x^2 + ny^2 の形で表せる素数 - めざせプライムマスター！

x2 + ny2の形で表せる素数
〜めざせプライムマスター〜
辻順平@tsujimotter
2019/03/30 関東すうがく徒のつどい #kantomath

スライドは期間限定で公開
#kantomath で検索（or @tsujimotter で検索）
2

今⽇のテーマ
p = x2 + ny2 の形で表される素数の法則
3
n を正の整数とする
「整数 x, y が存在して p = x2 + ny2」が成り⽴つ素数 p の条件は？
「整数 x, y が存在して p = x2 + ny2」が成り⽴つとき
「x2 + ny2 は p を表現する」という

今⽇の発表の趣旨
まるでポケモンをゲットするかのように
p = x2 + ny2 型の素数の法則を集めていきましょう

証明はほぼありません
法則を鑑賞して楽しみましょう
4

フィールドマップ（講演の流れ）
フェルマー地⽅
ガウス地⽅（今⽇のメインパート）
ヒルベルト地⽅（まくらさんの発表（昨⽇）に関連）
リング地⽅
5

参考⽂献
Cox, “Primes of the Form x2 + ny2”, WILLY.

フェルマー地⽅：
§1
ガウス地⽅：
§2, §3
ヒルベルト地⽅：
§5
リング地⽅：
§7, §9
6

めざせプライムマスター！
7

あっ！野⽣の
p = x2 + y2
が⾶び出してきた！
8

p = x2 + y2 （n = 1 のとき）
•  2 = 12 + 12（かける）
•  3（かけない）
•  5 = 22 + 12（かける）
•  11（かけない）
•  13 = 32 + 22（かける）
•  17 = 42 + 12（かける）
•  19（かけない）
9

p = x2 + y2 （n = 1 のとき）
•  2 = 12 + 12（かける）
•  5 = 22 + 12（かける）
•  13 = 32 + 22（かける）
•  17 = 42 + 12（かける）
•  11（かけない）
•  19（かけない）
p ≡ 1 (mod 4)
or
p = 2
p ≡ 3 (mod 4)
10

p = x2 + y2 （n = 1 のとき）
2 を除く素数 p に対して次が成り⽴つ．
整数 X, Y が存在して p = X2 + Y2 ⇔ p ≡ 1 (mod 4)
11
フェルマーの２平⽅定理
法則、ゲットだぜ！

平⽅剰余
p を奇素数、a を p で割り切れない整数とし
x2 ≡ a (mod p) (1)
という合同式を考える

式 (1) が解 x を持つ： a は p の平⽅剰余 (a/p) = 1
式 (1) が解 x を持たない： a は p の平⽅⾮剰余 (a/p) = –1

12

13
p, q を相異なる奇素数とする．
(i) 
(ii) 
(iii) 
平⽅剰余の相互法則（モンスターボール）
平⽅剰余の相互法則によって mod を⼊れ替えることができる
( 1/p) = 1 () p ⌘ 1 (mod 4)
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( 2/p) = 1 () p ⌘ 1, 7 (mod 8)
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(q/p) = ( 1)
p 1
2 ( 1)
q 1
2 (p/q)
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整数 a, b と素数 p に対して、が成り⽴つ（準同型性）(ab/p) = (a/p)(b/p)
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「フェルマーの2平⽅定理」証明の概略
p = x2 + y2 ⇒ p ≡ 1 (mod 4)

p = x2 + y2 ⇐ x2 + 1 ≡ 0 (mod p)

⇔ (–1/p) = 1

⇔ (–1)(p–1)/2 = 1

⇔ p ≡ 1 (mod 4)
14
（∵ 無限降下法より）
（∵ 平⽅剰余の相互法則 (i) より）
（∵ 平⽅剰余の定義より）
（∵ x, yの偶奇より）

p = x2 + 2y2 （n = 2 のとき）
整数 X, Y が存在して p = X2 + 2Y2 ⇔ p ≡ 1, 7 (mod 8)
15
x2 + 2y2 型素数の法則
(–2/p) = 1

p = x2 + 3y2 （n = 3 のとき）
16
(–3/p) = 1

p = x2 + 7y2 （n = 7 のとき）
整数 X, Y が存在して p = X2 + 7Y2 ⇔ p ≡ 1, 9, 11, 15, 23, 25 (mod 28)
17
(–7/p) = 1

フィールドマップ
フェルマー地⽅平⽅剰余の相互法則（n = 1, 2, 3, 7）
ガウス地⽅
ヒルベルト地⽅
リング地⽅
18

ガウス地⽅に⽣息している x2 + ny2
19
x2 + 5y2
x2 + 6y2
x2 + 10y2

2次形式
2次形式 ax2 + bxy + cy2 が表現する素数を考えよう
（ただし a, b, c は互いに素な整数）

D = b2 – 4ac を2次形式 ax2 + bxy + cy2 の判別式という

そもそも、x2 + ny2 の形に限る必要はない

2次形式
2次形式 ax2 + bxy + cy2 が表現する素数を考えよう
（ただし a, b, c は互いに素な整数）

D = b2 – 4ac を2次形式 ax2 + bxy + cy2 の判別式という

•  判別式 D < 0 かつ a > 0 の2次形式は、正の値だけを表現（正定値）
•  判別式が異なる2次形式が表現する数の集合は⼀致しない
※以降、2次形式といったら正定値2次形式を指す
そもそも、x2 + ny2 の形に限る必要はない

2次形式の変換
2つの2次形式 f(x, y) = ax2 + bxy + cy2 と g(X, Y) = AX2 + BXY + CY2

ただし，p, q, r, s は整数かつ ps – qr = 1

このとき、f と gは同値であるといい、f 〜 g で表す

22
✓
X
Y
◆
=
✓
p q
r s
◆ ✓
x
y
◆
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（ps – qr = ±1 のとき逆変換を持つ）

2次形式の変換
2つの2次形式 f(x, y) = ax2 + bxy + cy2 と g(X, Y) = AX2 + BXY + CY2

ただし，p, q, r, s は整数かつ ps – qr = 1

このとき、f と gは同値であるといい、f 〜 g で表す

•  「f にすべての整数の組 (x, y) を⼊れて作れる数の集合」と
「g にすべての整数の組 (x, y) を⼊れて作れる数の集合」が⼀致する
•  この変換で判別式は変化しない
23
✓
X
Y
◆
=
✓
p q
r s
◆ ✓
x
y
◆
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（ps – qr = ±1 のとき逆変換を持つ）

x2 + y2
〜 x2 + 4xy + 5y2
〜 2x2 + 6xy + 5y2
〜 10x2 + 34xy + 29y2
…
例：判別式 D = –4
24
•  判別式 –4 の2次形式はすべて同値
•  これらが表現する素数は、2または4で割って1あまる素数のみ

例：判別式 D = –20
25
x2 + 5y2
〜 x2 + 2xy + 6y2
〜 6x2 + 22xy + 21y2
〜 29x2 – 26xy + 6y2
…
2x2 + 2xy + 3y2
〜 7x2 + 22xy + 18y2
〜 3x2 – 10xy + 10y2
〜 18x2 – 14xy + 3y2
…
異なる素数を表現する

2次形式の同値類と類数
ax2 + bxy + cy2 に同値な2次形式全体の集合を
ax2 + bxy + cy2 の同値類といい [ax2 + bxy + cy2] と表記する
判別式 D の2次形式の同値類全体の集合を C(D) とかく
C(D) の位数を判別式 D の類数といい h(D) で表す
例：判別式 D = –4

C(–4) = { [x2 + y2] },

h(–4) = 1
例：判別式 D = –20

C(–20) = { [x2 + 5y2], [2x2 + 2xy + 3y2] },
h(–20) = 2
26

27
D を割り切らない素数 p について以下が成り⽴つ：
(D/p) = 1 ⇔ C(D) のいずれかの2次形式は p を表現する

C(D) のどの同値類がどの素数を表現するか？
•  類数 1 の場合は特定できる
（D = –4n に対し h(–4n) = 1 になるのは D = –4, –8, –12, –28 のときだけ）

•  類数 2 以上の場合は特定できるのか？ => ガウスの種の理論
28

合成
[ax2 + bxy + cy2], [a’x2 + b’xy + c’y2] ∈ C(D) に対して
(a, a’,(b+b’)/2) = 1 を満たすとき、以下の合成を定義する：

ただし，B は以下を満たす mod 4aa’ で⼀意的な整数
B ≡ b (mod 2a)
B ≡ b’ (mod 2a’)
B2 ≡ D (mod 4aa’)

C(D) は合成について群をなす
29

aa0
x2
+ Bxy +
B2
D
4aa0
y2
2 C(D)
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群・・・結合法則、単位元、逆元をもつ集合

例：
•  [x2 + 5y2] [x2 + 5y2]

= [x2 + 5y2]
•  [x2 + 5y2] [2x2 + 2xy + 3y2]

= [2x2 + 2xy + 3y2]
•  [2x2 + 2xy + 3y2] [x2 + 5y2]

= [2x2 + 2xy + 3y2]
•  [2x2 + 2xy + 3y2] [2x2 + 2xy + 3y2]
= [2x2 + 2xy + 3y2] [3x2 – 2xy + 2y2]
= [6x2 + 10xy + 5y2]

= [x2 + 5y2]
30
(x, y) 7! ( y, x)
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C(–20) = { [x2 + 5y2], [2x2 + 2xy + 3y2] }
(x, y) 7! (x, x + y)
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例：
31
合成 [x2 + 5y2] [2x2 + 2xy + 3y2]
[x2 + 5y2] [x2 + 5y2] [2x2 + 2xy + 3y2]
[2x2 + 2xy + 3y2] [2x2 + 2xy + 3y2] [x2 + 5y2]
C(–20) = { [x2 + 5y2], [2x2 + 2xy + 3y2] }

ガウスの種の理論（genus theory）
•  C(D)2 := C(D)の元を2乗してできる部分群
principal genus（主種）という
•  素数 p を表現する2次形式がprincipal genusに⼊る条件を与える
32

33
•  principal genus C(D)2 には [x2 + ny2] が必ず⼊る
•  principal genus の元が１つだけならば x2 + ny2 が p を表現する条件が
書ける
[ ]
principal genus C(D)2
判別式 D を持つ2次形式の同値類全体 C(D)
[ ] [ ] [ ][x2 + ny2] [ ]

素判別式分解

ただし、Di としては次のものだけを考える
•  –4
•  ±8
•  (–1)(p–1)/2p （p は奇素数）
34
D = D1D2 · · · Dg
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判別式 D を割らない素数 p について、
p が C(D) の2次形式のいずれかで表せるとする。
このとき、D の素判別式分解に対し次が成り⽴つ：

⇔ p は principal genus C(D)2 の2次形式で表せる
35
定理（スーパーボール）
D = D1D2 · · · Dg
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(D1/p) = 1, (D2/p) = 1, · · · , (Dg/p) = 1
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例：D = –20 の場合
素判別式分解 D = –4・5
(–4/p) = 1 かつ (5/p) = 1
平⽅剰余の相互法則より
p ≡ 1 (mod 4) かつ p ≡ 1, 4 (mod 5)
よって p ≡ 1, 9 (mod 20)
36

p = x2 + 5y2（n = 5 のとき）
37
2, 5 を除く素数 p に対して次が成り⽴つ．

38
整数 X, Y が存在して p = X2 + 10Y2 ⇔ p ≡ 1, 9, 11, 19 (mod 40)

オイラーの便利数
正整数 n に対して D = –4n とする
C(D)2 が [x2 + ny2] のみであるような n を便利数という
39

現在知られている便利数（65個）
• 1, 2, 3, 4, 7
• 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 22, 25, 28, 37, 58
• 21, 24, 30, 33, 40, 42, 45, 48, 57, 60, 70, 72, 78, 85, 88, 93, 102,
112, 130, 133, 177, 190, 232, 253
• 105, 120, 165, 168, 210, 240, 273, 280, 312, 330, 345, 357,
385, 408, 462, 520, 760
• 840, 1320, 1365, 1848
40

E715 On various ways of examining very large numbers, for whether or not they are primes
（便利数）

現在知られている便利数（65個）
• 1, 2, 3, 4, 7（←フェルマー地⽅）
• 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 22, 25, 28, 37, 58
• 21, 24, 30, 33, 40, 42, 45, 48, 57, 60, 70, 72, 78, 85, 88, 93, 102,
112, 130, 133, 177, 190, 232, 253
• 105, 120, 165, 168, 210, 240, 273, 280, 312, 330, 345, 357,
385, 408, 462, 520, 760
• 840, 1320, 1365, 1848
42

フィールドマップ
フェルマー地⽅平⽅剰余の相互法則（n = 1, 2, 3, 7）
ガウス地⽅ガウスの種の理論（65個）
ヒルベルト地⽅
リング地⽅
43

種の理論によって65個の法則は得られた
もっとたくさんの、無数に法則を得ることができるのか？

n が平⽅因⼦を持たないかつ n ≡ 3 (mod 4) でない
=> ヒルベルトの分岐理論
44

p = x2 + ny2 とはどういうことか？
45
p
x2 + ny2
= (x – y√–n)(x + y√–n)
（ℚ の）素数

46
p
x2 + ny2
= (x – y√–n)(x + y√–n)
ℚ
ℚ(√–n)
（ℚ の）素数
ℚ に √–n を加えた世界
（拡⼤体）
体・・・四則演算ができる集合

47
p
x2 + ny2
= (x – y√–n)(x + y√–n)
ℚ
ℚ(√–n)
（ℚ の）素数
素数 p が ℚ(√–n) で「素因数分解」する
ℚ に √–n を加えた世界
（拡⼤体）
体・・・四則演算ができる集合

ヒルベルトの分岐理論の設定
48
K （代数体）
L （Kのガロア拡⼤体）
K の素イデアル p
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ガロア拡⼤
Pe
1Pe
2 · · · Pe
g
<latexit sha1_base64="nKWoTIM6a2Zu6sKy1FYScD9sTNE=">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</latexit><latexit sha1_base64="nKWoTIM6a2Zu6sKy1FYScD9sTNE=">AAAGQHicpVTLbtNAFL0tBEp4tIUNEhuLEIQQqiYBxGOBIth0R19pKzUhst1JasUv2U7SYPkH+AEWrECwQPwFbBBbYNE/ALFAqEhIiAVnrkOp2yYssOWZO/d17j0zY8O3rTASYmts/NDh3JGjE8fyx0+cPDU5NX16OfQ6gSmrpmd7waqhh9K2XFmNrMiWq34gdcew5YrRvqfsK10ZhJbnLkV9X9YdveVaTcvUI6gaU3dqjh5tNAO9Hc8ljdKDWCYZTVlptJq57kWhlrG0lKUxVRAzgh9tv1AaCIWKIH7mvOnxDtVonTwyqUMOSXIpgmyTTiHeNSqRIB+6OsXQBZAstktKKI/YDrwkPHRo2xhbWK0NtC7WKmfI0SZQbHwBIjUqio/ipdgWb8Ur8Vn8Qq7ikGwxvg4jBiMwY8ZSNfcxGymG9BuTj84u/vhnlIM5oo2/USN7i6hJN7knCz36rFHdmml89+Hj7cXbC8X4ongmvqDPp2JLvEGnbve7+WJeLjwZUY+BWlSXw/Hv0xKV6QrPJa6iCVbdXTXkeU8l9DVaRFc6fGLIFjQOXo+tedSqQaviNjEqj2wm5ac6bMGWoibwUTWp3TCZh9RfA84sI6c5lUe4g7H7/V+M/E4WC54pU8NPTsS5+9RjrP5IZmOMAWQfjKkzvjnytIXI7fCN6ONbH/i6sPW4K4etLiyK1y7XkbAcwBrxeZPYbQ84Md3CLVPWC7xLLuZkkFFlacNT+StGIo4LMnvscNWqIo95sfFWkK24w3+NK1XIvYxHNtbksxJwjYp/g1fJnrVaWWBhg7v4k3V4/5eh38zgrGGX6xjre2pMz2FlV1YNvFwFO9dHdFnALUh4Lh/Imsc3G5zh71ja+y/cLyyXZ0qQ568VKnfT3yRN0Dk6T5eAcwOIszRHVfT4nN7Re/qQe537lPua+5a6jo8NYs5Q5sn9/A2lgVLq</latexit><latexit sha1_base64="nKWoTIM6a2Zu6sKy1FYScD9sTNE=">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</latexit><latexit sha1_base64="aqW+ebmq+5rtYT3nZ4gMyOdQDxo=">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</latexit>

49
K （代数体）
e = 1 のとき不分岐
g = [L : K] 完全分解
ガロア拡⼤
Pe
1Pe
2 · · · Pe
g

50
K （代数体）
K の素イデアルが L で素イデアル分解する様⼦を調べたいp
ガロア拡⼤
Pe
1Pe
2 · · · Pe
g
e = 1 のとき不分岐
g = [L : K] 完全分解

x^2 + ny^2 の形で表せる素数 - めざせプライムマスター！

x^2 + ny^2 の形で表せる素数 - めざせプライムマスター！

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x^2 + ny^2 の形で表せる素数 - めざせプライムマスター！