765,53519,52350,35966

,50034,50934,35535,53157,52895,33064,53915,52587,34402,5415

2,48660,35739,53850,52775,35539,48514,52944,35738,53995,530

74,33402,54350,51052,35786,54239,48470,36097,53604,52917,35

130,50037,53173,35851,54074,53541,33451,52184,52779,35959,5

4500,48772,36264,54418,51397,36242,50122,53400,35625,54480,

53572,32524,54643,53166,35856,54785,49534,36401,52904,52605

,36485,50551,53701,36177,54633,52038,33522,54292,53569,3622

6,54434,49734,35357,54717,53360,36650,50712,54014,36398,531

54,53326,33438,55064,53310,36612,55298,48340,36583,54934,53

741,36795,51108,53692,35048,55111,54048,33948,55086,53925,3

6714,53590,50072,36843,54804,53617,36989,50988,52746,36596,

55317,54300,34034,55569,54103,35182,55699,50113,36903,55377

,53994,37290,49938,54538,36633,55615,53910,34306,55835,5274

7,36910,55854,50175,36821,55741,54636,36113,51126,54688,368

78,55689,55110,34448,54074,54280,36929,55747,50437,36940,55

669,52886,37247,51832,54922,37221,56111,55215,33439,56112,5

3718,37205,56239,50898,37441,54360,54711,37338,52051,55258,

37310,55614,53744,34380,56464,54909,37383,56714,51069,36364

,56227,54937,37175,52285,55446,37428,54372,55517,34661,5664

1,55071,37651,56704,49057,37666,56586,55322,37785,52683,554

72,36426,56202,55755,34962,56271,55481,37748,55342,51497,37

777,56751,55324,37863,52933,54134,37327,56884,55849,34902,5

7007,55664,36630,57307,51438,38056,56609,55269,38246,51294,

56150,37846,57118,56128,35319,57388,54030,37896,56872,51469

,38060,57315,56324,37059,53052,56204,37929,57360,56478,3514

3,55729,56044,37784,57569,51920,38416,57527,54344,38407,531

96,56055,38151,57675,56817,34352,57236,56206,38187,57828,52

40

430,38401,55296,56327,38522,53532,56910,38363,57962,54682,3

5572,58155,56401,37984,58103,52662,37447,57860,56202,38619,

53798,57223,38398,56290,56538,35650,58180,56704,38627,58276

,51174,38633,58279,56925,38830,52878,57372,37399,58175,5737

0,35861,58435,57086,38789,56894,52843,38453,58290,56468,392

78,54400,55848,38659,58436,57574,35972,58820,56740,37798,58

833,53031,38809,58611,57507,39266,52649,57609,38938,58186,5

7602,36342,59250,55623,39028,58668,53430,39165,59035,57717,

37800,54472,57724,39164,59039,58251,36333,57259,56937,39084

,59369,53004,39483,59191,55942,39487,54827,58256,39240,5924

3,58459,35161,58788,57774,39144,59383,53807,39654,57427,579

09,39675,55020,58305,39123,59152,56575,36641,59628,58320,39

414,59945,53805,38606,59477,57333,39821,55365,58504,39640,5

7794,58626,36655,59792,58329,39643,59630,52477,39903,59683,

58616,39705,55553,59010,38367,60036,58853,36434,60118,58784

,39979,58408,54215,39994,59594,58615,40249,55767,56568,3987

9,60104,59185,37094,60292,58902,38857,60585,54096,40203,595

60,59114,40334,54175,58947,39958,60279,59257,37305,60904,57

309,39356,60520,54629,40404,60621,59138,39209,56151,59488,4

0206,60454,59191,37400,58530,59257,40256,60976,54957,40508,

60694,57530,40497,55751,59825,40195,60906,59666,36280,60948

,59275,40376,61205,55387,40346,59182,59363,40859,56631,6008

0,40328,61038,58244,37671,61304,58881,40630,61542,55604,396

30,61293,59693,40976,56432,60663,40744,58837,60221,37654,61

558,59801,40939,61680,53909,40898,61614,59706,40717,57319,6

0732,39603,61556,60603,37917,61815,60403,41150,60022,55260,

40550,61780,60239,41358,57379,58891,41029,61765,60850,38094

,61481,60538,39828,61751,56241,41328,62004,60421,41581,5549

5,60872,40784,62118,60932,38352,62371,58492,41183,62433,563

64,41561,62378,60345,40271,57506,61279,41379,62284,61286,38

210,60590,60807,41278,62073,56435,41678,62505,59207,41768,5

8084,61419,41347,62153,61741,36967,62827,61046,41481,63065,

56970,41848,60745,61235,41905,58059,60560,41820,62883,59839

,38660,62953,61454,41556,63436,57306,40723,62071,61470,4182

5,58347,62176,41945,61112,61980,38619,62994,61467,41639,636

23,55423,42095,62936,61871,42191,58780,62328,40653,63285,61

41

396,39014,63611,62210,41956,61539,57108,42209,63470,61804,4

2619,58255,60572,42127,63526,62471,39024,63837,62287,40624,

64016,57369,41918,63635,62204,42690,57131,62817,42236,63781

,62573,39463,63615,59731,42255,63934,57931,42626,63749,6258

1,41486,59227,63004,42415,63473,62728,39568,62261,62578,424

41,64255,57992,42808,64373,60642,42184,59431,63085,42316,64

345,63421,38413,64492,62721,42655,64731,57880,42950,62363,6

2940,43126,59206,63216,42837,64707,61614,39636,63970,62994,

42788,64921,58742,41820,64615,62614,43216,60220,63734,42492

,62624,63563,39814,64866,63270,43225,65267,56980,42930,6491

6,62862,43242,60150,64033,41873,64967,64007,39955,65342,637

85,43257,62239,58831,43467,65178,63364,43641,60563,62065,43

275,65170,64365,39696,65623,63283,42030,65781,59182,43564,6

5341,64075,43770,58875,63673,43443,65508,64471,40214,65867,

62058,43441,65550,59443,43742,64934,64415,42640,61025,64387

,43529,64985,65010,40665,63990,64207,43212,66105,59591,4385

8,66088,62465,44037,61071,64546,43808,66174,64484,39410,662

99,64418,43820,66179,60080,44039,64094,64351,43939,60683,65

140,44076,66617,63235,40759,66305,64767,43970,66753,60118,4

2340,66016,64743,44436,61403,65679,44144,64403,65332,41047,

66745,64457,44256,67080,58006,44417,66631,65194,44640,61732

,65950,42681,66014,65567,41148,67040,65565,44208,65104,6013

3,44592,67052,65257,44400,62164,63905,44439,66999,65505,412

23,66846,65728,43202,67392,60089,44670,67126,65928,45121,60

369,66212,44556,67164,65681,41476,66750,63669,44539,67555,6

1053,44898,67731,66071,43585,62509,66194,43890,67302,66556,

41734,65684,65994,44606,67835,61217,45038,67590,63346,45301

,62135,66624,44977,67802,66905,40370,67895,66208,45067,6751

1,61493,45263,65864,65901,45421,62988,66692,45267,68105,648

45,41385,68274,66361,45146,68604,62030,43712,67995,66398,45

557,63123,66624,45365,66072,67189,41852,68454,66600,45542,6

8493,60208,45561,67633,66830,45467,63401,67488,44086,68475,

67327,42184,68816,66716,45151,66642,62204,45776,68593,67016

,45997,63808,65414,45692,68757,67026,41907,69131,67433,4438

9,68973,62374,45911,68781,67493,46182,61303,67767,45749,682

89,67892,42486,69322,65394,45703,69598,62643,45687,69316,67

42

592,44832,64050,67482,45891,69289,68348,42617,67266,66986,4

5833,69620,62794,46550,69419,65744,46054,64285,68454,46016,

68901,68457,41435,69258,67716,46125,69990,63186,46564,67150

,68203,46039,64480,68356,46448,69922,66582,42925,70097,6827

3,46351,69930,62189,45137,70036,68301,46571,64823,69019,464

28,67635,68983,43060,69572,68426,46356,70651,61704,46529,70

163,68484,46676,65195,69213,44707,70315,69035,43299,70146,6

8882,46724,68251,63805,46945,70319,67635,47202,65511,67337,

46721,70238,69073,43406,70780,69106,45368,70226,63789,46922

,70786,69207,47421,63118,69499,46417,70755,69668,43234,7115

6,66917,46916,71213,64330,47183,71021,69355,46061,65354,690

69,46864,70940,70070,43863,68854,69459,47033,71281,64431,47

367,70656,66964,47546,65964,70266,47207,71300,70015,42470,7

1444,69771,46893,71715,64889,47357,69287,69750,47685,65933,

70216,47578,71611,67750,44059,71534,69996,47428,71565,65062

,46382,71698,69931,47766,65830,70855,47692,69674,70356,4419

1,72106,69573,47600,72402,63136,47297,71951,70399,47942,670

09,71175,46320,72089,70944,44039,72349,69972,47806,70182,64

944,47944,72078,70498,48510,67117,69054,47838,70905,71294,4

4422,72560,70908,46538,72758,65412,48204,72043,71031,48038,

64997,70746,48114,72723,71371,44682,72887,68672,48114,72958

,65055,48445,72890,71411,46617,67460,71414,48313,72810,7185

9,44951,70026,71112,48279,73302,65969,48740,72604,69037,488

30,67657,71707,47808,73089,72027,43513,73562,71647,48478,73

499,66088,48946,71040,70694,48824,67857,72285,48781,73245,6

9930,45156,73607,71844,48363,73072,66814,47281,73519,71711,

49000,68017,72647,48943,71487,72188,44745,73316,71953,48972

,74296,64843,49091,73775,71752,49255,68525,72087,47469,7385

8,72036,45485,74013,72254,49016,71991,66888,49295,73365,722

16,49599,68653,70746,48648,73976,72911,45521,74572,72619,47

274,74267,67182,49422,74145,72803,49695,66181,73164,49345,7

4398,72392,45753,74600,70521,49404,74671,67564,49398,74632,

72466,48221,68450,73303,49531,74680,73845,45934,72572,72884

,49420,75081,67601,49070,74999,70393,49963,69385,73645,4955

5,74825,74072,44767,75265,72482,49746,74718,68142,50180,727

28,73104,49928,69256,73881,49949,74543,71764,46150,75446,73

43

064,49865,75631,68416,48679,75430,73390,49736,69922,74284,4

9864,72834,74273,46013,75545,73487,50233,75965,65432,50209,

75514,73891,50530,70046,74585,48595,75089,74304,46568,75152

,74027,50312,73859,68543,50469,75773,73916,50622,70470,7192

0,49666,75907,74731,46795,76011,74488,48762,76332,68768,505

88,76063,73819,50437,68232,74864,50396,76046,74689,47024,76

461,72307,50279,75822,68944,50800,76022,74815,49454,70460,7

5021,50625,76460,75295,46680,74259,74723,50495,76888,68425,

51172,76679,72450,51096,70891,74753,50570,76737,75465,45916

,76900,74973,50412,77079,69816,51314,73372,74946,51192,7114

7,75710,51146,77049,73268,47221,76711,75288,50272,77361,699

47,49834,77090,75257,51360,71513,76027,51317,74738,74784,47

600,77557,75347,51460,77745,67968,51353,77046,75694,51645,7

1025,75880,49390,77326,76275,47647,77702,75874,51572,75473,

70323,51078,77608,75578,51970,71553,74066,51490,77294,76239

,47746,78130,75328,50064,78028,70529,51661,77948,75838,5212

1,70028,76684,51596,77146,76655,47765,78544,73984,51748,783

69,70783,51637,78335,76436,50091,72346,76797,51814,78210,77

353,48296,76195,76101,51862,77962,70167,52327,78499,74393,5

2418,72527,77150,52008,78704,77529,46898,77961,76108,52090,

78729,71421,52438,76160,76523,52511,73124,77286,51896,79078

,75149,48101,78820,77078,52214,79390,71604,50749,78755,7606

9,52685,73230,78037,52443,75820,77770,48626,79263,77364,524

87,78657,69458,52756,79155,77329,52832,73442,77601,51022,79

381,77949,48191,79303,77683,52726,77271,71966,52822,79347,7

7511,52707,73597,75049,52664,79416,78295,48840,79887,78074,

51096,79875,71924,53070,78274,77974,53427,71653,78406,52659

,79623,78263,49214,80470,75791,52397,80125,71855,53399,8020

2,78168,51604,74066,78699,53129,79988,77964,49338,77887,783

43,52682,80477,72697,53488,80183,75938,53663,73632,79000,53

181,80240,79425,47971,79711,78548,53252,80775,73158,53200,7

7978,78422,53806,74706,79503,53552,80972,76342,49548,80949,

77819,53293,81044,73329,52183,80726,78891,54049,74812,79973

,53401,77577,79551,49745,81036,79078,53758,81417,70801,5377

5,81125,78984,53457,74652,80106,52256,81160,79868,50036,815

12,79564,54007,79269,72789,54137,81129,78664,54506,75325,77

44

647,54045,81361,80198,49998,80869,79883,52335,81867,73534,5

4100,81625,79887,54334,73278,80179,53528,81618,80240,50616,

82343,77240,54113,81631,74151,54585,82002,79429,52774,75863

,80216,54429,81868,81069,50636,79740,79567,54183,81529,7435

5,54857,82265,77888