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ラマヌジャンやっぱりやばいじゃん - 第15回 #日曜数学会

2019年6月29日に開催された第15回日曜数学会というイベントで発表した資料です。ラマヌジャンが発見したタクシー数1729にまつわるホットなお話について発表しました。

発表時間に収まらなかった分は、ブログ記事でもまとめましたので、詳細が気になる方はぜひこちらもどうぞ。
http://tsujimotter.hatenablog.com/entry/the-1729-k3-surface

発表者:tsujimotter
http://tsujimotter.info

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ラマヌジャンやっぱりやばいじゃん - 第15回 #日曜数学会

  1. 1. ラマヌジャン やっぱりやばいじゃん 辻順平 @tsujimotter (今⽇の) やばい⼈
  2. 2. タクシー数 1729 = 13 + 123 = 93 + 103 <latexit sha1_base64="Mv6nB7ymJNFAxSg3DzF2l2ZFXXI=">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</latexit><latexit sha1_base64="Mv6nB7ymJNFAxSg3DzF2l2ZFXXI=">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</latexit><latexit sha1_base64="Mv6nB7ymJNFAxSg3DzF2l2ZFXXI=">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</latexit><latexit sha1_base64="kb7jrfF6fwBYZST2g4gcIGPxeNc=">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</latexit> 1729は、2つの3乗数の和として2通りに表すことができる最⼩の数 (by ラマヌジャン) 2
  3. 3. ラマヌジャンやばい︕ 3
  4. 4. そんなの知ってるし いやいや・・・ 4
  5. 5. ・・・なあなたに 5
  6. 6. Wikipedia “ラマヌジャン” 情報提供︓みずすましさん(感謝︕) 6
  7. 7. タクシー数とK3曲⾯ 『・・・ラマヌジャンは答を導出する過程で1729と楕円曲線か ら今⽇で⾔う K3曲⾯ を構成していたことを発⾒した。これ はアンドレ・ヴェイユによるK3曲⾯の再発⾒と命名に30年以上 先⾏する仕事である。』 7
  8. 8. なんかかっこいい︕ 8
  9. 9. 9
  10. 10. 読んでみました 10
  11. 11. ラマヌジャンの発⾒(1913) オイラーのディオファントス⽅程式 の無限個の解を与えるパラメータ解︓ X3 + Y 3 = Z3 + W3 <latexit 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4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A <latexit 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B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 11
  12. 12. 両辺を 33 で割ると A = 2, B = 1 <latexit 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303 + 273 <latexit 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sha1_base64="V5K2VRsWpe5Hw8rqp4Nvxl5lhZs=">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</latexit> 123 = ( 1)3 + 103 + 93 <latexit 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(6A2 4AB + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5A <latexit 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sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit> + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 (タクシー数) 12
  13. 13. (6A2 4AB + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5A <latexit 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sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit> + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 A, B <latexit 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無数の「タクシー数的な数」が得られる︕ 13
  14. 14. やばい︕︕ 14
  15. 15. が、まだまだこれから 15
  16. 16. Ono and Trebat-Leder の論⽂ 1. タクシー数と楕円曲線 2. タクシー数とK3曲⾯ 16
  17. 17. 両辺 B6 で割って T = A/B とおく (6T2 4T + 4)3 + ( 3T2 5T + 5)3 = (4T2 4T + 6)3 + (5T2 <latexit 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T + 5)3 = (4T2 4T + 6)3 + (5T2 5T 3)3 = k(T) <latexit 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5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB <latexit 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sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit> 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 とおく 17
  18. 18. 上の⽅程式 の2つの解 が得られた。 X3 + Y 3 = k(T) <latexit 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(6T2 4T + 4, 3T2 5T + 5) = (4T2 4T + 6, 5T2 5T + 3) <latexit 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Q(T) <latexit 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  19. 19. 上の楕円曲線 X3 + Y 3 = k(T) <latexit 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<latexit 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k(t2) <latexit 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sha1_base64="r+eV9/Jgcm5PnvobFAIxIqbNht8=">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</latexit><latexit 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k(t3) <latexit 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