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)
<latexit sha1_base64="9C1EhA6WhfFIH9SJg+cS9vMwSz0=">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</latexit><latexit sha1_base64="9C1EhA6WhfFIH9SJg+cS9vMwSz0=">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</latexit><latexit sha1_base64="9C1EhA6WhfFIH9SJg+cS9vMwSz0=">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</latexit><latexit sha1_base64="wJFncUT1rxm/XpS2D+GzupzoxlU=">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</latexit>
( 2/p) = 1 () p ⌘ 1, 7 (mod 8)
<latexit sha1_base64="OCbqvRVATk3NOfc4iXmWyHi2X7g=">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</latexit><latexit sha1_base64="OCbqvRVATk3NOfc4iXmWyHi2X7g=">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</latexit><latexit sha1_base64="OCbqvRVATk3NOfc4iXmWyHi2X7g=">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</latexit><latexit sha1_base64="QHCRtttHQ+c//R2dg5pOd+QodJY=">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</latexit>
(q/p) = ( 1)
p 1
2 ( 1)
q 1
2 (p/q)
<latexit sha1_base64="UayEn0wet1sb7YX9g8K2W5G1p5w=">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</latexit><latexit sha1_base64="UayEn0wet1sb7YX9g8K2W5G1p5w=">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</latexit><latexit sha1_base64="UayEn0wet1sb7YX9g8K2W5G1p5w=">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</latexit><latexit sha1_base64="69clUa5Lm8oyVhpEXH+6cwEcXHs=">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</latexit>
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)
<latexit sha1_base64="lOHEPIdAyGMCPAyIqlp5Etj3UVM=">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</latexit><latexit sha1_base64="lOHEPIdAyGMCPAyIqlp5Etj3UVM=">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</latexit><latexit sha1_base64="lOHEPIdAyGMCPAyIqlp5Etj3UVM=">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</latexit><latexit sha1_base64="3Z0Q/Lrqx+Qo+i1tNrFoYjntkaE=">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</latexit>

「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
◆
<latexit sha1_base64="V0Vo8rqXULovszjz39q97onJ81o=">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</latexit><latexit sha1_base64="V0Vo8rqXULovszjz39q97onJ81o=">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</latexit><latexit sha1_base64="V0Vo8rqXULovszjz39q97onJ81o=">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</latexit><latexit sha1_base64="6pUDejE2285qDuZ8TlGHWbZSSvw=">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</latexit>
（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
◆
<latexit sha1_base64="V0Vo8rqXULovszjz39q97onJ81o=">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</latexit><latexit sha1_base64="V0Vo8rqXULovszjz39q97onJ81o=">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</latexit><latexit sha1_base64="V0Vo8rqXULovszjz39q97onJ81o=">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</latexit><latexit sha1_base64="6pUDejE2285qDuZ8TlGHWbZSSvw=">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</latexit>
（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)
<latexit sha1_base64="zGP80Q0gH7mF5KbP6fwuJ9pIotI=">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</latexit><latexit sha1_base64="zGP80Q0gH7mF5KbP6fwuJ9pIotI=">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</latexit><latexit sha1_base64="zGP80Q0gH7mF5KbP6fwuJ9pIotI=">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</latexit><latexit sha1_base64="8ZyVF+Zeu+vgBzx7FFTx8BT+ArQ=">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</latexit>
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)
<latexit sha1_base64="hANuCg4TNigITUiAmNidsWvFqaE=">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</latexit><latexit sha1_base64="hANuCg4TNigITUiAmNidsWvFqaE=">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</latexit><latexit sha1_base64="hANuCg4TNigITUiAmNidsWvFqaE=">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</latexit><latexit sha1_base64="Nc583sq1eZ9CE5YwuZLdw2u/2i4=">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</latexit>
C(–20) = { [x2 + 5y2], [2x2 + 2xy + 3y2] }
(x, y) 7! (x, x + y)
<latexit sha1_base64="/xDjPlHUNu5RxSvhE6ZIasn2HWs=">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</latexit><latexit sha1_base64="/xDjPlHUNu5RxSvhE6ZIasn2HWs=">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</latexit><latexit sha1_base64="/xDjPlHUNu5RxSvhE6ZIasn2HWs=">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</latexit><latexit sha1_base64="EuxArsPO9ISuR0MrRZmNC17nDPw=">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</latexit>

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
<latexit sha1_base64="+aKJAuJ5GTjzashe/ty0wh0bMPc=">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</latexit><latexit sha1_base64="+aKJAuJ5GTjzashe/ty0wh0bMPc=">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</latexit><latexit sha1_base64="+aKJAuJ5GTjzashe/ty0wh0bMPc=">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</latexit><latexit sha1_base64="vqgaLawA7+EJgtqPeHfAzK4eLJE=">AAAGGHicpVRLaxNRFD6tRmt8tFUQwc1gjIhIuYmKDxCKdtGdfbfQhDAzuUmHzouZycsxf8A/IOjKggupP0JwoW4FF/0J4rKCGxd+98xYk7SJC+dy5557Xt85370zhm9bYSTE3tj4seOZEycnTmVPnzl7bnJq+vxa6DUCU66anu0FG4YeStty5WpkRbbc8AOpO4Yt143tx8q+3pRBaHnuStTxZdnR665Vs0w9gqoydXFOe6jNVQqYRa1kVr0ohFivTOXEjOBHOywUUiFH6bPgTY83qERV8sikBjkkyaUIsk06hRibVCBBPnRliqELIFlsl9SlLGIb8JLw0KHdxruO3WaqdbFXOUOONoFiYwaI1Cgvvoq3Yl98Ervim/iFXPkh2WLMBiMGIzBjxlI1d7AaCYb0K5PPLy3//GeUgzWirb9RI3uLqEb3uCcLPfqsUd2aSXzz6Yv95QdL+fia2BHf0edrsSc+oFO3+cN8syiXXo2ox0Atqsvh+E9ohYp0k9cCV1EDq25PDVk+Uwl9iZbRlQ6fGLIFjYPhsTWLWjVoVVwbb+XRn0n5qQ7rsCWoXfiomtRpmMxD4q8BZ56Rk5zKIzzA6B3/i5E9yGLBM2Fq+M2JOHeHWozVGclsjHcA2Qdj6o63R962ELkd/iI6mNXU14WtxV05bHVhUbw2uY4uywGsEd83idP2gBPTfXxlynqVT8nF2k0zqizb8FT+ipGI44K+M3a4alWRx7zYGLPIlj/gv8SVKuRWn0d/rMl3JeAaFf8G77oDe7WzwMIWd/En6/D+b0Df7sPZxCmX8S4P1Jjcw9merBp4uQV27ozoMoevoMtr8UjWPP6ywRn+joXBf+FhYa04U4C8eDs3+yj9T07QZbpC14FzF4jztECr6PEZ7dAuvcu8zLzPfMx8TlzHx9KYC9T3ZL78BguBPpA=</latexit>

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
<latexit sha1_base64="+aKJAuJ5GTjzashe/ty0wh0bMPc=">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</latexit><latexit sha1_base64="+aKJAuJ5GTjzashe/ty0wh0bMPc=">AAAGGHicpVRLaxNRFD6tRmt8NFUQwc1gjIhIuYmKDxCCdtGdbdMXNCHMTG7SofNiZpI0jvkD/gFBVxZcSP0Rggt1K7joTxCXFdy48LtnYk3SJi6cy5177nl953z3zhi+bYWREHsTk8eOp06cnDqVPn3m7LnpzMz51dBrBqZcMT3bC9YNPZS25cqVyIpsue4HUncMW64ZW4+Vfa0lg9Dy3OWo48uKozdcq26ZegRVNXNxTnuozVXzmAWtbNa8KITYqGayYlbwox0W8j0hWxTEz4I3M9mkMtXII5Oa5JAklyLINukUYmxQngT50FUohi6AZLFdUpfSiG3CS8JDh3YL7wZ2Gz2ti73KGXK0CRQbM0CkRjnxVbwV++KT2BXfxC/kyo3IFmM2GTEYgxkzlqq5g9VIMKRfnX5+qfTzn1EO1og2/0aN7S2iOt3jniz06LNGdWsm8a2nL/ZLD5Zy8TWxI76jz9diT3xAp27rh/lmUS69GlOPgVpUl6Pxn9AyFegmr3muog5W3b4a0nymEvoyldCVDp8YsgWNg+GxNY1aNWhV3DbeymMwk/JTHTZgS1C78FE1qdMwmYfEXwPOPCMnOZVHeIDRP/4XI32QxYJnwtTomxNx7g61GaszltkY7wCyD8bUHd8ee9tC5Hb4i+hg1nq+Lmxt7sphqwuL4rXFdXRZDmCN+L5JnLYHnJju4ytT1qt8Si7Wbi+jyrIFT+WvGIk4Lhg4Y4erVhV5zIuNUUS23AH/Za5UIbcHPAZjTb4rAdeo+Dd41x3aq50FFja5iz9ZR/d/A/rtAZwNnHIF78pQjck9LPZl1cDLLbBzZ0yXWXwFXV4LR7Lm8ZcNzvB3zA//Cw8Lq4XZPOTF29nio+Q3SVN0ma7QdeDcBeI8LdAKenxGO7RL71IvU+9TH1OfE9fJiV7MBRp4Ul9+A26fPwA=</latexit><latexit sha1_base64="+aKJAuJ5GTjzashe/ty0wh0bMPc=">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</latexit><latexit sha1_base64="vqgaLawA7+EJgtqPeHfAzK4eLJE=">AAAGGHicpVRLaxNRFD6tRmt8tFUQwc1gjIhIuYmKDxCKdtGdfbfQhDAzuUmHzouZycsxf8A/IOjKggupP0JwoW4FF/0J4rKCGxd+98xYk7SJC+dy5557Xt85370zhm9bYSTE3tj4seOZEycnTmVPnzl7bnJq+vxa6DUCU66anu0FG4YeStty5WpkRbbc8AOpO4Yt143tx8q+3pRBaHnuStTxZdnR665Vs0w9gqoydXFOe6jNVQqYRa1kVr0ohFivTOXEjOBHOywUUiFH6bPgTY83qERV8sikBjkkyaUIsk06hRibVCBBPnRliqELIFlsl9SlLGIb8JLw0KHdxruO3WaqdbFXOUOONoFiYwaI1Cgvvoq3Yl98Ervim/iFXPkh2WLMBiMGIzBjxlI1d7AaCYb0K5PPLy3//GeUgzWirb9RI3uLqEb3uCcLPfqsUd2aSXzz6Yv95QdL+fia2BHf0edrsSc+oFO3+cN8syiXXo2ox0Atqsvh+E9ohYp0k9cCV1EDq25PDVk+Uwl9iZbRlQ6fGLIFjYPhsTWLWjVoVVwbb+XRn0n5qQ7rsCWoXfiomtRpmMxD4q8BZ56Rk5zKIzzA6B3/i5E9yGLBM2Fq+M2JOHeHWozVGclsjHcA2Qdj6o63R962ELkd/iI6mNXU14WtxV05bHVhUbw2uY4uywGsEd83idP2gBPTfXxlynqVT8nF2k0zqizb8FT+ipGI44K+M3a4alWRx7zYGLPIlj/gv8SVKuRWn0d/rMl3JeAaFf8G77oDe7WzwMIWd/En6/D+b0Df7sPZxCmX8S4P1Jjcw9merBp4uQV27ozoMoevoMtr8UjWPP6ywRn+joXBf+FhYa04U4C8eDs3+yj9T07QZbpC14FzF4jztECr6PEZ7dAuvcu8zLzPfMx8TlzHx9KYC9T3ZL78BguBPpA=</latexit>
(D1/p) = 1, (D2/p) = 1, · · · , (Dg/p) = 1
<latexit sha1_base64="j1VzRTFwdQaO75Z70Xjs6vJe1AE=">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</latexit><latexit sha1_base64="j1VzRTFwdQaO75Z70Xjs6vJe1AE=">AAAGPHicpVQ7b9NQFD4tBEp4tIUFicUiBBVUlesAAoqQImDoRtv0JTVRZLu3qRW/ZDtpg5U/wB9gYAEkBsR/YEFCbLAwdGdBFVMRLAx89zgtSUvCgC3fe+55fed8916bgWNHsRDbQ8NHjmaOHR85kT156vSZ0bHxs0uR3wgtuWj5jh+umEYkHduTi7EdO3IlCKXhmo5cNusPlH25KcPI9r2FuBXIimvUPHvdtowYqurY9MTDqn4tuKLd0/RJrXxXw7rQvS5ba34cKWnPXOuYq2M5MSX40Q4LekfIFQXxM+uPDzeoTGvkk0UNckmSRzFkhwyK8K6SToIC6CqUQBdCstkuqU1ZxDbgJeFhQFvHWMNqtaP1sFY5I462gOLgCxGpUV58Fq/Frvgg3oiv4hdy5ftkS/A1GDEcgJkwlqq5hdlMMWRQHX1yvvTzn1Eu5pg2/kQN7C2mdbrNPdnoMWCN6tZK45uPn+6WpufzyWXxUuygzxdiW7xDp17zu/VqTs4/G1CPiVpUl/3xH9ECFWiSZ52rWAerXlcNWd5TCX2ZSujKgE8C2YbGxeuzNYtaNWhV3BZG5dGbSfmpDmuwpaht+Kia1G5YzEPqrwFnhpHTnMoj2sfofv8XI7ufxYZnylT/kxNz7hZtMlZrILMJxhByAMbUGd8aeNoi5Hb5RrTwrXV8Pdg2uSuXrR4sitcm19FmOYQ15vMmsds+cBK6g1umrJd4lzzM7U5GlaUOT+WvGIk5LuzZY5erVhX5zIuDt4hs+X3+y1ypQt7s8eiNtfishFyj4t/kVfvAWq1ssLDBXexl7d//Vei3enBWscsVjJUDNabnsNiVVQMv18HOzQFd5nAL2jwX/sqazzcbnOHvqB/8Fx4WlgpTOuS5G7ni/fQ3SSN0gS7SBHBuAXGGZmkRPT6n9/SRPmXeZr5kdjLfUtfhoU7MOep5Mj9+AzPJSAE=</latexit><latexit sha1_base64="j1VzRTFwdQaO75Z70Xjs6vJe1AE=">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</latexit><latexit sha1_base64="VTCgOE3nG34TtroH6qlWps91TWo=">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</latexit>

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
<latexit sha1_base64="1TNSKsC8m0+HkZgM4WQQHGQnbx4=">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</latexit><latexit sha1_base64="1TNSKsC8m0+HkZgM4WQQHGQnbx4=">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</latexit><latexit sha1_base64="1TNSKsC8m0+HkZgM4WQQHGQnbx4=">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</latexit><latexit sha1_base64="i7Q2TyadRVTPgK3S9cd+cyv9sRY=">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</latexit>
ガロア拡⼤
Pe
1Pe
2 · · · Pe
g
<latexit sha1_base64="nKWoTIM6a2Zu6sKy1FYScD9sTNE=">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</latexit><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="aqW+ebmq+5rtYT3nZ4gMyOdQDxo=">AAAGQHicpVTLbtNAFL0tBEp4tIUNEhuLEIQQqiYBxGOBKth0R9v0JTUhst1JasUv2U7aYPkH+AEWrECwQPwFbBBbYNE/ALFAqEhIiAVnrk2p2yYssOWZO/d17j0zY8O3rTASYmtk9NDhwpGjY8eKx0+cPDU+MXl6KfS6gSkXTc/2ghVDD6VtuXIxsiJbrviB1B3DlstG556yL/dkEFqeuxD1fdlw9LZrtSxTj6BqTtypO3q03gr0TjybNCsPYpnkNFWl0ermmheFWs7SVpbmRElMCX60/UIlE0qUPbPe5GiX6rRGHpnUJYckuRRBtkmnEO8qVUiQD12DYugCSBbbJSVURGwXXhIeOrQdjG2sVjOti7XKGXK0CRQbX4BIjcrio3gptsVb8Up8Fr+QqzwgW4yvy4jBEMyYsVTNfcxGiiH95vijs7Uf/4xyMEe0/jdqaG8Rtegm92ShR581qlszje89fLxduz1fji+KZ+IL+nwqtsQbdOr2vpsv5uT8kyH1GKhFdTkY/z4tUJWu8FzhKlpg1d1VQ5H3VEJfpxq60uETQ7agcfB6bC2iVg1aFbeJUXnkMyk/1WEbthQ1gY+qSe2GyTyk/hpwZhg5zak8wh2M3e//YhR3sljwTJkafHIizt2nDcbqD2U2xhhA9sGYOuObQ09biNwO34g+vrXM14Vtg7ty2OrConjtcR0JywGsEZ83id32gBPTLdwyZb3Au+RiTrKMKksHnspfMRJxXJDbY4erVhV5zIuNdxrZyjv817lShbyR88jHmnxWAq5R8W/wKtmzVisLLKxzF3+yDu7/MvSbOZxV7HIDY2NPjek5nN6VVQMvV8HO9SFdlnALEp6rB7Lm8c0GZ/g7Vvb+C/cLS9WpCuS5a6Xpu9l/cozO0Xm6BJwbQJyhWVpEj8/pHb2nD4XXhU+Fr4VvqevoSBZzhnJP4edvQmNSeg==</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！

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

Plus de Junpei Tsuji

Plus de Junpei Tsuji (20)

Dernier

Dernier (20)

x^2 + ny^2 の形で表せる素数 - めざせプライムマスター！