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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!

2019/03/30 関東すうがく徒のつどい #kantomath で講演予定のスライドです。

【2019/03/30 〜 2019/04/01 までの期間限定で公開します。】
スライドの内容チェックおよび修正の可能性があるため、公開終了後期間をおいて再アップロードします。

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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!

  1. 1. 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
  2. 2. スライドは期間限定で公開 #kantomath で検索(or @tsujimotter で検索) 2
  3. 3. 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
  4. 4. 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
  5. 5. 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
  6. 6. 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
  7. 7. めざせプライムマスター! 7
  8. 8. あっ!野⽣の  p = x2 + y2 が⾶び出してきた! 8
  9. 9. 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) •  7(かけない) •  11(かけない) •  13 = 32 + 22(かける) •  17 = 42 + 12(かける) •  19(かけない) 9
  10. 10. 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) •  3(かけない) •  7(かけない) •  11(かけない) •  19(かけない) p ≡ 1 (mod 4) or p = 2 p ≡ 3 (mod 4) 10
  11. 11. 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!
  12. 12. 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. 13. 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 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( 2/p) = 1 () p ⌘ 1, 7 (mod 8) <latexit 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(q/p) = ( 1) p 1 2 ( 1) q 1 2 (p/q) <latexit 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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) <latexit 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  14. 14. 「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)
  15. 15. p = x2 + 2y2 (n = 2 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 + 2Y2 ⇔ p ≡ 1, 7 (mod 8) 15 x2 + 2y2 型素数の法則 (–2/p) = 1 法則、ゲットだぜ!
  16. 16. p = x2 + 3y2 (n = 3 のとき) 3 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 + 3Y2 ⇔ p ≡ 1, 7 (mod 12) 16 x2 + 3y2 型素数の法則 (–3/p) = 1 法則、ゲットだぜ!
  17. 17. p = x2 + 7y2 (n = 7 のとき) 7 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 + 7Y2 ⇔ p ≡ 1, 9, 11, 15, 23, 25 (mod 28) 17 x2 + 7y2 型素数の法則 (–7/p) = 1 法則、ゲットだぜ!
  18. 18. 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
  19. 19. ガウス地⽅に⽣息している x2 + ny2 19 x2 + 5y2 x2 + 6y2 x2 + 10y2
  20. 20. 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
  21. 21. 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
  22. 22. 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)
  23. 23. 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=">AAAGe3icpVRLb9NAEJ4UGkqAPuCCxMUipEJVVW0KFQ8JqYJLb/TdoiaqbHebWvUL20ljLP8B/gAHTiBxQPwMLvwBDj1xRhyLxAUkvh2btnFJOGDLu7Mz38w3M7trw7etMBLisDR07vxw+cLIxcqly1dGx8Ynrq6HXjsw5Zrp2V6waeihtC1XrkVWZMtNP5C6Y9hyw9h/ouwbHRmElueuRrEvm47ecq1dy9QjqLbH44YhW5ab+I4eBVY31Ta1RkN7pjWku3OifKQVYb42qT1X0ABCWIAXwV0FjAug7fGqmBH8aGeFei5UKX8WvYmhNjVohzwyqU0OSXIpgmyTTiHeLaqTIB+6JiXQBZAstktKqQLfNlASCB3afYwtrLZyrYu1ihmytwkWG18AT41q4rN4L47EJ/FBfBU/EavWJ1qCr82MwQDOhLlUzjFmI+OQ/vbYy+srP/7p5WCOaO/Ea2BtEe3Sfa7JQo0+a1S1ZubfefHqaOXhci2ZFG/FN9T5RhyKj6jU7Xw33y3J5dcD8jGQi6qyP/9TWqVZmua5zlnsoqvuqRwqvKcS+gatoCodmASyBY2D12NrBblq0Cq/LkaF6I2kcKrCFmwZawqMyknthsl9yPAaeBaYOYupEOExx+n3fzkqx1EsILNO9T85EceO6YC54oGdTTAGkH10TJ3x7sDTFiK2wzcixreTY13YDrgqh60uLKqvHc4jZTmANeLzJrHbHngSeoBbpqy3eJdczGkeUUXZB1LhVUci9gt69tjhrFVGHvfFxjuPaLXj/jc4U8V80IPo9TX5rASco+q/wau0sFYrC13Y4yr+RO1f/xT03R6eLexyE2OzkGN2DudPRdXQlzvoztyAKqu4BSnPs3/tmsc3Gz3D37Fe/BeeFdZnZ+qQl+5W5x/n/8kRukE36TZ47oFxgRZpDTV+KQ2XRktjw7/K1fJUeTqDDpVyn2vU85TnfgM5+2GL</latexit> (ps – qr = ±1 のとき逆変換を持つ)
  24. 24. 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
  25. 25. 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
  26. 26. 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. 27. 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
  28. 28. 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
  29. 29. 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=">AAAGP3icpVTLbtNAFL0tBEp4tIUNEhuLECivahKKeEhIUdtFd/RdpDhUtjtJrfoR2U6aYPkH+AEWrACxQHwFYgNbEIt+AUJICKlICIkFZ65DaVoSFtjyzJ37OveembFZd+wwEmJrYPDAwcyhw0NHskePHT8xPDJ6cjn0G4Ellyzf8YN7phFKx/bkUmRHjrxXD6Thmo5cMTemlH2lKYPQ9r3FqF2XFdeoeXbVtowIqtWRO7ojq1HZMC607he1y9pkq41RrwaGFU9Cc1WbTuIJmJM2Vnpg19ajiqbbnjY1Nn1xdSQnxgU/2n6h0BFyJUH8zPqjgw3SaY18sqhBLknyKILskEEh3jIVSFAdugrF0AWQbLZLSiiL2Aa8JDwMaDcw1rAqd7Qe1ipnyNEWUBx8ASI1yosP4oXYFm/ES/FJ/ESufI9sMb4GIwZ9MGPGUjW3MZsphqyvDj88vfD9n1Eu5ojW/0T17S2iKt3knmz0WGeN6tZK45sPHm0v3J7Px+fFU/EZfT4RW+I1OvWa36znc3L+cZ96TNSiuuyNf5cWqUhXeC5wFVWw6u2qIct7KqHXaQFdGfCJIdvQuHh9tmZRqwatimthVB7dmZSf6rAGW4qawEfVpHbDYh5Sfw04M4yc5lQe4Q7G7vd/MbI7WWx4pkz1PjkR527TJmO1+zIbYwwg18GYOuOtvqctRG6Xb0Qb31rH14Ntk7ty2erBonhtch0JywGsEZ83id32gRPTLdwyZT3Hu+RhTjoZVZYNeCp/xUjEcUHXHrtctarIZ14cvCVky+/wr3OlCnmzy6M71uKzEnCNin+TV8metVrZYGGdu/idtXf/l6BvdeGUscsVjJU9NabnsLQrqwZeroGd6326zOEWJDwX/8qazzcbnOHvWNj7L9wvLBfHC5DnJnKlyfQ3SUN0hs7SGHBuAHGGZmkJPT6jt/SO3mdeZT5mvmS+pq6DA52YU9T1ZH78AiNRTWg=</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
  30. 30. 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=">AAAGG3icpVQ7b9NQFD4tBEp4NIUFxGIRggqU6iaAeEwRLN3ou5WSKLLd29aKX7KdNMGK2PkDDIgBpA4ICX5EF4RYO/QnIMYisTDw3WNTmrQJA7Z877nn9Z3z3Xtt+LYVRkLsjYyeOJk5dXrsTPbsufMXxnMTF5dDrxmYcsn0bC9YNfRQ2pYrlyIrsuWqH0jdMWy5YjSeKvtKSwah5bmLUceXNUffcK11y9QjqOq5K5PtKa1zs+rofhh5mlrdaWu3oarn8mJa8KMdFYqpkKf0mfUmRptUpTXyyKQmOSTJpQiyTTqFeCtUJEE+dDWKoQsgWWyX1KUsYpvwkvDQoW1g3MCqkmpdrFXOkKNNoNj4AkRqVBC74r3YF5/FB/FN/EKuwoBsMb4mIwZDMGPGUjV3MBsJhvTr4y8vL/z8Z5SDOaLNv1FDe4tonR5yTxZ69FmjujWT+NbzV/sLj+cL8Q3xTnxHn2/FnthBp27rh7k9J+dfD6nHQC2qy8H4z2iRSjTFc5GrWAer7qEasrynEvoqLaArHT4xZAsaB6/H1ixq1aBVcW2MyqM3k/JTHW7AlqB24aNqUrthMg+JvwacGUZOciqP8ADj8Pu/GNmDLBY8E6YGn5yIc3doi7E6Q5mNMQaQfTCmznh76GkLkdvhG9HBt5b6urBtcVcOW11YFK8trqPLcgBrxOdNYrc94MT0CLdMWa/zLrmYu2lGlaUBT+WvGIk4LujZY4erVhV5zIuNt4xshQP+q1ypQt7q8eiNNfmsBFyj4t/gVbdvrVYWWNjkLv5kHdz/LejbPTgV7HINY62vxuQclg9l1cDLXbBzf0iXedyCLs+lY1nz+GaDM/wdi/3/wqPCcmm6CHnuXr78JP1PjtFVukaTwHkAxBmapSX0+IK26SN9yrzJ7GS+ZL4mrqMjacwl6nkyu78B5qg/nQ==</latexit>
  31. 31. 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] }
  32. 32. 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. 33. 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] [   ]
  34. 34. 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=">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</latexit>
  35. 35. 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 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1, (D2/p) = 1, · · · , (Dg/p) = 1 <latexit 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  36. 36. 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
  37. 37. 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. 整数 X, Y が存在して p = X2 + 5Y2 ⇔ p ≡ 1, 9 (mod 20) x2 + 5y2 型素数の法則 法則、ゲットだぜ!
  38. 38. 38 2, 3 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 + 6Y2 ⇔ p ≡ 1, 7 (mod 24) x2 + 6y2 型素数の法則 2, 5 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 + 10Y2 ⇔ p ≡ 1, 9, 11, 19 (mod 40) x2 + 10y2 型素数の法則 法則、ゲットだぜ!
  39. 39. 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
  40. 40. 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
  41. 41. E715 On various ways of examining very large numbers, for whether or not they are primes (便利数)
  42. 42. 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 法則、ゲットだぜ!
  43. 43. 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
  44. 44. 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
  45. 45. 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. 46. p = x2 + ny2 とはどういうことか? 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. 47. p = x2 + ny2 とはどういうことか? 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
  48. 48. 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 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  49. 49. ヒルベルトの分岐理論の設定 49 K (代数体) L (Kのガロア拡⼤体) K の素イデアル p <latexit 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K]   完全分解 ガロア拡⼤ Pe 1Pe 2 · · · Pe g <latexit 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  50. 50. ヒルベルトの分岐理論の設定 50 K (代数体) L (Kのガロア拡⼤体) K の素イデアル p <latexit 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で素イデアル分解する様⼦を調べたいp <latexit 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e = 1 のとき 不分岐 g = [L : K]   完全分解

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