利用者:Inforiver/sandbox

カンペドフェリエの超幾何関数

$(u+v)^{6}+t(u+v)^{5}+s(u+v)^{4}=0$

$u^{6}+6u^{5}v+15u^{4}v^{2}+20u^{3}v^{3}+15u^{2}v^{4}+6uv^{5}+v^{6}+t(u^{5}+5u^{4}v+10u^{3}v^{2}+10u^{2}v^{3}+5uv^{4}+v^{5})+s(u^{4}v^{2}+4u^{3}v^{3}+6u^{2}v^{4}+4uv^{5}+v^{6})=0$

$u=A\cos(t\theta )+B\sin(t\theta )$

$v=C\cos(s\theta )+D\sin(s\theta )$

$x=(A\cos(t\theta )+B\sin(t\theta ))+(C\cos(s\theta )+D\sin(s\theta ))$

${\displaystyle }$

${\begin{aligned}\arctan z&=\textstyle \sum \limits _{n=0}^{\infty }{\dfrac {(-1)^{n}}{2n+1}}z^{2n+1}\\&=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\dotsb ;\quad |z|\leq 1,z\neq \pm i\end{aligned}}$

$\tan ^{-1}z=z\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}z^{2n}=z\cdot {_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},1\\{\frac {3}{2}}\end{matrix}};-z^{2}\right]$

${\frac {\tan ^{-1}z}{z}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}z^{2n}={_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},1\\{\frac {3}{2}}\end{matrix}};-z^{2}\right]$

$\left({\frac {\tan ^{-1}z}{z}}\right)^{1}={_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},1\\{\frac {3}{2}}\end{matrix}};-z^{2}\right]$

$\left({\frac {\tan ^{-1}z}{z}}\right)^{2}=F_{1:1;0}^{1:2;1}\left[{\begin{matrix}1:&{\frac {1}{2}},1;&1;\\2:&{\frac {3}{2}};&-;\end{matrix}}-z^{2},-z^{2}\right]$

$\left({\frac {\tan ^{-1}z}{z}}\right)^{3}=F^{(3)}\left[{\begin{matrix}{\frac {3}{2}}\colon \!\colon &-;\;1;\;-\colon &1;\;1;\;{\frac {1}{2}},1;\\{\frac {5}{2}}\colon \!\colon &-;\;2;\;-\colon &-;\;-;\;{\frac {3}{2}};\end{matrix}}-z^{2},-z^{2},-z^{2}\right]$

Certain Identities Involving the General Kampé de Fériet Function and Srivastava’s General Triple Hypergeometric Series

${\dfrac {\left({\dfrac {1}{2}}\right)_{n}}{\left({\dfrac {3}{2}}\right)_{n}}}={\dfrac {1}{2n+1}}$

${\dfrac {(1)_{n}}{(2)_{n}}}={\dfrac {1}{n+1}}$

入国管理局正字

亐 仝 俓 僿 內 卄 卨 吳 姬 娛 媤 嬅 尙 峀 嶪 帿 强 悅 慤 慽 戱 戶 敎 敭 敾 旣 旽 昐 昻 暳 枾 栒 桭 椧 橓 櫶 欌 歲 毁 浿 淃 淸 渽 湺 瀜 灐 炚 炡 熉 玧 珤 琠 琡 琸 瑥 璂 瓆 甛 畓 畵 癎 磵 礖 稅 稶 竗 筽 篒 綎 耉 耭 脫 𧥱 說 諪 𧴫 𨋓 迲 銳 鏶 鐥 閒 閱 靑 頀 頹 騈 髙 魽

㐇 㐈 㐉 乤 㐊 㐋 乧 㐍 㐎 㐏 乬 乫 㠰 乭 𭆁 㐐 㐑 㐒 㐓 㐕 㐔 𬼟 乮 乯 乲 㐗 㐘 𭆂 乶 乷 㐙 㐚 㐛 乺 乻 乼 乽 㐝 㐞 㐟 㐠 㐢 㐣 㐥 㐦 㕾 兺 哛 㖋 㖎 巼 㖌 㖍 唟 㖙 㖚 㖛 㖜 唜 㖝 𮂻 莻 㖯 㖰 㖲 㖳 𮇎 喸 蒊 㗡 嗭 旕 㗟 㗠 㗯 廤 㘏 㘒 㪲 巪 㔖 㫇 䎞 㪳 㔔 㫈 䜳 䪪

漢字	入管正字コード	入管外字コード	住基ネット統一文字コード	対応するUCS	備考
亐	4E90	E577	J+C150	U+4E90
仝	4EDD		J+4EDD	U+4EDD
俓
僿
內
卄
卨
吳
姬
娛
媤
嬅
尙
峀
嶪
帿
强
悅
慤
慽
戱
戶
敎
敭
敾
旣
旽
昐
昻
暳
枾
栒
桭
椧
橓
櫶
欌
歲
毁
浿
淃
淸
渽
湺
瀜
灐
炚
炡
熉
玧
珤
琠
琡
琸
瑥
璂
瓆
甛
畓
畵
癎
磵
礖
稅
稶
竗
筽
篒
綎
耉
耭
脫
𧥱
說
諪
𧴫
𨋓
迲
銳
鏶
鐥
閒
閱
靑
頀
頹
騈
髙
魽

漢字	入管正字コード	入管外字コード	住基ネット統一文字コード	対応するUCS	備考
㐇	3407	E530	J+C109	U+3407	[韓]音は「굴(クル)」
㐈	3408	E531	J+C10A	U+3408	[韓]音は「둘(トゥル)」
㐉	3409	E532	J+C10B	U+3409	[韓]音は「절(チョル)」
乤	4E64	E569	J+C142	U+4E64	[韓]音は「할(ハル)」
㐊	340A	E533	J+C10C	U+340A	[韓]音は「살(サル)」
㐋	340B	E534	J+C10D	U+340B	[韓]音は「톨(トル)」
乧	4E67	E56A	J+C143	U+4E67	[韓]音は「둘(トゥル)」
㐍
㐎
㐏
乬
乫
㠰
乭
𭆁
㐐
㐑
㐒
㐓
㐕
㐔
𬼟
乮
乯
乲
㐗
㐘
𭆂
乶
乷
㐙
㐚
㐛
乺
乻
乼
乽
㐝
㐞
㐟
㐠
㐢
㐣
㐥
㐦
㕾
兺
哛
㖋
㖎
巼
㖌
㖍
唟
㖙
㖚
㖛
㖜
唜
㖝
𮂻
莻
㖯
㖰
㖲
㖳
𮇎
喸
蒊
㗡
嗭
旕
㗟
㗠
㗯
廤
㘏
㘒
㪲
巪
㔖
㫇
䎞
㪳
㔔
㫈
䜳
䪪

正11角形

${\begin{aligned}\cos {\frac {2\pi }{11}}=&a_{0}\\&+(a_{1}+b_{1}i)\left({\sqrt[{5}]{-a_{2}-a_{3}-\left(9b_{2}-b_{3}\right)i}}+{\sqrt[{5}]{-a_{2}+a_{3}-\left(b_{2}+9b_{3}\right)i}}\right)\\&+(a_{1}-b_{1}i)\left({\sqrt[{5}]{-a_{2}-a_{3}+\left(9b_{2}-b_{3}\right)i}}+{\sqrt[{5}]{-a_{2}+a_{3}+\left(b_{2}+9b_{3}\right)i}}\right)\\=&0.841253\dots \end{aligned}}$

$\cos {\frac {2\pi }{11}}=a_{0}+(a_{1}+b_{1}i)\left({\sqrt[{5}]{-a_{2}-a_{3}-\left(9b_{2}-b_{3}\right)i}}+{\sqrt[{5}]{-a_{2}+a_{3}-\left(b_{2}+9b_{3}\right)i}}\right)+(a_{1}-b_{1}i)\left({\sqrt[{5}]{-a_{2}-a_{3}+\left(9b_{2}-b_{3}\right)i}}+{\sqrt[{5}]{-a_{2}+a_{3}+\left(b_{2}+9b_{3}\right)i}}\right)$

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正十九角形の余弦

$\cos(2\pi /19)$を平方根と立方根で表すことが可能であるが、三次方程式を2回解く必要である。 以下には、中間結果（三次方程式を1回解いた際の関係式）を示す。

${\begin{aligned}2\cos {\frac {2\pi }{19}}+2\cos {\frac {16\pi }{19}}+2\cos {\frac {14\pi }{19}}=&{\frac {-1+{\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}\omega ^{2}+{\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}\omega }{3}}\\=&{\frac {1}{3}}\left({-1+{\sqrt {19}}\cdot {\sqrt[{3}]{\frac {7+3{\sqrt {3}}i}{2{\sqrt {19}}}}}\omega ^{2}+{\sqrt {19}}\cdot {\sqrt[{3}]{\frac {7-3{\sqrt {3}}i}{2{\sqrt {19}}}}}\omega }\right)\\=&{\frac {1}{3}}\left({-1+{\sqrt {19}}\omega ^{2}\cdot {\sqrt[{3}]{\frac {3\omega +5}{\sqrt {19}}}}+{\sqrt {19}}\omega \cdot {\sqrt[{3}]{\frac {3\omega ^{2}+5}{\sqrt {19}}}}}\right)\\=&\alpha \end{aligned}}$

${\begin{aligned}2\cos {\frac {4\pi }{19}}+2\cos {\frac {6\pi }{19}}+2\cos {\frac {10\pi }{19}}=&{\frac {-1+{\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+{\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}\\=&{\frac {1}{3}}\left({-1+{\sqrt {19}}\cdot {\sqrt[{3}]{\frac {7+3{\sqrt {3}}i}{2{\sqrt {19}}}}}+{\sqrt {19}}\cdot {\sqrt[{3}]{\frac {7-3{\sqrt {3}}i}{2{\sqrt {19}}}}}}\right)\\=&{\frac {1}{3}}\left({-1+{\sqrt {19}}\cdot {\sqrt[{3}]{\frac {3\omega +5}{\sqrt {19}}}}+{\sqrt {19}}\cdot {\sqrt[{3}]{\frac {3\omega ^{2}+5}{\sqrt {19}}}}}\right)\\=&\beta \end{aligned}}$

${\begin{aligned}2\cos {\frac {8\pi }{19}}+2\cos {\frac {18\pi }{19}}+2\cos {\frac {12\pi }{19}}=&{\frac {-1+{\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}\omega +{\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}\omega ^{2}}{3}}\\=&{\frac {1}{3}}\left({-1+{\sqrt {19}}\cdot {\sqrt[{3}]{\frac {7+3{\sqrt {3}}i}{2{\sqrt {19}}}}}\omega +{\sqrt {19}}\cdot {\sqrt[{3}]{\frac {7-3{\sqrt {3}}i}{2{\sqrt {19}}}}}\omega ^{2}}\right)\\=&{\frac {1}{3}}\left({-1+{\sqrt {19}}\omega \cdot {\sqrt[{3}]{\frac {3\omega +5}{\sqrt {19}}}}+{\sqrt {19}}\omega ^{2}\cdot {\sqrt[{3}]{\frac {3\omega ^{2}+5}{\sqrt {19}}}}}\right)\\=&\gamma \end{aligned}}$

$\cos(2\pi /19)$は $\alpha ,\beta$ を用いた以下の三次方程式の解の一つである。

$x^{3}-{\frac {\alpha }{2}}x^{2}-{\frac {\beta +1}{4}}x-{\frac {\beta +2}{8}}=0$

変数変換、関係式より

$x=y+{\frac {\alpha }{6}},\quad \alpha ^{2}=4-\beta ,\quad \alpha \beta =-3-2\alpha -\beta ,\quad \alpha ^{3}=3+6\alpha +\beta$

整理すると

$y^{3}-{\frac {2\beta +7}{12}}y-{\frac {3\alpha +20\beta +33}{216}}=0$

三角関数を使用した解の1つは

$y={\frac {1}{3}}{\sqrt {2\beta +7}}\cos \left({\frac {1}{3}}\arccos {\frac {3\alpha +20\beta +33}{2(2\beta +7)^{\frac {3}{2}}}}\right)$

上記の三次方程式は以下のように変形できる。

$y^{3}-{\frac {2\beta +7}{12}}y-{\frac {(2\beta +7)(-\alpha +12\beta +15)}{216\cdot 7}}=0$

三角関数を使用した解の1つは

$y={\frac {1}{3}}{\sqrt {2\beta +7}}\cos \left({\frac {1}{3}}\arccos {\frac {-\alpha +12\beta +15}{14{\sqrt {(2\beta +7)}}}}\right)$

※（注意）計算ミスの可能性がある。

$y={\sqrt[{3}]{{\frac {(2\beta +7)(-\alpha +12\beta +15)}{2\cdot 216\cdot 7}}+{\frac {(2\beta +7)}{2\cdot 216\cdot 7}}\cdot {\sqrt {162\alpha +135\beta -351}}}}+{\sqrt[{3}]{{\frac {(2\beta +7)(-\alpha +12\beta +15)}{2\cdot 216\cdot 7}}-{\frac {(2\beta +7)}{2\cdot 216\cdot 7}}\cdot {\sqrt {162\alpha +135\beta -351}}}}$

$x={\frac {\alpha }{6}}+{\sqrt[{3}]{{\frac {(2\beta +7)(-\alpha +12\beta +15)}{2\cdot 216\cdot 7}}+{\frac {(2\beta +7)}{2\cdot 216\cdot 7}}\cdot {\sqrt {162\alpha +135\beta -351}}}}+{\sqrt[{3}]{{\frac {(2\beta +7)(-\alpha +12\beta +15)}{2\cdot 216\cdot 7}}-{\frac {(2\beta +7)}{2\cdot 216\cdot 7}}\cdot {\sqrt {162\alpha +135\beta -351}}}}$

$x={\frac {\alpha }{6}}+{\sqrt[{3}]{{\frac {(2\beta +7)(-\alpha +12\beta +15)}{2\cdot 216\cdot 7}}+i{\frac {(2\beta +7)}{2\cdot 216\cdot 7}}\cdot {\sqrt {-162\alpha -135\beta +351}}}}+{\sqrt[{3}]{{\frac {(2\beta +7)(-\alpha +12\beta +15)}{2\cdot 216\cdot 7}}-i{\frac {(2\beta +7)}{2\cdot 216\cdot 7}}\cdot {\sqrt {-162\alpha -135\beta +351}}}}$

$x={\frac {\alpha }{6}}+{\frac {\sqrt {2\beta +7}}{6}}{\sqrt[{3}]{{\frac {-\alpha +12\beta +15}{14{\sqrt {2\beta +7}}}}+i{\frac {\sqrt {-162\alpha -135\beta +351}}{14{\sqrt {2\beta +7}}}}}}+{\frac {\sqrt {2\beta +7}}{6}}{\sqrt[{3}]{{\frac {-\alpha +12\beta +15}{14{\sqrt {2\beta +7}}}}-i{\frac {\sqrt {-162\alpha -135\beta +351}}{14{\sqrt {2\beta +7}}}}}}$

見直し

${\begin{aligned}&2\cos {\frac {2\pi }{19}}+2\cos {\frac {16\pi }{19}}+2\cos {\frac {14\pi }{19}}=\alpha \\&2\cos {\frac {2\pi }{19}}+\omega \cdot 2\cos {\frac {16\pi }{19}}+\omega ^{2}\cdot 2\cos {\frac {14\pi }{19}}\\&2\cos {\frac {2\pi }{19}}+\omega ^{2}\cdot 2\cos {\frac {16\pi }{19}}+\omega \cdot 2\cos {\frac {14\pi }{19}}\\\end{aligned}}$

(a+b+c)^3の展開公式、ω、ω²はz^3=1の複素数解

${\begin{aligned}(a+b+c)^{3}=&a^{3}+b^{3}+c^{3}+3a^{2}b+3ab^{2}+3b^{2}c+3bc^{2}+3c^{2}a+3ca^{2}+6abc\\(a+\omega b+\omega ^{2}c)^{3}=&a^{3}+b^{3}+c^{3}+6abc+3\omega (a^{2}b+b^{2}c+c^{2}a)+3\omega ^{2}(ab^{2}+bc^{2}+ca^{2})\\(a+\omega ^{2}b+\omega c)^{3}=&a^{3}+b^{3}+c^{3}+6abc+3\omega ^{2}(a^{2}b+b^{2}c+c^{2}a)+3\omega (ab^{2}+bc^{2}+ca^{2})\\\end{aligned}}$

この式を利用して

${\begin{aligned}\left(2\cos {\frac {2\pi }{19}}+\omega \cdot 2\cos {\frac {16\pi }{19}}+\omega ^{2}\cdot 2\cos {\frac {14\pi }{19}}\right)^{3}=&3\alpha +7\beta +12+\omega (6\alpha +6\gamma )+\omega ^{2}(6\alpha +3\beta +3\gamma )\\=&3\alpha +7\beta +12-6\omega (\beta +1)+3\omega ^{2}(\alpha -1)\\\left(2\cos {\frac {2\pi }{19}}+\omega ^{2}\cdot 2\cos {\frac {16\pi }{19}}+\omega \cdot 2\cos {\frac {14\pi }{19}}\right)^{3}=&3\alpha +7\beta +12-6\omega ^{2}(\beta +1)+3\omega (\alpha -1)\\\end{aligned}}$

両辺の立方根を取ると

${\begin{aligned}2\cos {\frac {2\pi }{19}}+\omega \cdot 2\cos {\frac {16\pi }{19}}+\omega ^{2}\cdot 2\cos {\frac {14\pi }{19}}=&{\sqrt[{3}]{3\alpha +7\beta +12-6\omega (\beta +1)+3\omega ^{2}(\alpha -1)}}\\2\cos {\frac {2\pi }{19}}+\omega ^{2}\cdot 2\cos {\frac {16\pi }{19}}+\omega \cdot 2\cos {\frac {14\pi }{19}}=&{\sqrt[{3}]{3\alpha +7\beta +12-6\omega ^{2}(\beta +1)+3\omega (\alpha -1)}}\\\end{aligned}}$

よって

${\begin{aligned}6\cos {\frac {2\pi }{19}}=&\alpha +{\sqrt[{3}]{3\alpha +7\beta +12-6\omega (\beta +1)+3\omega ^{2}(\alpha -1)}}+{\sqrt[{3}]{3\alpha +7\beta +12-6\omega ^{2}(\beta +1)+3\omega (\alpha -1)}}\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left(\alpha +{\sqrt[{3}]{3\alpha +7\beta +12-6\omega (\beta +1)+3\omega ^{2}(\alpha -1)}}+{\sqrt[{3}]{3\alpha +7\beta +12-6\omega ^{2}(\beta +1)+3\omega (\alpha -1)}}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left(\alpha +{\sqrt[{3}]{3\alpha +7\beta -6\omega \beta +3\omega ^{2}\alpha +12-3\omega ^{2}-6\omega }}+{\sqrt[{3}]{3\alpha +7\beta -6\omega ^{2}\beta +3\omega \alpha +12-3\omega -6\omega ^{2}}}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left(\alpha +{\sqrt[{3}]{3\alpha +7\beta -6\omega \beta +3(-1-\omega )\alpha +12-3(-1-\omega )-6\omega }}+{\sqrt[{3}]{3\alpha +7\beta -6(-1-\omega )\beta +3\omega \alpha +12-3\omega -6(-1-\omega )}}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left(\alpha +{\sqrt[{3}]{3\alpha +7\beta -6\omega \beta -3\alpha -3\omega \alpha +12+3+3\omega -6\omega }}+{\sqrt[{3}]{3\alpha +7\beta +6\beta +6\omega \beta +3\omega \alpha +12-3\omega +6+6\omega }}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left(\alpha +{\sqrt[{3}]{7\beta -6\omega \beta -3\omega \alpha +15-3\omega }}+{\sqrt[{3}]{3\alpha +13\beta +6\omega \beta +3\omega \alpha +18+3\omega }}\right)\\\end{aligned}}$

以下のように置くと

${\begin{aligned}A={\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}\\B={\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}\\\end{aligned}}$

α、βは以下のように表される。

${\begin{aligned}&\alpha ={\frac {-1+\omega ^{2}A+\omega B}{3}}\\&\beta ={\frac {-1+A+B}{3}}\\\end{aligned}}$

-6ωβ、-3ωα、3αは以下のように表される。

${\begin{aligned}&-6\omega \beta =2\omega -2\omega A-2\omega B\\&-3\omega \alpha =\omega -A-\omega ^{2}B=\omega -A+(\omega +1)B=\omega -A+\omega B+B\\&3\alpha =-1-A-\omega A+\omega B\end{aligned}}$

これらの値を代入すると

${\begin{aligned}\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left({\frac {-1+\omega ^{2}A+\omega B}{3}}+{\sqrt[{3}]{7{\frac {-1+A+B}{3}}-2\omega A-2\omega B-A+\omega B+B+15}}+{\sqrt[{3}]{-1-A-\omega A+\omega B+13{\frac {-1+A+B}{3}}+2\omega A+2\omega B+A-\omega B-B+18}}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left({\frac {-1+\omega ^{2}A+\omega B}{3}}+{\sqrt[{3}]{7{\frac {-1+A+B}{3}}-2\omega A-\omega B-A+B+15}}+{\sqrt[{3}]{13{\frac {-1+A+B}{3}}+\omega A+2\omega B-B+17}}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left({\frac {-1+\omega ^{2}A+\omega B}{3}}+{\sqrt[{3}]{\frac {-7+7A+7B-6\omega A-3\omega B-3A+3B+45}{3}}}+{\sqrt[{3}]{\frac {-13+13A+13B+3\omega A+6\omega B-3B+51}{3}}}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left({\frac {-1+\omega ^{2}A+\omega B}{3}}+{\sqrt[{3}]{\frac {38+4A+10B-6\omega A-3\omega B}{3}}}+{\sqrt[{3}]{\frac {38+13A+10B+3\omega A+6\omega B}{3}}}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left({\frac {-1+\omega ^{2}A+\omega B}{3}}+{\sqrt[{3}]{\frac {38+(4-6\omega )A+(10-3\omega )B}{3}}}+{\sqrt[{3}]{\frac {38+(13+3\omega )A+(10+6\omega )B}{3}}}\right)\\\end{aligned}}$

A、Bを代入すると

${\begin{aligned}\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left({\frac {-1+\omega ^{2}{\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+\omega {\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}+{\sqrt[{3}]{\frac {38+(4-6\omega ){\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+(10-3\omega ){\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}}+{\sqrt[{3}]{\frac {38+(13+3\omega ){\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+(10+6\omega ){\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}}\right)\\\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left({\tfrac {-1+\omega ^{2}{\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+\omega {\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}+{\sqrt[{3}]{\tfrac {38+(10+6\omega ^{2}){\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+(10-3\omega ){\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}}+{\sqrt[{3}]{\tfrac {38+(10-3\omega ^{2}){\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+(10+6\omega ){\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}}\right)\\\end{aligned}}$

単純にωを複素数にして整理すると

${\begin{aligned}\cos {\frac {2\pi }{19}}=&{\frac {1}{6}}\left(\alpha +{\sqrt[{3}]{{\frac {3\alpha +20\beta +33}{2}}-i\cdot {\frac {3{\sqrt {3}}(\alpha +2\beta +1)}{2}}}}+{\sqrt[{3}]{{\frac {3\alpha +20\beta +33}{2}}+i\cdot {\frac {3{\sqrt {3}}(\alpha +2\beta +1)}{2}}}}\right)\\\end{aligned}}$

多角形

正(2n+1)角形、正2(2n+1)角形、正4(2n+1)角形のコサイン

${\begin{aligned}&\cos {\frac {2\pi }{2n+1}}\\&\cos {\frac {2\pi }{2(2n+1)}}=\cos {\frac {\pi }{2n+1}}=\cos \left(\pi -{\frac {2n\pi }{2n+1}}\right)=-\cos {\frac {2n\pi }{2n+1}}\\&\cos {\frac {2\pi }{4(2n+1)}}=\cos {\frac {\pi }{2(2n+1)}}={\frac {1}{2}}{\sqrt {2+2\cos {\frac {\pi }{2n+1}}}}={\frac {1}{2}}{\sqrt {2-2\cos {\frac {2n\pi }{2n+1}}}}\\&=\sin {\frac {n\pi }{2n+1}}\\&={\sqrt {1-\cos ^{2}{\frac {n\pi }{2n+1}}}}={\sqrt {1-\cos ^{2}{\frac {(n+1)\pi }{2n+1}}}}={\sqrt {1+\cos {\frac {n\pi }{2n+1}}\cos {\frac {(n+1)\pi }{2n+1}}}}\\\end{aligned}}$

正多角形

使い方の注意が必要な式変形

$\cos {\frac {\theta }{n}}={\frac {e^{i{\frac {\theta }{n}}}+e^{-i{\frac {\theta }{n}}}}{2}}={\frac {{\sqrt[{n}]{e^{i\theta }}}+{\sqrt[{n}]{e^{-i\theta }}}}{2}}={\frac {{\sqrt[{n}]{\cos \theta +i\sin \theta }}+{\sqrt[{n}]{\cos \theta -i\sin \theta }}}{2}}={\frac {{\sqrt[{n}]{\cos \theta +i{\sqrt {1-\cos ^{2}\theta }}}}+{\sqrt[{n}]{\cos \theta -i{\sqrt {1-\cos ^{2}\theta }}}}}{2}}$

$\cos {\frac {\theta }{n}}={\frac {{\sqrt[{n}]{{\frac {\sqrt {1+\cos {2\theta }}}{2}}+i{\frac {\sqrt {1-\cos {2\theta }}}{2}}}}+{\sqrt[{n}]{{\frac {\sqrt {1+\cos {2\theta }}}{2}}-i{\frac {\sqrt {1-\cos {2\theta }}}{2}}}}}{2}}$

系列

正三角形

• 正三角形

$\cos {\frac {2\pi }{3}}={\frac {\omega +\omega ^{2}}{2}}$

• 正九角形

$\cos {\frac {2\pi }{9}}={\frac {{\sqrt[{3}]{\omega }}+{\sqrt[{3}]{\omega ^{2}}}}{2}}$

• 正二十七角形

$\cos {\frac {2\pi }{27}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{\omega }}}+{\sqrt[{3}]{\sqrt[{3}]{\omega ^{2}}}}}{2}}$

• 正八十一角形

$\cos {\frac {2\pi }{81}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{\omega }}}}+{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{\omega ^{2}}}}}}{2}}$

正六角形

• 正六角形

$\cos {\frac {2\pi }{6}}={\frac {-\omega ^{2}-\omega }{2}}$

• 正十八角形

$\cos {\frac {2\pi }{18}}={\frac {{\sqrt[{3}]{-\omega ^{2}}}+{\sqrt[{3}]{-\omega }}}{2}}$

• 正五十四角形

$\cos {\frac {2\pi }{54}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{-\omega ^{2}}}}+{\sqrt[{3}]{\sqrt[{3}]{-\omega }}}}{2}}$

• 正百六十二角形

$\cos {\frac {2\pi }{162}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{-\omega ^{2}}}}}+{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{-\omega }}}}}{2}}$

正十二角形

• 正十二角形

$\cos {\frac {2\pi }{12}}={\frac {-i\omega +i\omega ^{2}}{2}}$

• 正三十六角形

$\cos {\frac {2\pi }{36}}={\frac {{\sqrt[{3}]{-i\omega }}+{\sqrt[{3}]{i\omega ^{2}}}}{2}}$

• 正百八角形

$\cos {\frac {2\pi }{109}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{-i\omega }}}+{\sqrt[{3}]{\sqrt[{3}]{i\omega ^{2}}}}}{2}}$

• 正三百二十四角形

$\cos {\frac {2\pi }{324}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{-i\omega }}}}+{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{i\omega ^{2}}}}}}{2}}$

正二十四角形

$\sigma _{8}=e^{{\frac {2\pi }{8}}i}={\tfrac {1+i}{\sqrt {2}}}$

• 正二十四角形

$\cos {\frac {2\pi }{24}}={\frac {i\omega ^{2}\sigma _{8}-\omega \sigma _{8}}{2}}$

• 正七十二角形

$\cos {\frac {2\pi }{72}}={\frac {{\sqrt[{3}]{i\omega ^{2}\sigma _{8}}}+{\sqrt[{3}]{-\omega \sigma _{8}}}}{2}}$

• 正二百十六角形

$\cos {\frac {2\pi }{216}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{i\omega ^{2}\sigma _{8}}}}+{\sqrt[{3}]{\sqrt[{3}]{-\omega \sigma _{8}}}}}{2}}$

• 正六百四十八角形

$\cos {\frac {2\pi }{648}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{i\omega ^{2}\sigma _{8}}}}}+{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{-\omega \sigma _{8}}}}}}{2}}$

正四十八角形

$\sigma _{16}=e^{{\frac {2\pi }{16}}i}={\tfrac {{\sqrt {2+{\sqrt {2}}}}+i{\sqrt {2-{\sqrt {2}}}}}{2}}$

• 正四十八角形

$\cos {\frac {2\pi }{48}}={\frac {-\omega \sigma _{16}^{3}+i\omega ^{2}\sigma _{16}}{2}}$

• 正百四十四角形

$\cos {\frac {2\pi }{144}}={\frac {{\sqrt[{3}]{-\omega \sigma _{16}^{3}}}+{\sqrt[{3}]{i\omega ^{2}\sigma _{16}}}}{2}}$

正九十六角形

$\sigma _{32}=e^{{\frac {2\pi }{32}}i}={\tfrac {{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}+i{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}{2}}$

• 正九十六角形

$\cos {\frac {2\pi }{96}}={\frac {i\omega ^{2}\sigma _{32}^{3}-\omega \sigma _{32}^{5}}{2}}$

• 正二百八十八角形

$\cos {\frac {2\pi }{288}}={\frac {{\sqrt[{3}]{i\omega ^{2}\sigma _{32}^{3}}}+{\sqrt[{3}]{-\omega \sigma _{32}^{5}}}}{2}}$

正十五角形

$\sigma _{5}=e^{{\frac {2\pi }{5}}i}={\tfrac {{\sqrt {5}}-1+i{\sqrt {10+2{\sqrt {5}}}}}{4}}$

• 正十五角形

$\cos {\frac {2\pi }{15}}={\frac {\omega ^{2}\sigma _{5}^{2}+\omega \sigma _{5}^{3}}{2}}$

• 正四十五角形

$\cos {\frac {2\pi }{45}}={\frac {{\sqrt[{3}]{\omega ^{2}\sigma _{5}^{2}}}+{\sqrt[{3}]{\omega \sigma _{5}^{3}}}}{2}}$

• 正百三十五角形

$\cos {\frac {2\pi }{135}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{\omega ^{2}\sigma _{5}^{2}}}}+{\sqrt[{3}]{\sqrt[{3}]{\omega \sigma _{5}^{3}}}}}{2}}$

正三十角形

$\sigma _{5}=e^{{\frac {2\pi }{5}}i}={\tfrac {{\sqrt {5}}-1+i{\sqrt {10+2{\sqrt {5}}}}}{4}}$

• 正三十角形

$\cos {\frac {2\pi }{30}}={\frac {-\omega \sigma _{5}-\omega ^{2}\sigma _{5}^{4}}{2}}$

• 正九十角形

$\cos {\frac {2\pi }{90}}={\frac {{\sqrt[{3}]{-\omega \sigma _{5}}}+{\sqrt[{3}]{-\omega ^{2}\sigma _{5}^{4}}}}{2}}$

• 正二百七十角形

$\cos {\frac {2\pi }{270}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{-\omega \sigma _{5}}}}+{\sqrt[{3}]{\sqrt[{3}]{-\omega ^{2}\sigma _{5}^{4}}}}}{2}}$

正六十角形

$\sigma _{5}=e^{{\frac {2\pi }{5}}i}={\tfrac {{\sqrt {5}}-1+i{\sqrt {10+2{\sqrt {5}}}}}{4}}$

• 正六十角形

$\cos {\frac {2\pi }{60}}={\frac {-i\omega ^{2}\sigma _{5}^{3}+i\omega \sigma _{5}^{2}}{2}}$

• 正百八十角形

$\cos {\frac {2\pi }{180}}={\frac {{\sqrt[{3}]{-i\omega ^{2}\sigma _{5}^{3}}}+{\sqrt[{3}]{i\omega \sigma _{5}^{2}}}}{2}}$

正百二十角形

$\sigma _{5}=e^{{\frac {2\pi }{5}}i}={\tfrac {{\sqrt {5}}-1+i{\sqrt {10+2{\sqrt {5}}}}}{4}}$

$\sigma _{8}=e^{{\frac {2\pi }{8}}i}={\tfrac {1+i}{\sqrt {2}}}$

• 正百二十角形

$\cos {\frac {2\pi }{120}}={\frac {-i\omega \sigma _{5}^{4}\sigma _{8}+\omega ^{2}\sigma _{5}\sigma _{8}}{2}}$

• 正三百六十角形

$\cos {\frac {2\pi }{360}}={\frac {{\sqrt[{3}]{-i\omega \sigma _{5}^{4}\sigma _{8}}}+{\sqrt[{3}]{\omega ^{2}\sigma _{5}\sigma _{8}}}}{2}}$

正二百四十角形

$\sigma _{5}=e^{{\frac {2\pi }{5}}i}={\tfrac {{\sqrt {5}}-1+i{\sqrt {10+2{\sqrt {5}}}}}{4}}$

$\sigma _{16}=e^{{\frac {2\pi }{16}}i}={\tfrac {{\sqrt {2+{\sqrt {2}}}}+i{\sqrt {2-{\sqrt {2}}}}}{2}}$

• 正二百四十角形

$\cos {\frac {2\pi }{240}}={\frac {-i\omega ^{2}\sigma _{5}^{2}\sigma _{16}^{3}+\omega \sigma _{5}^{3}\sigma _{16}}{2}}$

• 正七百二十角形

$\cos {\frac {2\pi }{720}}={\frac {{\sqrt[{3}]{-i\omega ^{2}\sigma _{5}^{2}\sigma _{16}^{3}}}+{\sqrt[{3}]{\omega \sigma _{5}^{3}\sigma _{16}}}}{2}}$

正五十一角形

$\sigma _{17}=e^{{\frac {2\pi }{17}}i}={\tfrac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}+i\cdot 2{\sqrt {34-2{\sqrt {17}}+2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}}}{16}}$

• 正五十一角形

$\cos {\frac {2\pi }{51}}={\frac {\omega ^{2}\sigma _{17}^{6}+\omega \sigma _{17}^{11}}{2}}$

• 正百五十三角形

$\cos {\frac {2\pi }{153}}={\frac {{\sqrt[{3}]{\omega ^{2}\sigma _{17}^{6}}}+{\sqrt[{3}]{\omega \sigma _{17}^{11}}}}{2}}$

正百二角形

• 正百二角形

$\cos {\frac {2\pi }{102}}={\frac {-\omega \sigma _{17}^{3}-\omega ^{2}\sigma _{17}^{14}}{2}}$

• 正三百六角形

$\cos {\frac {2\pi }{306}}={\frac {{\sqrt[{3}]{-\omega \sigma _{17}^{3}}}+{\sqrt[{3}]{-\omega ^{2}\sigma _{17}^{14}}}}{2}}$

正二百五十五角形

$\sigma _{5}=e^{{\frac {2\pi }{5}}i}={\tfrac {{\sqrt {5}}-1+i{\sqrt {10+2{\sqrt {5}}}}}{4}}$

$\sigma _{17}=e^{{\frac {2\pi }{17}}i}$

• 正二百五十五角形

$\cos {\frac {2\pi }{255}}={\frac {\omega \sigma _{5}\sigma _{17}^{8}+\omega ^{2}\sigma _{5}^{4}\sigma _{17}^{9}}{2}}$

• 正七百六十五角形

$\cos {\frac {2\pi }{765}}={\frac {{\sqrt[{3}]{\omega \sigma _{5}\sigma _{17}^{8}}}+{\sqrt[{3}]{\omega ^{2}\sigma _{5}^{4}\sigma _{17}^{9}}}}{2}}$

正五百十角形

• 正五百十角形

$\cos {\frac {2\pi }{510}}={\frac {-\omega ^{2}\sigma _{5}^{3}\sigma _{17}^{4}-\omega \sigma _{5}^{2}\sigma _{17}^{13}}{2}}$

• 正千五百三十角形

$\cos {\frac {2\pi }{1530}}={\frac {{\sqrt[{3}]{-\omega ^{2}\sigma _{5}^{3}\sigma _{17}^{4}}}+{\sqrt[{3}]{-\omega \sigma _{5}^{2}\sigma _{17}^{13}}}}{2}}$

正五角形

$\sigma _{5}=e^{{\frac {2\pi }{5}}i}={\tfrac {{\sqrt {5}}-1+i{\sqrt {10+2{\sqrt {5}}}}}{4}}$

• 正五角形

$\cos {\frac {2\pi }{5}}={\frac {\sigma _{5}+\sigma _{5}^{4}}{2}}$

• 正二十五角形

$\cos {\frac {2\pi }{25}}={\frac {{\sqrt[{5}]{\sigma _{5}}}+{\sqrt[{5}]{\sigma _{5}^{4}}}}{2}}$

• 正五十角形

$\cos {\frac {2\pi }{50}}={\frac {{\sqrt[{5}]{-\sigma _{5}^{3}}}+{\sqrt[{5}]{-\sigma _{5}^{2}}}}{2}}$

• 正七十五角形

$\cos {\frac {2\pi }{75}}={\frac {{\sqrt[{5}]{\omega ^{2}\sigma _{5}^{2}}}+{\sqrt[{5}]{\omega \sigma _{5}^{3}}}}{2}}$

• 正百角形

$\cos {\frac {2\pi }{100}}={\frac {{\sqrt[{5}]{i\sigma _{5}^{4}}}+{\sqrt[{5}]{-i\sigma _{5}}}}{2}}$

• 正百二十五角形

$\cos {\frac {2\pi }{125}}={\frac {{\sqrt[{5}]{\sqrt[{5}]{\sigma _{5}}}}+{\sqrt[{5}]{\sqrt[{5}]{\sigma _{5}^{4}}}}}{2}}$

• 正百五十角形

$\cos {\frac {2\pi }{150}}={\frac {{\sqrt[{5}]{-\omega \sigma _{5}}}+{\sqrt[{5}]{-\omega ^{2}\sigma _{5}^{4}}}}{2}}$

• 正百七十五角形

$\cos {\frac {2\pi }{175}}={\frac {{\sqrt[{5}]{\sigma _{5}^{3}\sigma _{7}^{3}}}+{\sqrt[{5}]{\sigma _{5}^{2}\sigma _{7}^{4}}}}{2}}$

• 正二百角形

$\cos {\frac {2\pi }{200}}={\frac {{\sqrt[{5}]{-\sigma _{5}^{2}\sigma _{8}}}+{\sqrt[{5}]{i\sigma _{5}^{3}\sigma _{8}}}}{2}}$

• 正二百二十五角形

$\cos {\frac {2\pi }{225}}={\frac {{\sqrt[{5}]{\sqrt[{3}]{\omega ^{2}\sigma _{5}^{2}}}}+{\sqrt[{5}]{\sqrt[{3}]{\omega \sigma _{5}^{3}}}}}{2}}$

• 正二百五十角形

$\cos {\frac {2\pi }{250}}={\frac {{\sqrt[{5}]{\sqrt[{5}]{-\sigma _{5}^{3}}}}+{\sqrt[{5}]{\sqrt[{5}]{-\sigma _{5}^{2}}}}}{2}}$

• 正二百七十五角形

$\cos {\frac {2\pi }{275}}={\frac {{\sqrt[{5}]{\sigma _{5}\sigma _{11}^{9}}}+{\sqrt[{5}]{\sigma _{5}^{4}\sigma _{11}^{2}}}}{2}}$

• 正三百角形

$\cos {\frac {2\pi }{300}}={\frac {{\sqrt[{5}]{\omega ^{2}\sigma _{4}^{3}\sigma _{5}^{3}}}+{\sqrt[{5}]{\omega \sigma _{4}\sigma _{5}^{2}}}}{2}}$

• 正三百二十五角形

$\cos {\frac {2\pi }{325}}={\frac {{\sqrt[{5}]{\sigma _{5}^{2}\sigma _{13}^{8}}}+{\sqrt[{5}]{\sigma _{5}^{3}\sigma _{13}^{5}}}}{2}}$

• 正三百五十角形

$\cos {\frac {2\pi }{350}}={\frac {{\sqrt[{5}]{-\sigma _{5}^{4}\sigma _{7}^{5}}}+{\sqrt[{5}]{-\sigma _{5}\sigma _{7}^{2}}}}{2}}$

• 正三百七十五角形

$\cos {\frac {2\pi }{375}}={\frac {{\sqrt[{5}]{\sqrt[{5}]{\omega ^{2}\sigma _{5}^{2}}}}+{\sqrt[{5}]{\sqrt[{5}]{\omega \sigma _{5}^{3}}}}}{2}}$

• 正四百角形

$\cos {\frac {2\pi }{400}}={\frac {{\sqrt[{5}]{-i\sigma _{5}\sigma _{16}}}+{\sqrt[{5}]{\sigma _{5}^{4}\sigma _{16}^{3}}}}{2}}$

• 正四百二十五角形

$\cos {\frac {2\pi }{425}}={\frac {{\sqrt[{5}]{\sigma _{5}^{3}\sigma _{17}^{7}}}+{\sqrt[{5}]{\sigma _{5}^{2}\sigma _{17}^{10}}}}{2}}$

• 正四百五十角形

$\cos {\frac {2\pi }{450}}={\frac {{\sqrt[{5}]{\sqrt[{3}]{-\omega \sigma _{5}}}}+{\sqrt[{5}]{\sqrt[{3}]{-\omega ^{2}\sigma _{5}^{4}}}}}{2}}$

• 正四百七十五角形

$\cos {\frac {2\pi }{475}}={\frac {{\sqrt[{5}]{\sigma _{5}^{4}\sigma _{19}^{4}}}+{\sqrt[{5}]{\sigma _{5}\sigma _{19}^{15}}}}{2}}$

• 正五百角形

$\cos {\frac {2\pi }{500}}={\frac {{\sqrt[{5}]{\sqrt[{5}]{i\sigma _{5}^{4}}}}+{\sqrt[{5}]{\sqrt[{5}]{-i\sigma _{5}}}}}{2}}$

• 正五百二十五角形

$\cos {\frac {2\pi }{525}}={\frac {{\sqrt[{5}]{\omega ^{2}\sigma _{5}\sigma _{7}}}+{\sqrt[{5}]{\omega \sigma _{5}^{4}\sigma _{7}^{6}}}}{2}}$

• 正五百五十角形

$\cos {\frac {2\pi }{550}}={\frac {{\sqrt[{5}]{-\sigma _{5}^{3}\sigma _{11}^{10}}}+{\sqrt[{5}]{-\sigma _{5}^{2}\sigma _{11}}}}{2}}$

• 正五百七十五角形（575のトーシェント関数は440=2^3・5・11のため、以下の式はおかしい）

$\cos {\frac {2\pi }{575}}={\frac {{\sqrt[{5}]{\sigma _{5}^{2}\sigma _{23}^{14}}}+{\sqrt[{5}]{\sigma _{5}^{3}\sigma _{23}^{9}}}}{2}}$

• 正六百角形

$\cos {\frac {2\pi }{600}}={\frac {{\sqrt[{5}]{-i\omega \sigma _{5}^{4}\sigma _{8}}}+{\sqrt[{5}]{\omega ^{2}\sigma _{5}\sigma _{8}}}}{2}}$

• 正六百二十五角形

$\cos {\frac {2\pi }{625}}={\frac {{\sqrt[{5}]{\sqrt[{5}]{\sqrt[{5}]{\sigma _{5}}}}}+{\sqrt[{5}]{\sqrt[{5}]{\sqrt[{5}]{\sigma _{5}^{4}}}}}}{2}}$

正十七角形

$\sigma _{17}=e^{{\frac {2\pi }{17}}i}={\tfrac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}+i\cdot 2{\sqrt {34-2{\sqrt {17}}+2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}}}{16}}$

• 正十七角形

$\cos {\frac {2\pi }{17}}={\frac {\sigma _{17}+\sigma _{17}^{16}}{2}}$

• 正二百八十九角形（1）

$\cos {\frac {2\pi }{289}}={\frac {{\sqrt[{17}]{\sigma _{17}}}+{\sqrt[{17}]{\sigma _{17}^{16}}}}{2}}$

• 正五百七十八角形（2）

$\cos {\frac {2\pi }{577}}={\frac {{\sqrt[{17}]{-\sigma _{17}^{9}}}+{\sqrt[{17}]{-\sigma _{17}^{8}}}}{2}}$

• 正八百六十七角形（3）

$\cos {\frac {2\pi }{867}}={\frac {{\sqrt[{17}]{\omega ^{2}\sigma _{17}^{6}}}+{\sqrt[{17}]{\omega \sigma _{17}^{11}}}}{2}}$

• 正千百五十六角形（4）

$\cos {\frac {2\pi }{1156}}={\frac {{\sqrt[{17}]{i\sigma _{17}^{13}}}+{\sqrt[{17}]{-i\sigma _{17}^{4}}}}{2}}$

• 正千四百四十五角形（5）

$\cos {\frac {2\pi }{1445}}={\frac {{\sqrt[{17}]{\sigma _{5}^{3}\sigma _{17}^{7}}}+{\sqrt[{17}]{\sigma _{5}^{2}\sigma _{17}^{10}}}}{2}}$

• 正千七百三十四角形（6）

$\cos {\frac {2\pi }{1734}}={\frac {{\sqrt[{17}]{-\omega \sigma _{17}^{3}}}+{\sqrt[{17}]{-\omega ^{2}\sigma _{17}^{14}}}}{2}}$

• 正二千三百十二角形（8）

$\cos {\frac {2\pi }{2312}}={\frac {{\sqrt[{17}]{\sigma _{8}\sigma _{17}^{15}}}+{\sqrt[{17}]{-i\sigma _{8}\sigma _{17}^{2}}}}{2}}$

• 正二千八百九十角形（10）

$\cos {\frac {2\pi }{2890}}={\frac {{\sqrt[{17}]{-\sigma _{5}^{4}\sigma _{17}^{12}}}+{\sqrt[{17}]{-\sigma _{5}\sigma _{17}^{5}}}}{2}}$

• 正三千四百六十八角形（12）

$\cos {\frac {2\pi }{3468}}={\frac {{\sqrt[{17}]{-i\omega ^{2}\sigma _{17}^{10}}}+{\sqrt[{17}]{i\omega \sigma _{17}^{7}}}}{2}}$

• 正四千三百三十五角形（15）

$\cos {\frac {2\pi }{4335}}={\frac {{\sqrt[{17}]{\omega \sigma _{5}\sigma _{17}^{8}}}+{\sqrt[{17}]{\omega ^{2}\sigma _{5}^{4}\sigma _{17}^{9}}}}{2}}$

• 正四千六百二十四角形（16）

$\cos {\frac {2\pi }{4624}}={\frac {{\sqrt[{17}]{\sigma _{16}\sigma _{17}^{16}}}+{\sqrt[{17}]{-i\sigma _{16}^{3}\sigma _{17}}}}{2}}$

• 正四千九百十三角形（17）

$\cos {\frac {2\pi }{4913}}={\frac {{\sqrt[{17}]{\sqrt[{17}]{\sigma _{17}}}}+{\sqrt[{17}]{\sqrt[{17}]{\sigma _{17}^{16}}}}}{2}}$

8次方程式

• 十七角形
• 三十二角形
• 三十四角形
• 四十角形
• 四十八角形
• 六十角形

9次方程式

• 十九角形
• 二十七角形
• 三十八角形
• 五十四角形

12次方程式

• 三十五角形
• 三十九角形
• 四十五角形
• 五十二角形
• 五十六角形
• 七十角形
• 七十二角形
• 七十八角形
• 八十四角形
• 九十角形

16次方程式

• 五十一角形
• 六十四角形
• 六十八角形
• 八十角形
• 九十六角形
• 百二角形
• 百二十角形

18次方程式

1の原始冪根の数とオイラーのφ関数

• 三十七角形

37 ピアポント素数 37(1-1/37)=36=2(2×3×3)

• 五十七角形

57 3×19 → 57(1-1/3)(1-1/19)=36=2(2×3×3)

• 六十三角形

63 3×3×7 → 63(1-1/3)(1-1/7)=36=2(2×3×3)

• 七十四角形

74 2×37 → 74(1-1/2)(1-1/37)=36=2(2×3×3)

• 七十六角形

76 2×2×19 → 76(1-1/2)(1-1/19)=36=2(2×3×3)

• 百八角形

108 2×2×3×3×3 → 108(1-1/2)(1-1/3)=36=2(2×3×3)

• 百十四角形

114 2×3×19 → 114(1-1/2)(1-1/3)(1-1/19)=36=2(2×3×3)

• 百二十六角形

126 2×3×3×7 → 126(1-1/2)(1-1/3)(1-1/7)=36=2(2×3×3)

37角形

以下のように定義すると

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{37}}+2\cos {\frac {20\pi }{37}}+2\cos {\frac {22\pi }{37}}\\&x_{2}=2\cos {\frac {4\pi }{37}}+2\cos {\frac {34\pi }{37}}+2\cos {\frac {30\pi }{37}}\\&x_{3}=2\cos {\frac {8\pi }{37}}+2\cos {\frac {6\pi }{37}}+2\cos {\frac {14\pi }{37}}\\&x_{4}=2\cos {\frac {16\pi }{37}}+2\cos {\frac {12\pi }{37}}+2\cos {\frac {28\pi }{37}}\\&x_{5}=2\cos {\frac {32\pi }{37}}+2\cos {\frac {24\pi }{37}}+2\cos {\frac {18\pi }{37}}\\&x_{6}=2\cos {\frac {10\pi }{37}}+2\cos {\frac {26\pi }{37}}+2\cos {\frac {36\pi }{37}}\\\end{aligned}}$

以下の関係がある。

${\begin{aligned}&x_{1}+x_{3}+x_{5}={\frac {-1+{\sqrt {37}}}{2}}=\alpha \\&x_{2}+x_{4}+x_{6}={\frac {-1-{\sqrt {37}}}{2}}=\beta \\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(x_{3}+\omega \cdot x_{5}+\omega ^{2}\cdot x_{1}\right)^{3}=&{\frac {-37-8{\sqrt {37}}-3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}\\\left(x_{3}+\omega ^{2}\cdot x_{5}+\omega \cdot x_{1}\right)^{3}=&{\frac {-37-8{\sqrt {37}}+3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}\\\left(x_{4}+\omega \cdot x_{6}+\omega ^{2}\cdot x_{2}\right)^{3}=&{\frac {-37+8{\sqrt {37}}-3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}\\\left(x_{4}+\omega ^{2}\cdot x_{6}+\omega \cdot x_{2}\right)^{3}=&{\frac {-37+8{\sqrt {37}}+3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}\\\end{aligned}}$

両辺の立方根を取ると

${\begin{aligned}x_{3}+\omega \cdot x_{5}+\omega ^{2}\cdot x_{1}=&{\sqrt[{3}]{\frac {-37-8{\sqrt {37}}-3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}}=A\\x_{3}+\omega ^{2}\cdot x_{5}+\omega \cdot x_{1}=&{\sqrt[{3}]{\frac {-37-8{\sqrt {37}}+3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}}=B\\x_{4}+\omega \cdot x_{6}+\omega ^{2}\cdot x_{2}=&{\sqrt[{3}]{\frac {-37+8{\sqrt {37}}-3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}}=C\\x_{4}+\omega ^{2}\cdot x_{6}+\omega \cdot x_{2}=&{\sqrt[{3}]{\frac {-37+8{\sqrt {37}}+3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}}=D\\\end{aligned}}$

よって

${\begin{aligned}3x_{1}=&\alpha +\omega A+\omega ^{2}B\\3x_{3}=&\alpha +A+B\\3x_{5}=&\alpha +\omega ^{2}A+\omega B\\3x_{2}=&\beta +\omega C+\omega ^{2}D\\3x_{4}=&\beta +C+D\\3x_{6}=&\beta +\omega ^{2}C+\omega D\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(2\cos {\frac {2\pi }{37}}+\omega \cdot 2\cos {\frac {20\pi }{37}}+\omega ^{2}\cdot 2\cos {\frac {22\pi }{37}}\right)^{3}=&3x_{1}+x_{3}+6(x_{2}+2)+3\omega (2x_{1}+x_{4}+x_{5})+3\omega ^{2}(2x_{1}+x_{5}+x_{6})\\\left(2\cos {\frac {2\pi }{37}}+\omega ^{2}\cdot 2\cos {\frac {20\pi }{37}}+\omega \cdot 2\cos {\frac {22\pi }{37}}\right)^{3}=&3x_{1}+x_{3}+6(x_{2}+2)+3\omega ^{2}(2x_{1}+x_{4}+x_{5})+3\omega (2x_{1}+x_{5}+x_{6})\\\end{aligned}}$

37角形の2

${\begin{aligned}&x_{1}={\frac {3\cdot \left(6\cos {\frac {2\pi }{37}}+6\cos {\frac {52\pi }{37}}+6\cos {\frac {20\pi }{37}}-{\frac {-1+{\sqrt {37}}}{2}}\right)}{{\frac {-1+{\sqrt {37}}}{2}}\cdot {\sqrt {\frac {74+11{\sqrt {37}}}{3}}}}}\\&x_{2}={\frac {3\cdot \left(6\cos {\frac {32\pi }{37}}+6\cos {\frac {18\pi }{37}}+6\cos {\frac {24\pi }{37}}-{\frac {-1+{\sqrt {37}}}{2}}\right)}{{\frac {-1+{\sqrt {37}}}{2}}\cdot {\sqrt {\frac {74+11{\sqrt {37}}}{3}}}}}\\&x_{3}={\frac {3\cdot \left(6\cos {\frac {68\pi }{37}}+6\cos {\frac {66\pi }{37}}+6\cos {\frac {14\pi }{37}}-{\frac {-1+{\sqrt {37}}}{2}}\right)}{{\frac {-1+{\sqrt {37}}}{2}}\cdot {\sqrt {\frac {74+11{\sqrt {37}}}{3}}}}}\\&w_{1}={\frac {3\cdot \left(6\cos {\frac {4\pi }{37}}+6\cos {\frac {30\pi }{37}}+6\cos {\frac {40\pi }{37}}-{\frac {-1-{\sqrt {37}}}{2}}\right)}{{\frac {-1-{\sqrt {37}}}{2}}\cdot {\sqrt {\frac {74-11{\sqrt {37}}}{3}}}}}\\&w_{2}={\frac {3\cdot \left(6\cos {\frac {64\pi }{37}}+6\cos {\frac {36\pi }{37}}+6\cos {\frac {48\pi }{37}}-{\frac {-1-{\sqrt {37}}}{2}}\right)}{{\frac {-1-{\sqrt {37}}}{2}}\cdot {\sqrt {\frac {74-11{\sqrt {37}}}{3}}}}}\\&w_{3}={\frac {3\cdot \left(6\cos {\frac {62\pi }{37}}+6\cos {\frac {58\pi }{37}}+6\cos {\frac {28\pi }{37}}-{\frac {-1-{\sqrt {37}}}{2}}\right)}{{\frac {-1-{\sqrt {37}}}{2}}\cdot {\sqrt {\frac {74-11{\sqrt {37}}}{3}}}}}\\\end{aligned}}$

${\begin{aligned}&x_{1}+x_{2}+x_{3}=0\\&x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1}=-3=p\\&x_{1}x_{2}x_{3}=-{\frac {\sqrt {74-11{\sqrt {37}}}}{\sqrt {37}}}=-q\\&\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}={\frac {-1-{\frac {11}{2{\sqrt {37}}}}}{2}}=-{\frac {74+11{\sqrt {37}}}{4\cdot 37}}\\\end{aligned}}$

${\begin{aligned}&w_{1}+w_{2}+w_{3}=0\\&w_{1}w_{2}+w_{2}w_{3}+w_{3}w_{1}=-3=p\\&w_{1}w_{2}w_{3}=-{\frac {\sqrt {74+11{\sqrt {37}}}}{\sqrt {37}}}=-q\\&\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}={\frac {-1+{\frac {11}{2{\sqrt {37}}}}}{2}}=-{\frac {74-11{\sqrt {37}}}{4\cdot 37}}\\\end{aligned}}$

${\begin{aligned}&x_{1}=\omega ^{2}{\sqrt[{3}]{\tfrac {{\sqrt {74-11{\sqrt {37}}}}+i{\sqrt {74+11{\sqrt {37}}}}}{2{\sqrt {37}}}}}+\omega {\sqrt[{3}]{\tfrac {{\sqrt {74-11{\sqrt {37}}}}-i{\sqrt {74+11{\sqrt {37}}}}}{2{\sqrt {37}}}}}\\&x_{2}=\omega {\sqrt[{3}]{\tfrac {{\sqrt {74-11{\sqrt {37}}}}+i{\sqrt {74+11{\sqrt {37}}}}}{2{\sqrt {37}}}}}+\omega ^{2}{\sqrt[{3}]{\tfrac {{\sqrt {74-11{\sqrt {37}}}}-i{\sqrt {74+11{\sqrt {37}}}}}{2{\sqrt {37}}}}}\\&x_{3}={\sqrt[{3}]{\tfrac {{\sqrt {74-11{\sqrt {37}}}}+i{\sqrt {74+11{\sqrt {37}}}}}{2{\sqrt {37}}}}}+{\sqrt[{3}]{\tfrac {{\sqrt {74-11{\sqrt {37}}}}-i{\sqrt {74+11{\sqrt {37}}}}}{2{\sqrt {37}}}}}\\&w_{1}=\omega ^{2}{\sqrt[{3}]{\tfrac {{\sqrt {74+11{\sqrt {37}}}}+i{\sqrt {74-11{\sqrt {37}}}}}{2{\sqrt {37}}}}}+\omega {\sqrt[{3}]{\tfrac {{\sqrt {74+11{\sqrt {37}}}}-i{\sqrt {74-11{\sqrt {37}}}}}{2{\sqrt {37}}}}}\\&w_{2}={\sqrt[{3}]{\tfrac {{\sqrt {74+11{\sqrt {37}}}}+i{\sqrt {74-11{\sqrt {37}}}}}{2{\sqrt {37}}}}}+{\sqrt[{3}]{\tfrac {{\sqrt {74+11{\sqrt {37}}}}-i{\sqrt {74-11{\sqrt {37}}}}}{2{\sqrt {37}}}}}\\&w_{3}=\omega {\sqrt[{3}]{\tfrac {{\sqrt {74+11{\sqrt {37}}}}+i{\sqrt {74-11{\sqrt {37}}}}}{2{\sqrt {37}}}}}+\omega ^{2}{\sqrt[{3}]{\tfrac {{\sqrt {74+11{\sqrt {37}}}}-i{\sqrt {74-11{\sqrt {37}}}}}{2{\sqrt {37}}}}}\\\end{aligned}}$

${\begin{aligned}6\cos {\tfrac {2\pi }{37}}+6\cos {\tfrac {52\pi }{37}}+6\cos {\tfrac {20\pi }{37}}={\tfrac {-1+{\sqrt {37}}}{2}}+{\tfrac {-1+{\sqrt {37}}}{2}}\cdot {\tfrac {\omega ^{2}}{3}}\cdot {\sqrt {\tfrac {74+11{\sqrt {37}}}{3}}}{\sqrt[{3}]{\tfrac {{\sqrt {74-11{\sqrt {37}}}}+i{\sqrt {74+11{\sqrt {37}}}}}{2{\sqrt {37}}}}}+{\tfrac {-1+{\sqrt {37}}}{2}}\cdot {\tfrac {\omega }{3}}\cdot {\sqrt {\tfrac {74+11{\sqrt {37}}}{3}}}{\sqrt[{3}]{\tfrac {{\sqrt {74-11{\sqrt {37}}}}-i{\sqrt {74+11{\sqrt {37}}}}}{2{\sqrt {37}}}}}\\\end{aligned}}$

3つの積和公式

${\begin{aligned}8\cos \alpha \cos \beta \cos \gamma &=2\cos(\alpha +\beta +\gamma )+2\cos(\alpha +\beta -\gamma )+2\cos(\alpha -\beta +\gamma )+2\cos(-\alpha +\beta +\gamma )\\8\cos \alpha \cos \beta \cos \gamma &=2\cos(\alpha +\beta +\gamma )+2\cos(\alpha +\beta +\gamma -2\gamma )+2\cos(\alpha +\beta +\gamma -2\beta )+2\cos(\alpha +\beta +\gamma -2\alpha )\\8\cos ^{2}\alpha \cos \beta &=2\cos(2\alpha +\beta )+2\cos(2\alpha -\beta )+4\cos \beta \\8\cos ^{3}\alpha &=2\cos {3\alpha }+6\cos \alpha \\\end{aligned}}$

$\alpha +\beta +\gamma =2\pi$の場合

${\begin{aligned}&\left(2\cos \alpha +2\cos \beta +2\cos \gamma \right)^{3}=(2\cos {3\alpha }+2\cos {3\beta }+2\cos {3\gamma })+3(2\cos \alpha +2\cos \beta +2\cos \gamma )+12+6(2\cos {2\alpha }+2\cos {2\beta }+2\cos {2\gamma })\\&+3\left\{2\cos(2\alpha +\beta )+2\cos(2\alpha -\beta )+4\cos \beta +2\cos(2\beta +\gamma )+2\cos(2\beta -\gamma )+4\cos \gamma +2\cos(2\gamma +\alpha )+2\cos(2\gamma -\alpha )+4\cos \alpha \right\}\\&+3\left\{2\cos(2\beta +\alpha )+2\cos(2\beta -\alpha )+4\cos \alpha +2\cos(2\gamma +\beta )+2\cos(2\gamma -\beta )+4\cos \beta +2\cos(2\alpha +\gamma )+2\cos(2\alpha -\gamma )+4\cos \gamma \right\}\\&=(2\cos {3\alpha }+2\cos {3\beta }+2\cos {3\gamma })+6(2\cos {2\alpha }+2\cos {2\beta }+2\cos {2\gamma })+3(2\cos \alpha +2\cos \beta +2\cos \gamma )+12\\&+3\left\{2(2\cos \alpha +2\cos \beta +2\cos \gamma )+2\cos(\alpha -\beta )+2\cos(\beta -\gamma )+2\cos(\gamma -\alpha )+2\cos(2\alpha -\beta )+2\cos(2\beta -\gamma )+2\cos(2\gamma -\alpha )\right\}\\&+3\left\{2(2\cos \alpha +2\cos \beta +2\cos \gamma )+2\cos(\alpha -\beta )+2\cos(\beta -\gamma )+2\cos(\gamma -\alpha )+2\cos(\alpha -2\beta )+2\cos(\beta -2\gamma )+2\cos(\gamma -2\alpha )\right\}\\\end{aligned}}$

$\alpha +\beta +\gamma =2\pi$の場合

${\begin{aligned}&\left(6\cos \alpha +6\cos \beta +6\cos \gamma \right)^{3}=9(6\cos {3\alpha }+6\cos {3\beta }+6\cos {3\gamma })+54(6\cos {2\alpha }+6\cos {2\beta }+6\cos {2\gamma })+27(6\cos \alpha +6\cos \beta +6\cos \gamma )+324\\&+27\left\{2(6\cos \alpha +6\cos \beta +6\cos \gamma )+6\cos(\alpha -\beta )+6\cos(\beta -\gamma )+6\cos(\gamma -\alpha )+6\cos(2\alpha -\beta )+6\cos(2\beta -\gamma )+6\cos(2\gamma -\alpha )\right\}\\&+27\left\{2(6\cos \alpha +6\cos \beta +6\cos \gamma )+6\cos(\alpha -\beta )+6\cos(\beta -\gamma )+6\cos(\gamma -\alpha )+6\cos(\alpha -2\beta )+6\cos(\beta -2\gamma )+6\cos(\gamma -2\alpha )\right\}\\\end{aligned}}$

${\begin{aligned}&\left(2\cos \alpha +\omega \cdot 2\cos \beta +\omega ^{2}\cdot 2\cos \gamma \right)^{3}=\\&\left(2\cos \alpha +\omega ^{2}\cdot 2\cos \beta +\omega \cdot 2\cos \gamma \right)^{3}=\\\end{aligned}}$

57角形

以下のように定義すると

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{57}}+2\cos {\frac {14\pi }{57}}+2\cos {\frac {16\pi }{57}}\\&x_{2}=2\cos {\frac {10\pi }{57}}+2\cos {\frac {44\pi }{57}}+2\cos {\frac {34\pi }{57}}\\&x_{3}=2\cos {\frac {50\pi }{57}}+2\cos {\frac {8\pi }{57}}+2\cos {\frac {56\pi }{57}}\\&x_{4}=2\cos {\frac {22\pi }{57}}+2\cos {\frac {40\pi }{57}}+2\cos {\frac {52\pi }{57}}\\&x_{5}=2\cos {\frac {4\pi }{57}}+2\cos {\frac {28\pi }{57}}+2\cos {\frac {32\pi }{57}}\\&x_{6}=2\cos {\frac {20\pi }{57}}+2\cos {\frac {26\pi }{57}}+2\cos {\frac {46\pi }{57}}\\\end{aligned}}$

以下の関係がある。

${\begin{aligned}&x_{1}+x_{3}+x_{5}={\frac {1+{\sqrt {57}}}{2}}=\alpha \\&x_{2}+x_{4}+x_{6}={\frac {1-{\sqrt {57}}}{2}}=\beta \\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}\right)^{3}=&{\frac {-11+3{\sqrt {57}}}{2}}+3\omega \left({\frac {-29-3{\sqrt {57}}}{2}}\right)+3\omega ^{2}\left({\frac {47+7{\sqrt {57}}}{2}}\right)\\=&{\frac {-38-3{\sqrt {57}}-3{\sqrt {3}}(38+5{\sqrt {57}})i}{2}}\\\left(x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}\right)^{3}=&{\frac {-11+3{\sqrt {57}}}{2}}+3\omega ^{2}\left({\frac {-29-3{\sqrt {57}}}{2}}\right)+3\omega \left({\frac {47+7{\sqrt {57}}}{2}}\right)\\=&{\frac {-38-3{\sqrt {57}}+3{\sqrt {3}}(38+5{\sqrt {57}})i}{2}}\\\left(x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}\right)^{3}=&{\frac {-11-3{\sqrt {57}}}{2}}+3\omega \left({\frac {-29+3{\sqrt {57}}}{2}}\right)+3\omega ^{2}\left({\frac {47-7{\sqrt {57}}}{2}}\right)\\=&{\frac {-38+3{\sqrt {57}}-3{\sqrt {3}}(38-5{\sqrt {57}})i}{2}}\\\left(x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}\right)^{3}=&{\frac {-11-3{\sqrt {57}}}{2}}+3\omega ^{2}\left({\frac {-29+3{\sqrt {57}}}{2}}\right)+3\omega \left({\frac {47-7{\sqrt {57}}}{2}}\right)\\=&{\frac {-38+3{\sqrt {57}}+3{\sqrt {3}}(38-5{\sqrt {57}})i}{2}}\\\end{aligned}}$

両辺の立方根を取ると

${\begin{aligned}x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}=&{\sqrt[{3}]{\frac {-38-3{\sqrt {57}}-3{\sqrt {3}}(38+5{\sqrt {57}})i}{2}}}=A\\x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}=&{\sqrt[{3}]{\frac {-38-3{\sqrt {57}}+3{\sqrt {3}}(38+5{\sqrt {57}})i}{2}}}=B\\x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}=&{\sqrt[{3}]{\frac {-38+3{\sqrt {57}}-3{\sqrt {3}}(38-5{\sqrt {57}})i}{2}}}=C\\x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}=&{\sqrt[{3}]{\frac {-38+3{\sqrt {57}}+3{\sqrt {3}}(38-5{\sqrt {57}})i}{2}}}=D\\\end{aligned}}$

よって

${\begin{aligned}3x_{1}=&\alpha +A+B\\3x_{3}=&\alpha +\omega ^{2}A+\omega B\\3x_{5}=&\alpha +\omega A+\omega ^{2}B\\3x_{2}=&\beta +C+D\\3x_{4}=&\beta +\omega ^{2}C+\omega D\\3x_{6}=&\beta +\omega C+\omega ^{2}D\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}&\left(2\cos {\frac {2\pi }{57}}+\omega \cdot 2\cos {\frac {14\pi }{57}}+\omega ^{2}\cdot 2\cos {\frac {16\pi }{57}}\right)^{3}\\&=3x_{1}+2\cos {\frac {2\pi }{19}}+2\cos {\frac {16\pi }{19}}+2\cos {\frac {14\pi }{19}}+6(x_{5}+2)+3\omega \left(2x_{1}+x_{2}+2\cos {\frac {4\pi }{19}}+2\cos {\frac {6\pi }{19}}+2\cos {\frac {10\pi }{19}}\right)+3\omega ^{2}\left(2x_{1}+x_{6}+2\cos {\frac {4\pi }{19}}+2\cos {\frac {6\pi }{19}}+2\cos {\frac {10\pi }{19}}\right)\\&=3x_{1}+{\frac {-1+\omega ^{2}E+\omega F}{3}}+6(x_{5}+2)+3\omega \left(2x_{1}+x_{2}+{\frac {-1+E+F}{3}}\right)+3\omega ^{2}\left(2x_{1}+x_{6}+{\frac {-1+E+F}{3}}\right)\\&\left(2\cos {\frac {2\pi }{57}}+\omega ^{2}\cdot 2\cos {\frac {14\pi }{57}}+\omega \cdot 2\cos {\frac {16\pi }{57}}\right)^{3}\\&=3x_{1}+2\cos {\frac {2\pi }{19}}+2\cos {\frac {16\pi }{19}}+2\cos {\frac {14\pi }{19}}+6(x_{5}+2)+3\omega ^{2}\left(2x_{1}+x_{2}+2\cos {\frac {4\pi }{19}}+2\cos {\frac {6\pi }{19}}+2\cos {\frac {10\pi }{19}}\right)+3\omega \left(2x_{1}+x_{6}+2\cos {\frac {4\pi }{19}}+2\cos {\frac {6\pi }{19}}+2\cos {\frac {10\pi }{19}}\right)\\&=3x_{1}+{\frac {-1+\omega ^{2}E+\omega F}{3}}+6(x_{5}+2)+3\omega ^{2}\left(2x_{1}+x_{2}+{\frac {-1+E+F}{3}}\right)+3\omega \left(2x_{1}+x_{6}+{\frac {-1+E+F}{3}}\right)\\\end{aligned}}$

ここでE,Fは

${\begin{aligned}E={\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}\\F={\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}\\\end{aligned}}$

両辺の立方根を取り

${\begin{aligned}2\cos {\frac {2\pi }{57}}+\omega \cdot 2\cos {\frac {14\pi }{57}}+\omega ^{2}\cdot 2\cos {\frac {16\pi }{57}}=&\\2\cos {\frac {2\pi }{57}}+\omega ^{2}\cdot 2\cos {\frac {14\pi }{57}}+\omega \cdot 2\cos {\frac {16\pi }{57}}=&\\\end{aligned}}$

$\cos(2\pi /57)$を求めることができる。

63角形

正六十三角形

六十三角形（ろくじゅうさんかくけい、ろくじゅうさんかっけい、hexacontatrigon）は、多角形の一つで、63本の辺と63個の頂点を持つ図形である。内角の和は10980°、対角線の本数は1890本である。

正六十三角形においては、中心角と外角は5.714…°で、内角は174.285…°となる。一辺の長さが a の正六十三角形の面積 S は

$S={\frac {63}{4}}a^{2}\cot {\frac {\pi }{63}}\simeq 315.58114a^{2}$

関係式

以下のように定義すると

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{63}}+2\cos {\frac {8\pi }{63}}+2\cos {\frac {32\pi }{63}}\\&x_{2}=2\cos {\frac {22\pi }{63}}+2\cos {\frac {38\pi }{63}}+2\cos {\frac {26\pi }{63}}\\&x_{3}=2\cos {\frac {10\pi }{63}}+2\cos {\frac {40\pi }{63}}+2\cos {\frac {34\pi }{63}}\\&x_{4}=2\cos {\frac {4\pi }{63}}+2\cos {\frac {16\pi }{63}}+2\cos {\frac {62\pi }{63}}\\&x_{5}=2\cos {\frac {44\pi }{63}}+2\cos {\frac {50\pi }{63}}+2\cos {\frac {52\pi }{63}}\\&x_{6}=2\cos {\frac {20\pi }{63}}+2\cos {\frac {46\pi }{63}}+2\cos {\frac {58\pi }{63}}\\\end{aligned}}$

以下の関係がある。

${\begin{aligned}&x_{1}+x_{3}+x_{5}=0\\&x_{2}+x_{4}+x_{6}=0\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}\right)^{3}=&{\frac {-27(7+{\sqrt {21}})+9{\sqrt {3}}(21+5{\sqrt {21}})i}{2}}\\\left(x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}\right)^{3}=&{\frac {-27(7+{\sqrt {21}})-9{\sqrt {3}}(21+5{\sqrt {21}})i}{2}}\\\left(x_{4}+\omega \cdot x_{6}+\omega ^{2}\cdot x_{2}\right)^{3}=&{\frac {-27(7-{\sqrt {21}})-9{\sqrt {3}}(-21+5{\sqrt {21}})i}{2}}\\\left(x_{4}+\omega ^{2}\cdot x_{6}+\omega \cdot x_{2}\right)^{3}=&{\frac {-27(7-{\sqrt {21}})+9{\sqrt {3}}(-21+5{\sqrt {21}})i}{2}}\\\end{aligned}}$

両辺の立方根を取ると

${\begin{aligned}x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}=&{\sqrt[{3}]{\frac {-27(7+{\sqrt {21}})+9{\sqrt {3}}(21+5{\sqrt {21}})i}{2}}}=A\\x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}=&{\sqrt[{3}]{\frac {-27(7+{\sqrt {21}})-9{\sqrt {3}}(21+5{\sqrt {21}})i}{2}}}=B\\x_{4}+\omega \cdot x_{6}+\omega ^{2}\cdot x_{2}=&{\sqrt[{3}]{\frac {-27(7-{\sqrt {21}})-9{\sqrt {3}}(-21+5{\sqrt {21}})i}{2}}}=C\\x_{4}+\omega ^{2}\cdot x_{6}+\omega \cdot x_{2}=&{\sqrt[{3}]{\frac {-27(7-{\sqrt {21}})+9{\sqrt {3}}(-21+5{\sqrt {21}})i}{2}}}=D\\\end{aligned}}$

よって

${\begin{aligned}3x_{1}=&A+B\\3x_{3}=&\omega ^{2}A+\omega B\\3x_{5}=&\omega A+\omega ^{2}B\\3x_{2}=&\omega C+\omega ^{2}D\\3x_{4}=&C+D\\3x_{6}=&\omega ^{2}C+\omega D\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(2\cos {\frac {2\pi }{63}}+\omega \cdot 2\cos {\frac {8\pi }{63}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{63}}\right)^{3}=&3x_{1}+{\frac {1+{\sqrt {21}}}{2}}+6(x_{5}-1)+3\omega \left(2x_{1}+x_{4}+{\frac {1-{\sqrt {21}}}{2}}\right)+3\omega ^{2}(2x_{1}-1)\\\left(2\cos {\frac {2\pi }{63}}+\omega ^{2}\cdot 2\cos {\frac {8\pi }{63}}+\omega \cdot 2\cos {\frac {32\pi }{63}}\right)^{3}=&3x_{1}+{\frac {1+{\sqrt {21}}}{2}}+6(x_{5}-1)+3\omega ^{2}\left(2x_{1}+x_{4}+{\frac {1-{\sqrt {21}}}{2}}\right)+3\omega (2x_{1}-1)\\\end{aligned}}$

関係式2

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{63}}+2\cos {\frac {40\pi }{63}}+2\cos {\frac {44\pi }{63}}=0\\&x_{2}=2\cos {\frac {4\pi }{63}}+2\cos {\frac {46\pi }{63}}+2\cos {\frac {38\pi }{63}}=0\\&x_{3}=2\cos {\frac {8\pi }{63}}+2\cos {\frac {34\pi }{63}}+2\cos {\frac {50\pi }{63}}=0\\&x_{4}=2\cos {\frac {16\pi }{63}}+2\cos {\frac {58\pi }{63}}+2\cos {\frac {26\pi }{63}}=0\\&x_{5}=2\cos {\frac {32\pi }{63}}+2\cos {\frac {10\pi }{63}}+2\cos {\frac {52\pi }{63}}=0\\&x_{6}=2\cos {\frac {62\pi }{63}}+2\cos {\frac {20\pi }{63}}+2\cos {\frac {22\pi }{63}}=0\\\end{aligned}}$

74角形

正七十四角形

七十四角形（ななじゅうよんかくけい、ななじゅうよんかっけい、heptacontatetragon）は、多角形の一つで、74本の辺と74個の頂点を持つ図形である。内角の和は12960°、対角線の本数は2627本である。

正七十四角形においては、中心角と外角は4.864…°で、内角は175.135…°となる。一辺の長さが a の正七十四角形の面積 S は

$S={\frac {74}{4}}a^{2}\cot {\frac {\pi }{74}}\simeq 435.5044a^{2}$

関係式

以下のように定義すると

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{74}}+2\cos {\frac {54\pi }{74}}+2\cos {\frac {22\pi }{74}}\\&x_{2}=2\cos {\frac {10\pi }{74}}+2\cos {\frac {26\pi }{74}}+2\cos {\frac {38\pi }{74}}\\&x_{3}=2\cos {\frac {6\pi }{74}}+2\cos {\frac {14\pi }{74}}+2\cos {\frac {66\pi }{74}}\\&x_{4}=2\cos {\frac {30\pi }{74}}+2\cos {\frac {70\pi }{74}}+2\cos {\frac {34\pi }{74}}\\&x_{5}=2\cos {\frac {18\pi }{74}}+2\cos {\frac {42\pi }{74}}+2\cos {\frac {50\pi }{74}}\\&x_{6}=2\cos {\frac {58\pi }{74}}+2\cos {\frac {62\pi }{74}}+2\cos {\frac {46\pi }{74}}\\\end{aligned}}$

以下の関係がある。

${\begin{aligned}&x_{1}+x_{3}+x_{5}={\frac {1+{\sqrt {37}}}{2}}=\alpha \\&x_{2}+x_{4}+x_{6}={\frac {1-{\sqrt {37}}}{2}}=\beta \\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}\right)^{3}=&{\frac {37-8{\sqrt {37}}+3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}\\\left(x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}\right)^{3}=&{\frac {37-8{\sqrt {37}}-3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}\\\left(x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}\right)^{3}=&{\frac {37+8{\sqrt {37}}+3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}\\\left(x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}\right)^{3}=&{\frac {37+8{\sqrt {37}}-3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}\\\end{aligned}}$

両辺の立方根を取ると

${\begin{aligned}x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}=&{\sqrt[{3}]{\frac {37-8{\sqrt {37}}+3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}}=A\\x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}=&{\sqrt[{3}]{\frac {37-8{\sqrt {37}}-3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}}=B\\x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}=&{\sqrt[{3}]{\frac {37+8{\sqrt {37}}+3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}}=C\\x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{8}=&{\sqrt[{3}]{\frac {37+8{\sqrt {37}}-3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}}=D\\\end{aligned}}$

よって

${\begin{aligned}3x_{1}=&\alpha +A+B\\3x_{3}=&\alpha +\omega ^{2}A+\omega B\\3x_{5}=&\alpha +\omega A+\omega ^{2}B\\3x_{2}=&\beta +C+D\\3x_{4}=&\beta +\omega ^{2}C+\omega D\\3x_{6}=&\beta +\omega C+\omega ^{2}D\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(2\cos {\frac {2\pi }{74}}+\omega \cdot 2\cos {\frac {54\pi }{74}}+\omega ^{2}\cdot 2\cos {\frac {22\pi }{74}}\right)^{3}=&3x_{1}+x_{3}+6(x_{4}-2)+3\omega (2x_{1}+x_{5}+x_{6})+3\omega ^{2}(2x_{1}+x_{2}+x_{5})\\\left(2\cos {\frac {2\pi }{74}}+\omega ^{2}\cdot 2\cos {\frac {54\pi }{74}}+\omega \cdot 2\cos {\frac {22\pi }{74}}\right)^{3}=&3x_{1}+x_{3}+6(x_{4}-2)+3\omega ^{2}(2x_{1}+x_{5}+x_{6})+3\omega (2x_{1}+x_{2}+x_{5})\\\end{aligned}}$

76角形

正七十六角形

七十六角形（ななじゅうろくかくけい、ななじゅうろっかっけい、heptacontahexagon）は、多角形の一つで、76本の辺と76個の頂点を持つ図形である。内角の和は13320°、対角線の本数は2774本である。

正七十六角形においては、中心角と外角は4.736…°で、内角は175.263…°となる。一辺の長さが a の正七十六角形の面積 S は

$S={\frac {76}{4}}a^{2}\cot {\frac {\pi }{76}}\simeq 459.37765a^{2}$

関係式

以下のように定義すると

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{76}}+2\cos {\frac {54\pi }{76}}+2\cos {\frac {62\pi }{76}}\\&x_{2}=2\cos {\frac {14\pi }{76}}+2\cos {\frac {74\pi }{76}}+2\cos {\frac {22\pi }{76}}\\&x_{3}=2\cos {\frac {6\pi }{76}}+2\cos {\frac {10\pi }{76}}+2\cos {\frac {34\pi }{76}}\\&x_{4}=2\cos {\frac {42\pi }{76}}+2\cos {\frac {70\pi }{76}}+2\cos {\frac {66\pi }{76}}\\&x_{5}=2\cos {\frac {18\pi }{76}}+2\cos {\frac {30\pi }{76}}+2\cos {\frac {50\pi }{76}}\\&x_{6}=2\cos {\frac {26\pi }{76}}+2\cos {\frac {58\pi }{76}}+2\cos {\frac {46\pi }{76}}\\\end{aligned}}$

以下の関係がある。

${\begin{aligned}&x_{1}+x_{3}+x_{5}={\sqrt {19}}=\alpha \\&x_{2}+x_{4}+x_{6}=-{\sqrt {19}}=\beta \\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(x_{3}+\omega \cdot x_{5}+\omega ^{2}\cdot x_{1}\right)^{3}=&{\frac {22{\sqrt {19}}+3{\sqrt {3}}(42{\sqrt {19}})i}{4}}={\frac {11{\sqrt {19}}+63{\sqrt {57}}i}{2}}\\\left(x_{3}+\omega ^{2}\cdot x_{5}+\omega \cdot x_{1}\right)^{3}=&{\frac {22{\sqrt {19}}-3{\sqrt {3}}(42{\sqrt {19}})i}{4}}={\frac {11{\sqrt {19}}-63{\sqrt {57}}i}{2}}\\\left(x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}\right)^{3}=&{\frac {-22{\sqrt {19}}-3{\sqrt {3}}(42{\sqrt {19}})i}{4}}={\frac {-11{\sqrt {19}}-63{\sqrt {57}}i}{2}}\\\left(x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}\right)^{3}=&{\frac {-22{\sqrt {19}}+3{\sqrt {3}}(42{\sqrt {19}})i}{4}}={\frac {-11{\sqrt {19}}+63{\sqrt {57}}i}{2}}\\\end{aligned}}$

両辺の立方根を取ると

${\begin{aligned}x_{3}+\omega \cdot x_{5}+\omega ^{2}\cdot x_{1}=&{\sqrt[{3}]{\frac {11{\sqrt {19}}+63{\sqrt {57}}i}{2}}}=A\\x_{3}+\omega ^{2}\cdot x_{5}+\omega \cdot x_{1}=&{\sqrt[{3}]{\frac {11{\sqrt {19}}-63{\sqrt {57}}i}{2}}}=B\\x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}=&{\sqrt[{3}]{\frac {-11{\sqrt {19}}-63{\sqrt {57}}i}{2}}}=C\\x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}=&{\sqrt[{3}]{\frac {-11{\sqrt {19}}+63{\sqrt {57}}i}{2}}}=D\\\end{aligned}}$

よって

${\begin{aligned}3x_{1}=&\alpha +\omega A+\omega ^{2}B\\3x_{3}=&\alpha +A+B\\3x_{5}=&\alpha +\omega ^{2}A+\omega B\\3x_{2}=&\beta +C+D\\3x_{4}=&\beta +\omega ^{2}C+\omega D\\3x_{6}=&\beta +\omega C+\omega ^{2}D\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(2\cos {\frac {2\pi }{76}}+\omega \cdot 2\cos {\frac {54\pi }{76}}+\omega ^{2}\cdot 2\cos {\frac {62\pi }{76}}\right)^{3}=&3x_{1}+x_{3}+6(x_{3})+3\omega (2x_{1}+x_{5}+x_{6})+3\omega ^{2}(2x_{1}+x_{4}+x_{6})\\\left(2\cos {\frac {2\pi }{76}}+\omega ^{2}\cdot 2\cos {\frac {54\pi }{76}}+\omega \cdot 2\cos {\frac {62\pi }{76}}\right)^{3}=&3x_{1}+x_{3}+6(x_{3})+3\omega ^{2}(2x_{1}+x_{5}+x_{6})+3\omega (2x_{1}+x_{4}+x_{6})\\\end{aligned}}$

108角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{108}}+2\cos {\frac {70\pi }{108}}+2\cos {\frac {74\pi }{108}}=0\\&x_{2}=2\cos {\frac {10\pi }{108}}+2\cos {\frac {82\pi }{108}}+2\cos {\frac {62\pi }{108}}=0\\&x_{3}=2\cos {\frac {50\pi }{108}}+2\cos {\frac {22\pi }{108}}+2\cos {\frac {94\pi }{108}}=0\\&x_{4}=2\cos {\frac {34\pi }{108}}+2\cos {\frac {106\pi }{108}}+2\cos {\frac {38\pi }{108}}=0\\&x_{5}=2\cos {\frac {46\pi }{108}}+2\cos {\frac {98\pi }{108}}+2\cos {\frac {26\pi }{108}}=0\\&x_{6}=2\cos {\frac {14\pi }{108}}+2\cos {\frac {58\pi }{108}}+2\cos {\frac {86\pi }{108}}=0\\\end{aligned}}$

三次方程式の係数を求めると

${\begin{aligned}&2\cos {\frac {2\pi }{108}}\cdot 2\cos {\frac {70\pi }{108}}+2\cos {\frac {70\pi }{108}}\cdot 2\cos {\frac {74\pi }{108}}+2\cos {\frac {74\pi }{108}}\cdot 2\cos {\frac {2\pi }{108}}=-3\\&2\cos {\frac {10\pi }{108}}\cdot 2\cos {\frac {82\pi }{108}}+2\cos {\frac {82\pi }{108}}\cdot 2\cos {\frac {62\pi }{108}}+2\cos {\frac {62\pi }{108}}\cdot 2\cos {\frac {10\pi }{108}}=-3\\&2\cos {\frac {50\pi }{108}}\cdot 2\cos {\frac {22\pi }{108}}+2\cos {\frac {22\pi }{108}}\cdot 2\cos {\frac {94\pi }{108}}+2\cos {\frac {94\pi }{108}}\cdot 2\cos {\frac {50\pi }{108}}=-3\\&2\cos {\frac {34\pi }{108}}\cdot 2\cos {\frac {106\pi }{108}}+2\cos {\frac {106\pi }{108}}\cdot 2\cos {\frac {38\pi }{108}}+2\cos {\frac {38\pi }{108}}\cdot 2\cos {\frac {34\pi }{108}}=-3\\&2\cos {\frac {46\pi }{108}}\cdot 2\cos {\frac {98\pi }{108}}+2\cos {\frac {98\pi }{108}}\cdot 2\cos {\frac {26\pi }{108}}+2\cos {\frac {26\pi }{108}}\cdot 2\cos {\frac {46\pi }{108}}=-3\\&2\cos {\frac {14\pi }{108}}\cdot 2\cos {\frac {58\pi }{108}}+2\cos {\frac {58\pi }{108}}\cdot 2\cos {\frac {86\pi }{108}}+2\cos {\frac {86\pi }{108}}\cdot 2\cos {\frac {14\pi }{108}}=-3\\&2\cos {\frac {2\pi }{108}}\cdot 2\cos {\frac {70\pi }{108}}\cdot 2\cos {\frac {74\pi }{108}}=2\cos {\frac {2\pi }{36}}\\&2\cos {\frac {10\pi }{108}}\cdot 2\cos {\frac {82\pi }{108}}\cdot 2\cos {\frac {62\pi }{108}}=2\cos {\frac {10\pi }{36}}\\&2\cos {\frac {50\pi }{108}}\cdot 2\cos {\frac {22\pi }{108}}\cdot 2\cos {\frac {94\pi }{108}}=2\cos {\frac {22\pi }{36}}\\&2\cos {\frac {34\pi }{108}}\cdot 2\cos {\frac {106\pi }{108}}\cdot 2\cos {\frac {38\pi }{108}}=2\cos {\frac {34\pi }{36}}\\&2\cos {\frac {46\pi }{108}}\cdot 2\cos {\frac {98\pi }{108}}\cdot 2\cos {\frac {26\pi }{108}}=2\cos {\frac {26\pi }{36}}\\&2\cos {\frac {14\pi }{108}}\cdot 2\cos {\frac {58\pi }{108}}\cdot 2\cos {\frac {86\pi }{108}}=2\cos {\frac {14\pi }{36}}\\\end{aligned}}$

解と係数の関係より

${\begin{aligned}u^{3}-3u-2\cos {\frac {2\pi }{36}}=0\end{aligned}}$

三次方程式を解いて、整理すると$\cos(2\pi /108)$が求められる。

${\begin{aligned}2\cos {\frac {2\pi }{108}}=&{\sqrt[{3}]{\cos {\frac {2\pi }{36}}+i\sin {\frac {2\pi }{36}}}}+{\sqrt[{3}]{\cos {\frac {2\pi }{36}}-i\sin {\frac {2\pi }{36}}}}\\\cos {\frac {2\pi }{108}}=&{\frac {{\sqrt[{3}]{\sqrt[{3}]{\frac {{\sqrt {3}}+i}{2}}}}+{\sqrt[{3}]{\sqrt[{3}]{\frac {{\sqrt {3}}-i}{2}}}}}{2}}\end{aligned}}$

114角形

以下のように定義すると

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{114}}+2\cos {\frac {14\pi }{114}}+2\cos {\frac {98\pi }{114}}\\&x_{2}=2\cos {\frac {10\pi }{114}}+2\cos {\frac {70\pi }{114}}+2\cos {\frac {34\pi }{114}}\\&x_{3}=2\cos {\frac {50\pi }{114}}+2\cos {\frac {106\pi }{114}}+2\cos {\frac {58\pi }{114}}\\&x_{4}=2\cos {\frac {22\pi }{114}}+2\cos {\frac {74\pi }{114}}+2\cos {\frac {62\pi }{114}}\\&x_{5}=2\cos {\frac {110\pi }{114}}+2\cos {\frac {86\pi }{114}}+2\cos {\frac {82\pi }{114}}\\&x_{6}=2\cos {\frac {94\pi }{114}}+2\cos {\frac {26\pi }{114}}+2\cos {\frac {46\pi }{114}}\\\end{aligned}}$

以下の関係がある。

${\begin{aligned}&x_{1}+x_{3}+x_{5}={\frac {-1-{\sqrt {57}}}{2}}=\alpha \\&x_{2}+x_{4}+x_{6}={\frac {-1+{\sqrt {57}}}{2}}=\beta \\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}\right)^{3}=&{\frac {38+3{\sqrt {57}}+3{\sqrt {3}}(38+5{\sqrt {57}})i}{2}}\\\left(x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}\right)^{3}=&{\frac {38+3{\sqrt {57}}-3{\sqrt {3}}(38+5{\sqrt {57}})i}{2}}\\\left(x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}\right)^{3}=&{\frac {38-3{\sqrt {57}}+3{\sqrt {3}}(38-5{\sqrt {57}})i}{2}}\\\left(x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}\right)^{3}=&{\frac {38-3{\sqrt {57}}-3{\sqrt {3}}(38-5{\sqrt {57}})i}{2}}\\\end{aligned}}$

両辺の立方根を取ると

${\begin{aligned}x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}=&{\sqrt[{3}]{\frac {38+3{\sqrt {57}}+3{\sqrt {3}}(38+5{\sqrt {57}})i}{2}}}=A\\x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}=&{\sqrt[{3}]{\frac {38+3{\sqrt {57}}-3{\sqrt {3}}(38+5{\sqrt {57}})i}{2}}}=B\\x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}=&{\sqrt[{3}]{\frac {38-3{\sqrt {57}}+3{\sqrt {3}}(38-5{\sqrt {57}})i}{2}}}=C\\x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}=&{\sqrt[{3}]{\frac {38-3{\sqrt {57}}-3{\sqrt {3}}(38-5{\sqrt {57}})i}{2}}}=D\\\end{aligned}}$

よって

${\begin{aligned}3x_{1}=&\alpha +A+B\\3x_{3}=&\alpha +\omega ^{2}A+\omega B\\3x_{5}=&\alpha +\omega A+\omega ^{2}B\\3x_{2}=&\beta +C+D\\3x_{4}=&\beta +\omega ^{2}C+\omega D\\3x_{6}=&\beta +\omega C+\omega ^{2}D\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(2\cos {\frac {2\pi }{114}}+\omega \cdot 2\cos {\frac {14\pi }{114}}+\omega ^{2}\cdot 2\cos {\frac {98\pi }{114}}\right)^{3}=&3x_{1}+{\tfrac {1+E+F}{3}}+6(x_{5}-2)+3\omega \left(2x_{1}+x_{2}+{\tfrac {1+\omega ^{2}E+\omega F}{3}}\right)+3\omega ^{2}\left(2x_{1}+x_{6}+{\tfrac {1+\omega ^{2}E+\omega F}{3}}\right)\\\left(2\cos {\frac {2\pi }{114}}+\omega \cdot 2\cos {\frac {14\pi }{114}}+\omega ^{2}\cdot 2\cos {\frac {98\pi }{114}}\right)^{3}=&3x_{1}+{\tfrac {1+E+F}{3}}+6(x_{5}-2)+3\omega ^{2}\left(2x_{1}+x_{2}+{\tfrac {1+\omega ^{2}E+\omega F}{3}}\right)+3\omega \left(2x_{1}+x_{6}+{\tfrac {1+\omega ^{2}E+\omega F}{3}}\right)\\\end{aligned}}$

ここでE,Fは

${\begin{aligned}E={\sqrt[{3}]{\frac {-133+57{\sqrt {3}}i}{2}}}\\F={\sqrt[{3}]{\frac {-133-57{\sqrt {3}}i}{2}}}\\\end{aligned}}$

126角形

以下のように定義すると

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{126}}+2\cos {\frac {10\pi }{126}}+2\cos {\frac {50\pi }{126}}\\&x_{2}=2\cos {\frac {38\pi }{126}}+2\cos {\frac {62\pi }{126}}+2\cos {\frac {58\pi }{126}}\\&x_{3}=2\cos {\frac {34\pi }{126}}+2\cos {\frac {82\pi }{126}}+2\cos {\frac {94\pi }{126}}\\&x_{4}=2\cos {\frac {22\pi }{126}}+2\cos {\frac {110\pi }{126}}+2\cos {\frac {46\pi }{126}}\\&x_{5}=2\cos {\frac {86\pi }{126}}+2\cos {\frac {74\pi }{126}}+2\cos {\frac {118\pi }{126}}\\&x_{6}=2\cos {\frac {122\pi }{126}}+2\cos {\frac {106\pi }{126}}+2\cos {\frac {26\pi }{126}}\\\end{aligned}}$

以下の関係がある。

${\begin{aligned}&x_{1}+x_{3}+x_{5}=0\\&x_{2}+x_{4}+x_{6}=0\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}\right)^{3}=&{\frac {27(7+2{\sqrt {21}})+9{\sqrt {3}}(21+4{\sqrt {21}})i}{2}}\\\left(x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}\right)^{3}=&{\frac {27(7+2{\sqrt {21}})-9{\sqrt {3}}(21+4{\sqrt {21}})i}{2}}\\\left(x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}\right)^{3}=&{\frac {-27(-7+2{\sqrt {21}})+9{\sqrt {3}}(21-4{\sqrt {21}})i}{2}}\\\left(x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}\right)^{3}=&{\frac {-27(-7+2{\sqrt {21}})-9{\sqrt {3}}(21-5{\sqrt {21}})i}{2}}\\\end{aligned}}$

両辺の立方根を取ると

${\begin{aligned}x_{1}+\omega \cdot x_{3}+\omega ^{2}\cdot x_{5}=&{\sqrt[{3}]{\frac {27(7+2{\sqrt {21}})+9{\sqrt {3}}(21+4{\sqrt {21}})i}{2}}}=A\\x_{1}+\omega ^{2}\cdot x_{3}+\omega \cdot x_{5}=&{\sqrt[{3}]{\frac {27(7+2{\sqrt {21}})-9{\sqrt {3}}(21+4{\sqrt {21}})i}{2}}}=B\\x_{2}+\omega \cdot x_{4}+\omega ^{2}\cdot x_{6}=&{\sqrt[{3}]{\frac {-27(-7+2{\sqrt {21}})+9{\sqrt {3}}(21-4{\sqrt {21}})i}{2}}}=C\\x_{2}+\omega ^{2}\cdot x_{4}+\omega \cdot x_{6}=&{\sqrt[{3}]{\frac {-27(-7+2{\sqrt {21}})-9{\sqrt {3}}(21-4{\sqrt {21}})i}{2}}}=D\\\end{aligned}}$

よって

${\begin{aligned}3x_{1}=&A+B\\3x_{3}=&\omega ^{2}A+\omega B\\3x_{5}=&\omega A+\omega ^{2}B\\3x_{2}=&C+D\\3x_{4}=&\omega ^{2}C+\omega D\\3x_{6}=&\omega C+\omega ^{2}D\\\end{aligned}}$

さらに、以下のような関係式が得られる。

${\begin{aligned}\left(2\cos {\frac {2\pi }{126}}+\omega \cdot 2\cos {\frac {10\pi }{126}}+\omega ^{2}\cdot 2\cos {\frac {50\pi }{126}}\right)^{3}=&3x_{1}+{\tfrac {-1+{\sqrt {21}}}{2}}+6(x_{2}+1)+3\omega \left(2x_{1}+{\tfrac {-1+{\sqrt {21}}}{2}}\right)+3\omega ^{2}(2x_{1}+x_{4}+1)\\\left(2\cos {\frac {2\pi }{126}}+\omega ^{2}\cdot 2\cos {\frac {10\pi }{126}}+\omega \cdot 2\cos {\frac {50\pi }{126}}\right)^{3}=&3x_{1}+{\tfrac {-1+{\sqrt {21}}}{2}}+6(x_{2}+1)+3\omega ^{2}\left(2x_{1}+{\tfrac {-1+{\sqrt {21}}}{2}}\right)+3\omega (2x_{1}+x_{4}+1)\\\end{aligned}}$

24次方程式

• 六十五角形
• 百四角形
• 百五角形
• 百十二角形
• 百三十角形
• 百四十角形
• 百四十四角形
• 百五十六角形
• 百六十八角形
• 百八十角形
• 二百十角形

正六十五角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{65}}+2\cos {\frac {32\pi }{65}}+2\cos {\frac {8\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}+{\sqrt {\frac {35+5{\sqrt {13}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65+15{\sqrt {13}}}{2}}-{\sqrt {\frac {715+195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {4\pi }{65}}+2\cos {\frac {64\pi }{65}}+2\cos {\frac {16\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}+{\sqrt {\frac {35-5{\sqrt {13}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65-15{\sqrt {13}}}{2}}+{\sqrt {\frac {715-195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {6\pi }{65}}+2\cos {\frac {34\pi }{65}}+2\cos {\frac {24\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}-{\sqrt {\frac {35+5{\sqrt {13}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65+15{\sqrt {13}}}{2}}+{\sqrt {\frac {715+195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {12\pi }{65}}+2\cos {\frac {62\pi }{65}}+2\cos {\frac {48\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}-{\sqrt {\frac {35-5{\sqrt {13}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65-15{\sqrt {13}}}{2}}-{\sqrt {\frac {715-195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {18\pi }{65}}+2\cos {\frac {28\pi }{65}}+2\cos {\frac {58\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}+{\sqrt {\frac {35+5{\sqrt {13}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65+15{\sqrt {13}}}{2}}-{\sqrt {\frac {715+195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {36\pi }{65}}+2\cos {\frac {56\pi }{65}}+2\cos {\frac {14\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}+{\sqrt {\frac {35-5{\sqrt {13}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65-15{\sqrt {13}}}{2}}+{\sqrt {\frac {715-195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {54\pi }{65}}+2\cos {\frac {46\pi }{65}}+2\cos {\frac {44\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}-{\sqrt {\frac {35+5{\sqrt {13}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65+15{\sqrt {13}}}{2}}+{\sqrt {\frac {715+195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {22\pi }{65}}+2\cos {\frac {38\pi }{65}}+2\cos {\frac {42\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}-{\sqrt {\frac {35-5{\sqrt {13}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65-15{\sqrt {13}}}{2}}-{\sqrt {\frac {715-195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\\end{aligned}}$

正百四角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{104}}+2\cos {\frac {18\pi }{104}}+2\cos {\frac {46\pi }{104}}={\frac {{\frac {{\sqrt {2}}+{\sqrt {26}}}{2}}+{\sqrt {\frac {26+6{\sqrt {13}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {6\pi }{104}}+2\cos {\frac {54\pi }{104}}+2\cos {\frac {70\pi }{104}}={\frac {-{\frac {{\sqrt {2}}+{\sqrt {26}}}{2}}+{\sqrt {\frac {26+6{\sqrt {13}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {14\pi }{104}}+2\cos {\frac {82\pi }{104}}+2\cos {\frac {94\pi }{104}}={\frac {{\frac {{\sqrt {2}}-{\sqrt {26}}}{2}}-{\sqrt {\frac {26-6{\sqrt {13}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {42\pi }{104}}+2\cos {\frac {38\pi }{104}}+2\cos {\frac {74\pi }{104}}={\frac {-{\frac {{\sqrt {2}}-{\sqrt {26}}}{2}}-{\sqrt {\frac {26-6{\sqrt {13}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {98\pi }{104}}+2\cos {\frac {50\pi }{104}}+2\cos {\frac {34\pi }{104}}={\frac {{\frac {{\sqrt {2}}+{\sqrt {26}}}{2}}-{\sqrt {\frac {26+6{\sqrt {13}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {86\pi }{104}}+2\cos {\frac {58\pi }{104}}+2\cos {\frac {102\pi }{104}}={\frac {-{\frac {{\sqrt {2}}+{\sqrt {26}}}{2}}-{\sqrt {\frac {26+6{\sqrt {13}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {62\pi }{104}}+2\cos {\frac {66\pi }{104}}+2\cos {\frac {30\pi }{104}}={\frac {{\frac {{\sqrt {2}}-{\sqrt {26}}}{2}}+{\sqrt {\frac {26-6{\sqrt {13}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {22\pi }{104}}+2\cos {\frac {10\pi }{104}}+2\cos {\frac {90\pi }{104}}={\frac {-{\frac {{\sqrt {2}}-{\sqrt {26}}}{2}}+{\sqrt {\frac {26-6{\sqrt {13}}}{2}}}}{2}}\\\end{aligned}}$

正百五角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{105}}+2\cos {\frac {32\pi }{105}}+2\cos {\frac {92\pi }{105}}={\frac {{\frac {{\frac {-1+{\sqrt {105}}}{2}}-{\sqrt {\frac {13-{\sqrt {105}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {25-{\sqrt {105}}}{2}}-{\sqrt {\frac {325-25{\sqrt {105}}}{2}}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {22\pi }{105}}+2\cos {\frac {68\pi }{105}}+2\cos {\frac {38\pi }{105}}={\frac {{\frac {{\frac {-1-{\sqrt {105}}}{2}}+{\sqrt {\frac {13+{\sqrt {105}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {25+{\sqrt {105}}}{2}}+{\sqrt {\frac {325+25{\sqrt {105}}}{2}}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {4\pi }{105}}+2\cos {\frac {64\pi }{105}}+2\cos {\frac {26\pi }{105}}={\frac {{\frac {{\frac {-1+{\sqrt {105}}}{2}}+{\sqrt {\frac {13-{\sqrt {105}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {25-{\sqrt {105}}}{2}}+{\sqrt {\frac {325-25{\sqrt {105}}}{2}}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {44\pi }{105}}+2\cos {\frac {74\pi }{105}}+2\cos {\frac {76\pi }{105}}={\frac {{\frac {{\frac {-1-{\sqrt {105}}}{2}}-{\sqrt {\frac {13+{\sqrt {105}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {25+{\sqrt {105}}}{2}}-{\sqrt {\frac {325+25{\sqrt {105}}}{2}}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {8\pi }{105}}+2\cos {\frac {82\pi }{105}}+2\cos {\frac {52\pi }{105}}={\frac {{\frac {{\frac {-1+{\sqrt {105}}}{2}}-{\sqrt {\frac {13-{\sqrt {105}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {25-{\sqrt {105}}}{2}}-{\sqrt {\frac {325-25{\sqrt {105}}}{2}}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {88\pi }{105}}+2\cos {\frac {62\pi }{105}}+2\cos {\frac {58\pi }{105}}={\frac {{\frac {{\frac {-1-{\sqrt {105}}}{2}}+{\sqrt {\frac {13+{\sqrt {105}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {25+{\sqrt {105}}}{2}}+{\sqrt {\frac {325+25{\sqrt {105}}}{2}}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {16\pi }{105}}+2\cos {\frac {46\pi }{105}}+2\cos {\frac {104\pi }{105}}={\frac {{\frac {{\frac {-1+{\sqrt {105}}}{2}}+{\sqrt {\frac {13-{\sqrt {105}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {25-{\sqrt {105}}}{2}}+{\sqrt {\frac {325-25{\sqrt {105}}}{2}}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {34\pi }{105}}+2\cos {\frac {86\pi }{105}}+2\cos {\frac {94\pi }{105}}={\frac {{\frac {{\frac {-1-{\sqrt {105}}}{2}}-{\sqrt {\frac {13+{\sqrt {105}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {25+{\sqrt {105}}}{2}}-{\sqrt {\frac {325+25{\sqrt {105}}}{2}}}}{2}}}}{2}}\\\end{aligned}}$

正百十二角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{112}}+2\cos {\frac {62\pi }{112}}+2\cos {\frac {94\pi }{112}}=-{\frac {\sqrt {\frac {32-{\sqrt {\frac {1024+384{\sqrt {7}}}{2}}}}{2}}}{2}}=-{\sqrt {4-{\sqrt {8+3{\sqrt {7}}}}}}=-{\sqrt {\frac {8-3{\sqrt {2}}-{\sqrt {14}}}{2}}}\\&x_{2}=2\cos {\frac {10\pi }{112}}+2\cos {\frac {86\pi }{112}}+2\cos {\frac {22\pi }{112}}={\frac {\sqrt {\frac {32+{\sqrt {\frac {1024-384{\sqrt {7}}}{2}}}}{2}}}{2}}={\sqrt {4+{\sqrt {8-3{\sqrt {7}}}}}}\\&x_{3}=2\cos {\frac {6\pi }{112}}+2\cos {\frac {38\pi }{112}}+2\cos {\frac {58\pi }{112}}={\frac {\sqrt {\frac {32+{\sqrt {\frac {1024+384{\sqrt {7}}}{2}}}}{2}}}{2}}={\sqrt {4+{\sqrt {8+3{\sqrt {7}}}}}}\\&x_{4}=2\cos {\frac {30\pi }{112}}+2\cos {\frac {34\pi }{112}}+2\cos {\frac {66\pi }{112}}={\frac {\sqrt {\frac {32-{\sqrt {\frac {1024-384{\sqrt {7}}}{2}}}}{2}}}{2}}={\sqrt {4-{\sqrt {8-3{\sqrt {7}}}}}}\\&x_{5}=2\cos {\frac {18\pi }{112}}+2\cos {\frac {110\pi }{112}}+2\cos {\frac {50\pi }{112}}={\frac {\sqrt {\frac {32-{\sqrt {\frac {1024+384{\sqrt {7}}}{2}}}}{2}}}{2}}={\sqrt {4-{\sqrt {8+3{\sqrt {7}}}}}}\\&x_{6}=2\cos {\frac {90\pi }{112}}+2\cos {\frac {102\pi }{112}}+2\cos {\frac {26\pi }{112}}=-{\frac {\sqrt {\frac {32+{\sqrt {\frac {1024-384{\sqrt {7}}}{2}}}}{2}}}{2}}=-{\sqrt {4+{\sqrt {8-3{\sqrt {7}}}}}}\\&x_{7}=2\cos {\frac {54\pi }{112}}+2\cos {\frac {106\pi }{112}}+2\cos {\frac {74\pi }{112}}=-{\frac {\sqrt {\frac {32+{\sqrt {\frac {1024+384{\sqrt {7}}}{2}}}}{2}}}{2}}=-{\sqrt {4+{\sqrt {8+3{\sqrt {7}}}}}}\\&x_{8}=2\cos {\frac {46\pi }{112}}+2\cos {\frac {82\pi }{112}}+2\cos {\frac {78\pi }{112}}=-{\frac {\sqrt {\frac {32-{\sqrt {\frac {1024-384{\sqrt {7}}}{2}}}}{2}}}{2}}=-{\sqrt {4-{\sqrt {8-3{\sqrt {7}}}}}}\\\end{aligned}}$

正百三十角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{130}}+2\cos {\frac {98\pi }{130}}+2\cos {\frac {122\pi }{130}}={\frac {{\frac {{\frac {-1+{\sqrt {13}}}{2}}-{\sqrt {\frac {35-5{\sqrt {13}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65-15{\sqrt {13}}}{2}}+{\sqrt {\frac {715-195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {14\pi }{130}}+2\cos {\frac {94\pi }{130}}+2\cos {\frac {74\pi }{130}}={\frac {{\frac {{\frac {-1-{\sqrt {13}}}{2}}-{\sqrt {\frac {35+5{\sqrt {13}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65+15{\sqrt {13}}}{2}}-{\sqrt {\frac {715+195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {6\pi }{130}}+2\cos {\frac {34\pi }{130}}+2\cos {\frac {106\pi }{130}}={\frac {{\frac {{\frac {-1+{\sqrt {13}}}{2}}+{\sqrt {\frac {35-5{\sqrt {13}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65-15{\sqrt {13}}}{2}}-{\sqrt {\frac {715-195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {42\pi }{130}}+2\cos {\frac {22\pi }{130}}+2\cos {\frac {38\pi }{130}}={\frac {{\frac {{\frac {-1-{\sqrt {13}}}{2}}+{\sqrt {\frac {35+5{\sqrt {13}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65+15{\sqrt {13}}}{2}}+{\sqrt {\frac {715+195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {18\pi }{130}}+2\cos {\frac {102\pi }{130}}+2\cos {\frac {58\pi }{130}}={\frac {{\frac {{\frac {-1+{\sqrt {13}}}{2}}-{\sqrt {\frac {35-5{\sqrt {13}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65-15{\sqrt {13}}}{2}}+{\sqrt {\frac {715-195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {126\pi }{130}}+2\cos {\frac {66\pi }{130}}+2\cos {\frac {114\pi }{130}}={\frac {{\frac {{\frac {-1-{\sqrt {13}}}{2}}-{\sqrt {\frac {35+5{\sqrt {13}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65+15{\sqrt {13}}}{2}}-{\sqrt {\frac {715+195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {54\pi }{130}}+2\cos {\frac {46\pi }{130}}+2\cos {\frac {86\pi }{130}}={\frac {{\frac {{\frac {-1+{\sqrt {13}}}{2}}+{\sqrt {\frac {35-5{\sqrt {13}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65-15{\sqrt {13}}}{2}}-{\sqrt {\frac {715-195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {118\pi }{130}}+2\cos {\frac {62\pi }{130}}+2\cos {\frac {82\pi }{130}}={\frac {{\frac {{\frac {-1-{\sqrt {13}}}{2}}+{\sqrt {\frac {35+5{\sqrt {13}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65+15{\sqrt {13}}}{2}}+{\sqrt {\frac {715+195{\sqrt {13}}}{2}}}}{2}}}}{2}}\\\end{aligned}}$

正百四十角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{140}}+2\cos {\frac {118\pi }{140}}+2\cos {\frac {38\pi }{140}}={\frac {{\frac {{\sqrt {7}}+{\sqrt {35}}}{2}}-{\sqrt {\frac {5-{\sqrt {5}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {22\pi }{140}}+2\cos {\frac {102\pi }{140}}+2\cos {\frac {138\pi }{140}}=-{\frac {{\frac {{\sqrt {7}}+{\sqrt {35}}}{2}}-{\sqrt {\frac {5-{\sqrt {5}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {6\pi }{140}}+2\cos {\frac {74\pi }{140}}+2\cos {\frac {114\pi }{140}}={\frac {{\frac {{\sqrt {7}}-{\sqrt {35}}}{2}}+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {66\pi }{140}}+2\cos {\frac {26\pi }{140}}+2\cos {\frac {134\pi }{140}}=-{\frac {{\frac {{\sqrt {7}}-{\sqrt {35}}}{2}}+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {18\pi }{140}}+2\cos {\frac {58\pi }{140}}+2\cos {\frac {62\pi }{140}}={\frac {{\frac {{\sqrt {7}}+{\sqrt {35}}}{2}}+{\sqrt {\frac {5-{\sqrt {5}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {82\pi }{140}}+2\cos {\frac {78\pi }{140}}+2\cos {\frac {122\pi }{140}}=-{\frac {{\frac {{\sqrt {7}}+{\sqrt {35}}}{2}}+{\sqrt {\frac {5-{\sqrt {5}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {54\pi }{140}}+2\cos {\frac {106\pi }{140}}+2\cos {\frac {94\pi }{140}}={\frac {{\frac {{\sqrt {7}}-{\sqrt {35}}}{2}}-{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {34\pi }{140}}+2\cos {\frac {46\pi }{140}}+2\cos {\frac {86\pi }{140}}=-{\frac {{\frac {{\sqrt {7}}-{\sqrt {35}}}{2}}-{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}{2}}\\\end{aligned}}$

正百四十四角形

関係式

${\begin{aligned}&\\\end{aligned}}$

正百五十六角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{156}}+2\cos {\frac {122\pi }{156}}+2\cos {\frac {46\pi }{156}}={\frac {{\frac {{\sqrt {39}}-{\sqrt {3}}}{2}}+{\sqrt {\frac {13-3{\sqrt {13}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {22\pi }{156}}+2\cos {\frac {94\pi }{156}}+2\cos {\frac {118\pi }{156}}={\frac {-{\frac {{\sqrt {39}}+{\sqrt {3}}}{2}}+{\sqrt {\frac {13+3{\sqrt {13}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {14\pi }{156}}+2\cos {\frac {82\pi }{156}}+2\cos {\frac {10\pi }{156}}={\frac {{\frac {{\sqrt {39}}+{\sqrt {3}}}{2}}+{\sqrt {\frac {13+3{\sqrt {13}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {154\pi }{156}}+2\cos {\frac {34\pi }{156}}+2\cos {\frac {110\pi }{156}}={\frac {-{\frac {{\sqrt {39}}-{\sqrt {3}}}{2}}-{\sqrt {\frac {13-3{\sqrt {13}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {98\pi }{156}}+2\cos {\frac {50\pi }{156}}+2\cos {\frac {70\pi }{156}}={\frac {{\frac {{\sqrt {39}}-{\sqrt {3}}}{2}}-{\sqrt {\frac {13-3{\sqrt {13}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {142\pi }{156}}+2\cos {\frac {74\pi }{156}}+2\cos {\frac {146\pi }{156}}={\frac {-{\frac {{\sqrt {39}}+{\sqrt {3}}}{2}}-{\sqrt {\frac {13+3{\sqrt {13}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {62\pi }{156}}+2\cos {\frac {38\pi }{156}}+2\cos {\frac {134\pi }{156}}={\frac {{\frac {{\sqrt {39}}+{\sqrt {3}}}{2}}-{\sqrt {\frac {13+3{\sqrt {13}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {58\pi }{156}}+2\cos {\frac {106\pi }{156}}+2\cos {\frac {86\pi }{156}}={\frac {-{\frac {{\sqrt {39}}-{\sqrt {3}}}{2}}+{\sqrt {\frac {13-3{\sqrt {13}}}{2}}}}{2}}\\\end{aligned}}$

正百六十八角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{168}}+2\cos {\frac {50\pi }{168}}+2\cos {\frac {94\pi }{168}}={\frac {{\frac {{\sqrt {6}}+{\sqrt {14}}}{2}}+{\sqrt {\frac {22-2{\sqrt {21}}}{2}}}}{2}}={\frac {{\frac {{\sqrt {6}}+{\sqrt {14}}}{2}}+{\frac {-{\sqrt {2}}+{\sqrt {42}}}{2}}}{2}}\\&x_{2}=2\cos {\frac {22\pi }{168}}+2\cos {\frac {122\pi }{168}}+2\cos {\frac {26\pi }{168}}={\frac {{\frac {-{\sqrt {6}}+{\sqrt {14}}}{2}}+{\sqrt {\frac {22+2{\sqrt {21}}}{2}}}}{2}}={\frac {{\frac {-{\sqrt {6}}+{\sqrt {14}}}{2}}+{\frac {{\sqrt {2}}+{\sqrt {42}}}{2}}}{2}}\\&x_{3}=2\cos {\frac {38\pi }{168}}+2\cos {\frac {58\pi }{168}}+2\cos {\frac {106\pi }{168}}={\frac {{\frac {{\sqrt {6}}-{\sqrt {14}}}{2}}+{\sqrt {\frac {22+2{\sqrt {21}}}{2}}}}{2}}={\frac {{\frac {{\sqrt {6}}-{\sqrt {14}}}{2}}+{\frac {{\sqrt {2}}+{\sqrt {42}}}{2}}}{2}}\\&x_{4}=2\cos {\frac {34\pi }{168}}+2\cos {\frac {158\pi }{168}}+2\cos {\frac {82\pi }{168}}={\frac {{\frac {-{\sqrt {6}}-{\sqrt {14}}}{2}}+{\sqrt {\frac {22-2{\sqrt {21}}}{2}}}}{2}}={\frac {{\frac {-{\sqrt {6}}-{\sqrt {14}}}{2}}+{\frac {-{\sqrt {2}}+{\sqrt {42}}}{2}}}{2}}\\&x_{5}=2\cos {\frac {10\pi }{168}}+2\cos {\frac {86\pi }{168}}+2\cos {\frac {134\pi }{168}}={\frac {{\frac {{\sqrt {6}}+{\sqrt {14}}}{2}}-{\sqrt {\frac {22-2{\sqrt {21}}}{2}}}}{2}}={\frac {{\frac {{\sqrt {6}}+{\sqrt {14}}}{2}}-{\frac {-{\sqrt {2}}+{\sqrt {42}}}{2}}}{2}}\\&x_{6}=2\cos {\frac {110\pi }{168}}+2\cos {\frac {62\pi }{168}}+2\cos {\frac {130\pi }{168}}={\frac {{\frac {-{\sqrt {6}}+{\sqrt {14}}}{2}}-{\sqrt {\frac {22+2{\sqrt {21}}}{2}}}}{2}}={\frac {{\frac {-{\sqrt {6}}+{\sqrt {14}}}{2}}-{\frac {{\sqrt {2}}+{\sqrt {42}}}{2}}}{2}}\\&x_{7}=2\cos {\frac {146\pi }{168}}+2\cos {\frac {46\pi }{168}}+2\cos {\frac {142\pi }{168}}={\frac {{\frac {{\sqrt {6}}-{\sqrt {14}}}{2}}-{\sqrt {\frac {22+2{\sqrt {21}}}{2}}}}{2}}={\frac {{\frac {{\sqrt {6}}-{\sqrt {14}}}{2}}-{\frac {{\sqrt {2}}+{\sqrt {42}}}{2}}}{2}}\\&x_{8}=2\cos {\frac {166\pi }{168}}+2\cos {\frac {118\pi }{168}}+2\cos {\frac {74\pi }{168}}={\frac {{\frac {-{\sqrt {6}}-{\sqrt {14}}}{2}}-{\sqrt {\frac {22-2{\sqrt {21}}}{2}}}}{2}}={\frac {{\frac {-{\sqrt {6}}-{\sqrt {14}}}{2}}-{\frac {-{\sqrt {2}}+{\sqrt {42}}}{2}}}{2}}\\\end{aligned}}$

正百八十角形

関係式

${\begin{aligned}&\\\end{aligned}}$

正二百十角形

関係式

${\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{210}}+2\cos {\frac {118\pi }{210}}+2\cos {\frac {178\pi }{210}}={\frac {{\frac {{\frac {1-{\sqrt {21}}}{2}}-{\sqrt {\frac {55-5{\sqrt {21}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {25+5{\sqrt {21}}}{2}}-{\sqrt {\frac {115+25{\sqrt {21}}}{2}}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {22\pi }{210}}+2\cos {\frac {38\pi }{210}}+2\cos {\frac {142\pi }{210}}={\frac {{\frac {{\frac {1+{\sqrt {21}}}{2}}+{\sqrt {\frac {55+5{\sqrt {21}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {25-5{\sqrt {21}}}{2}}-{\sqrt {\frac {115-25{\sqrt {21}}}{2}}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {34\pi }{210}}+2\cos {\frac {94\pi }{210}}+2\cos {\frac {86\pi }{210}}={\frac {{\frac {{\frac {1-{\sqrt {21}}}{2}}+{\sqrt {\frac {55-5{\sqrt {21}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {25+5{\sqrt {21}}}{2}}+{\sqrt {\frac {115+25{\sqrt {21}}}{2}}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {46\pi }{210}}+2\cos {\frac {194\pi }{210}}+2\cos {\frac {106\pi }{210}}={\frac {{\frac {{\frac {1+{\sqrt {21}}}{2}}-{\sqrt {\frac {55+5{\sqrt {21}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {25-5{\sqrt {21}}}{2}}+{\sqrt {\frac {115-25{\sqrt {21}}}{2}}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {158\pi }{210}}+2\cos {\frac {82\pi }{210}}+2\cos {\frac {202\pi }{210}}={\frac {{\frac {{\frac {1-{\sqrt {21}}}{2}}-{\sqrt {\frac {55-5{\sqrt {21}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {25+5{\sqrt {21}}}{2}}-{\sqrt {\frac {115+25{\sqrt {21}}}{2}}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {58\pi }{210}}+2\cos {\frac {62\pi }{210}}+2\cos {\frac {122\pi }{210}}={\frac {{\frac {{\frac {1+{\sqrt {21}}}{2}}+{\sqrt {\frac {55+5{\sqrt {21}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {25-5{\sqrt {21}}}{2}}-{\sqrt {\frac {115-25{\sqrt {21}}}{2}}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {166\pi }{210}}+2\cos {\frac {134\pi }{210}}+2\cos {\frac {74\pi }{210}}={\frac {{\frac {{\frac {1-{\sqrt {21}}}{2}}+{\sqrt {\frac {55-5{\sqrt {21}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {25+5{\sqrt {21}}}{2}}+{\sqrt {\frac {115+25{\sqrt {21}}}{2}}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {146\pi }{210}}+2\cos {\frac {206\pi }{210}}+2\cos {\frac {26\pi }{210}}={\frac {{\frac {{\frac {1+{\sqrt {21}}}{2}}-{\sqrt {\frac {55+5{\sqrt {21}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {25-5{\sqrt {21}}}{2}}+{\sqrt {\frac {115-25{\sqrt {21}}}{2}}}}{2}}}}{2}}\\\end{aligned}}$

27次方程式

• 八十一角形

$\cos {\frac {2\pi }{81}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{\omega }}}}+{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{\omega ^{2}}}}}}{2}}$

• 百六十二角形

$\cos {\frac {2\pi }{162}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{-\omega ^{2}}}}}+{\sqrt[{3}]{\sqrt[{3}]{\sqrt[{3}]{-\omega }}}}}{2}}$

32次方程式

• 八十五角形
• 百二十八角形
• 百三十六角形

$2\cos {\frac {2\pi }{136}}={\frac {{\frac {{\frac {{\frac {{\sqrt {34}}-{\sqrt {2}}}{2}}-{\sqrt {\frac {34-2{\sqrt {17}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {34+6{\sqrt {17}}}{2}}+{\sqrt {\frac {340+76{\sqrt {17}}}{2}}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {34-2{\sqrt {17}}}{2}}-{\sqrt {\frac {68-4{\sqrt {17}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {68+12{\sqrt {17}}}{2}}-{\sqrt {\frac {1360+304{\sqrt {17}}}{2}}}}{2}}}}{2}}}}{2}}$

• 百六十角形

$2\cos {\frac {2\pi }{160}}={\frac {\sqrt {8+{\sqrt {10+3{\sqrt {10}}}}+{\sqrt {10-{\sqrt {10}}}}+{\sqrt {12+2{\sqrt {10}}-{\sqrt {152+48{\sqrt {10}}}}}}}}{2}}$

$\cos \left({\frac {\pi }{80}}\right)=\cos \left(2.25^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}}}$

• 百七十角形

$2\cos {\frac {2\pi }{170}}={\frac {{\frac {{\frac {{\frac {{\frac {-1+{\sqrt {85}}}{2}}-{\sqrt {\frac {11-{\sqrt {85}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {51-{\sqrt {85}}}{2}}-{\sqrt {\frac {17(47-3{\sqrt {85}})}{2}}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {3(17+{\sqrt {85}})}{2}}-{\sqrt {\frac {17(83+9{\sqrt {85}})}{2}}}}{2}}+{\sqrt {\frac {{\frac {595+57{\sqrt {85}}}{2}}-{\sqrt {\frac {17(18407+1995{\sqrt {85}})}{2}}}}{2}}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {{\frac {85+{\sqrt {85}}}{2}}+{\sqrt {\frac {85(11+{\sqrt {85}})}{2}}}}{2}}+{\sqrt {\frac {{\frac {5(51+{\sqrt {85}})}{2}}+{\sqrt {\frac {425(47+3{\sqrt {85}})}{2}}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {15(17-{\sqrt {85}})}{2}}+{\sqrt {\frac {425(83-9{\sqrt {85}})}{2}}}}{2}}-{\sqrt {\frac {{\frac {25(595-57{\sqrt {85}})}{2}}+{\sqrt {\frac {10625(18407-1995{\sqrt {85}})}{2}}}}{2}}}}{2}}}}{2}}}}{2}}$

$2\cos {\frac {2\pi }{170}}={\frac {{\frac {{\frac {{\frac {{\frac {-1+{\sqrt {85}}}{2}}-{\frac {{\sqrt {17}}-{\sqrt {5}}}{2}}}{2}}-{\sqrt {\frac {{\frac {51-{\sqrt {85}}}{2}}-{\frac {17{\sqrt {5}}-3{\sqrt {17}}}{2}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {3(17+{\sqrt {85}})}{2}}-{\frac {17{\sqrt {5}}+9{\sqrt {17}}}{2}}}{2}}+{\sqrt {\frac {{\frac {595+57{\sqrt {85}}}{2}}-{\sqrt {\frac {17(18407+1995{\sqrt {85}})}{2}}}}{2}}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {{\frac {85+{\sqrt {85}}}{2}}+{\frac {17{\sqrt {5}}+5{\sqrt {17}}}{2}}}{2}}+{\sqrt {\frac {{\frac {5(51+{\sqrt {85}})}{2}}+{\frac {5(17{\sqrt {5}}+3{\sqrt {17}})}{2}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {15(17-{\sqrt {85}})}{2}}+{\frac {5(17{\sqrt {5}}-9{\sqrt {17}})}{2}}}{2}}-{\sqrt {\frac {{\frac {25(595-57{\sqrt {85}})}{2}}+{\sqrt {\frac {10625(18407-1995{\sqrt {85}})}{2}}}}{2}}}}{2}}}}{2}}}}{2}}$

$2\cos {\frac {2\pi }{170}}={\frac {{\frac {{\frac {{\frac {{\frac {-1+{\sqrt {85}}}{2}}-{\frac {{\sqrt {17}}-{\sqrt {5}}}{2}}}{2}}-{\sqrt {\frac {{\frac {51-{\sqrt {85}}}{2}}-{\frac {17{\sqrt {5}}-3{\sqrt {17}}}{2}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {3(17+{\sqrt {85}})}{2}}-{\frac {17{\sqrt {5}}+9{\sqrt {17}}}{2}}}{2}}+{\sqrt {\frac {{\frac {595+57{\sqrt {85}}}{2}}-{\sqrt {\frac {17(18407+1995{\sqrt {85}})}{2}}}}{2}}}}{2}}}}{2}}+{\sqrt {{\sqrt {5}}\cdot {\frac {{\frac {{\frac {{\frac {17{\sqrt {5}}+{\sqrt {17}}}{2}}+{\frac {17+{\sqrt {85}}}{2}}}{2}}+{\sqrt {\frac {{\frac {51+{\sqrt {85}}}{2}}+{\frac {17{\sqrt {5}}+3{\sqrt {17}}}{2}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {3(17-{\sqrt {85}})}{2}}+{\frac {17{\sqrt {5}}-9{\sqrt {17}}}{2}}}{2}}-{\sqrt {\frac {{\frac {595-57{\sqrt {85}}}{2}}+{\sqrt {\frac {17(18407-1995{\sqrt {85}})}{2}}}}{2}}}}{2}}}}{2}}}}}{2}}$

• 百九十二角形

$2\cos {\frac {2\pi }{192}}={\frac {\sqrt {2(4+{\sqrt {2(4+{\sqrt {2(4+{\sqrt {6}}+{\sqrt {2}})}})}})}}{2}}$

$\cos \left({\frac {\pi }{96}}\right)=\cos \left(1.875^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}$

• 二百四角形

$2\cos {\frac {2\pi }{204}}={\frac {{\frac {{\frac {{\frac {{\sqrt {51}}+{\sqrt {3}}}{2}}+{\sqrt {\frac {3(17+{\sqrt {17}})}{2}}}}{2}}+{\sqrt {\frac {{\frac {3(17-3{\sqrt {17}})}{2}}-{\sqrt {\frac {9(85-19{\sqrt {17}})}{2}}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {17+{\sqrt {17}}}{2}}-{\sqrt {\frac {17+{\sqrt {17}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {17-3{\sqrt {17}}}{2}}+{\sqrt {\frac {85-19{\sqrt {17}}}{2}}}}{2}}}}{2}}}}{2}}$

$2\cos {\frac {2\pi }{204}}={\frac {{\sqrt {3}}\cdot {\frac {{\frac {{\frac {{\sqrt {17}}+1}{2}}+{\sqrt {\frac {17+{\sqrt {17}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {17-3{\sqrt {17}}}{2}}-{\sqrt {\frac {85-19{\sqrt {17}}}{2}}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {{\frac {17+{\sqrt {17}}}{2}}-{\sqrt {\frac {17+{\sqrt {17}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {17-3{\sqrt {17}}}{2}}+{\sqrt {\frac {85-19{\sqrt {17}}}{2}}}}{2}}}}{2}}}}{2}}$

• 二百四十角形

36次方程式

• 七十三角形
• 九十一角形
• 九十五角形
• 百十一角形
• 百十七角形
• 百三十五角形
• 百四十六角形
• 百四十八角形
• 百五十二角形
• 百八十二角形
• 百九十角形
• 二百十六角形
• 二百二十二角形
• 二百二十八角形
• 二百三十四角形
• 二百五十二角形
• 二百七十角形

48次方程式

• 九十七角形
• 百十九角形
• 百五十三角形
• 百九十四角形
• 百九十五角形
• 二百八角形
• 二百二十四角形
• 二百三十八角形
• 二百六十角形
• 二百八十角形
• 二百八十八角形
• 三百六角形
• 三百十二角形
• 三百三十六角形
• 三百六十角形
• 三百九十角形
• 四百二十角形

54次方程式

• 百九角形
• 百三十三角形
• 百七十一角形
• 百八十九角形
• 二百十八角形
• 二百六十六角形
• 三百二十四角形
• 三百四十二角形
• 三百七十八角形

64次方程式

• 二百五十五角形
• 二百五十六角形
• 二百七十二角形
• 三百二十角形
• 三百四十角形
• 三百八十四角形
• 四百八角形
• 四百八十角形
• 五百十角形

72次方程式

• 百八十五角形
• 二百十九角形
• 二百七十三角形
• 二百八十五角形
• 二百九十二角形
• 二百九十六角形
• 三百四角形
• 三百十五角形
• 三百六十四角形
• 三百七十角形
• 三百八十角形
• 四百三十二角形
• 四百三十八角形
• 四百四十四角形
• 四百五十六角形
• 四百六十八角形
• 五百四角形
• 五百四十角形
• 五百四十六角形
• 五百七十角形
• 六百三十角形

81次方程式

• 百六十三角形
• 二百四十三角形
• 三百二十六角形
• 四百八十六角形

96次方程式

• 百九十三角形
• 二百二十一角形
• 二百九十一角形
• 三百五十七角形
• 三百八十六角形
• 三百八十八角形
• 四百十六角形
• 四百四十二角形
• 四百四十八角形
• 四百七十六角形
• 五百二十角形
• 五百六十角形
• 五百七十六角形
• 五百八十二角形
• 六百十二角形
• 六百二十四角形
• 六百七十二角形
• 七百十四角形
• 七百二十角形
• 七百八十角形
• 八百四十角形

108次方程式

• 二百四十七角形
• 二百五十九角形
• 三百二十七角形
• 三百三十三角形
• 三百五十一角形
• 三百九十九角形
• 四百五角形
• 四百三十六角形
• 四百九十四角形
• 五百十八角形
• 五百三十二角形
• 六百四十八角形
• 六百五十四角形
• 六百六十六角形
• 六百八十四角形
• 七百二角形
• 七百五十六角形
• 七百九十八角形
• 八百十角形

128次方程式

• 二百五十七角形
• 五百十二角形
• 五百十四角形
• 五百四十四角形
• 六百四十角形
• 六百八十角形
• 七百六十八角形
• 八百十六角形
• 九百六十角形
• 千二十角形

正七角形

$c_{1}=6\cos {\frac {2\pi }{7}}+1$

$c_{2}=6\cos {\frac {4\pi }{7}}+1$

$c_{3}=6\cos {\frac {6\pi }{7}}+1$

$c_{4}=6\cos {\frac {8\pi }{7}}+1$

$c_{5}=6\cos {\frac {10\pi }{7}}+1$

$c_{6}=6\cos {\frac {12\pi }{7}}+1$

$s_{1}=6\sin {\frac {2\pi }{7}}-{\sqrt {7}}$

$s_{2}=6\sin {\frac {4\pi }{7}}-{\sqrt {7}}$

$s_{3}=6\sin {\frac {6\pi }{7}}+{\sqrt {7}}$

$s_{4}=6\sin {\frac {8\pi }{7}}-{\sqrt {7}}$

$s_{5}=6\sin {\frac {10\pi }{7}}+{\sqrt {7}}$

$s_{6}=6\sin {\frac {12\pi }{7}}+{\sqrt {7}}$

$c_{1}+c_{2}+c_{4}=0$

$c_{1}c_{2}+c_{2}c_{4}+c_{4}c_{1}=-21$

$c_{1}c_{2}c_{4}=7$

$c_{3}+c_{5}+c_{6}=0$

$c_{3}c_{5}+c_{5}c_{6}+c_{6}c_{3}=-21$

$c_{3}c_{5}c_{6}=7$

$s_{1}+s_{2}+s_{4}=0$

$s_{1}s_{2}+s_{2}s_{4}+s_{4}s_{1}=-21$

$s_{1}s_{2}s_{4}=-13{\sqrt {7}}$

$s_{3}+s_{5}+s_{6}=0$

$s_{3}s_{5}+s_{5}s_{6}+s_{6}s_{3}=-21$

$s_{3}s_{5}s_{6}=13{\sqrt {7}}$

$c_{x}=\omega ^{3-k}{\sqrt[{3}]{{\frac {7}{2}}+{\frac {21{\sqrt {3}}i}{2}}}}+\omega ^{k}{\sqrt[{3}]{{\frac {7}{2}}-{\frac {21{\sqrt {3}}i}{2}}}}$

$s_{x}=\omega ^{3-k}{\sqrt[{3}]{{\frac {-13{\sqrt {7}}}{2}}+{\frac {3{\sqrt {21}}i}{2}}}}+\omega ^{k}{\sqrt[{3}]{{\frac {-13{\sqrt {7}}}{2}}-{\frac {3{\sqrt {21}}i}{2}}}}$

$s_{x}=\omega ^{3-k}{\sqrt[{3}]{{\frac {13{\sqrt {7}}}{2}}+{\frac {3{\sqrt {21}}i}{2}}}}+\omega ^{k}{\sqrt[{3}]{{\frac {13{\sqrt {7}}}{2}}-{\frac {3{\sqrt {21}}i}{2}}}}$

$c_{x}=\omega ^{3-k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {1+3{\sqrt {3}}i}{2{\sqrt {7}}}}}+\omega ^{k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {1-3{\sqrt {3}}i}{2{\sqrt {7}}}}}=\omega ^{3-k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {3\omega +2}{\sqrt {7}}}}+\omega ^{k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {3\omega ^{2}+2}{\sqrt {7}}}}$

$s_{x}=\omega ^{3-k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {-13+3{\sqrt {3}}i}{14}}}+\omega ^{k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {-13-3{\sqrt {3}}i}{14}}}=\omega ^{3-k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {3\omega -5}{7}}}+\omega ^{k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {3\omega ^{2}-5}{7}}}$

$s_{x}=\omega ^{3-k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {13+3{\sqrt {3}}i}{14}}}+\omega ^{k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {13-3{\sqrt {3}}i}{14}}}=\omega ^{3-k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {3\omega +8}{7}}}+\omega ^{k}{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {3\omega ^{2}+8}{7}}}$

正十三角形

$c_{1}=6\cos {\frac {2\pi }{13}}-{\frac {-1+{\sqrt {13}}}{2}}$

$c_{3}=6\cos {\frac {6\pi }{13}}-{\frac {-1+{\sqrt {13}}}{2}}$

$c_{9}=6\cos {\frac {18\pi }{13}}-{\frac {-1+{\sqrt {13}}}{2}}$

$s_{1}=6\sin {\frac {2\pi }{13}}-{\sqrt {\frac {13-3{\sqrt {13}}}{2}}}$

$s_{3}=6\sin {\frac {6\pi }{13}}-{\sqrt {\frac {13-3{\sqrt {13}}}{2}}}$

$s_{9}=6\sin {\frac {18\pi }{13}}-{\sqrt {\frac {13-3{\sqrt {13}}}{2}}}$

$c_{1}+c_{3}+c_{9}=0$

$c_{1}c_{3}+c_{3}c_{9}+c_{9}c_{1}={\frac {-39+3{\sqrt {13}}}{2}}=p$

$c_{1}c_{3}c_{9}={\frac {52-10{\sqrt {13}}}{2}}=26-5{\sqrt {13}}=-q$

$\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}={\frac {-351}{4}}={\frac {-27\cdot 13}{4}}$

$c_{x}=\omega ^{3-k}{\sqrt[{3}]{\tfrac {26-5{\sqrt {13}}+3{\sqrt {3}}i\cdot {\sqrt {13}}}{2}}}+\omega ^{k}{\sqrt[{3}]{\tfrac {26-5{\sqrt {13}}-3{\sqrt {3}}i\cdot {\sqrt {13}}}{2}}}=\omega ^{3-k}{\sqrt {13}}\cdot {\sqrt[{3}]{\tfrac {2{\sqrt {13}}-5+3{\sqrt {3}}i}{26}}}+\omega ^{k}{\sqrt {13}}\cdot {\sqrt[{3}]{\tfrac {2{\sqrt {13}}-5-3{\sqrt {3}}i}{26}}}$

$s_{1}+s_{3}+s_{9}=0$

$s_{1}s_{3}+s_{3}s_{9}+s_{9}s_{1}={\frac {-39+9{\sqrt {13}}}{2}}=p$

$s_{1}s_{3}s_{9}=-5\cdot {\sqrt {\frac {130+36{\sqrt {13}}}{2}}}=-5\cdot {\sqrt {65+18{\sqrt {13}}}}=-q$

$\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}={\frac {-27\cdot (65+18{\sqrt {13}})}{4}}$

$s_{x}=\omega ^{3-k}{\sqrt[{3}]{\tfrac {-5\cdot {\sqrt {65+18{\sqrt {13}}}}+3{\sqrt {3}}i\cdot {\sqrt {65+18{\sqrt {13}}}}}{2}}}+\omega ^{k}{\sqrt[{3}]{\tfrac {-5\cdot {\sqrt {65+18{\sqrt {13}}}}-3{\sqrt {3}}i\cdot {\sqrt {65+18{\sqrt {13}}}}}{2}}}$

$s_{x}=\omega ^{3-k}\cdot {\sqrt {\frac {13+3{\sqrt {13}}}{2}}}\cdot {\sqrt[{3}]{\frac {-5+3{\sqrt {3}}i}{2{\sqrt {13}}}}}+\omega ^{k}\cdot {\sqrt {\frac {13+3{\sqrt {13}}}{2}}}\cdot {\sqrt[{3}]{\frac {-5-3{\sqrt {3}}i}{2{\sqrt {13}}}}}$

正十九角形

$u_{1}=6\sin {\frac {2\pi }{19}}+6\sin {\frac {14\pi }{19}}+6\sin {\frac {22\pi }{19}}-{\sqrt {19}}$

$u_{2}=6\sin {\frac {4\pi }{19}}+6\sin {\frac {28\pi }{19}}+6\sin {\frac {6\pi }{19}}+{\sqrt {19}}$

$u_{3}=6\sin {\frac {8\pi }{19}}+6\sin {\frac {18\pi }{19}}+6\sin {\frac {12\pi }{19}}-{\sqrt {19}}$

$u_{4}=6\sin {\frac {16\pi }{19}}+6\sin {\frac {36\pi }{19}}+6\sin {\frac {24\pi }{19}}+{\sqrt {19}}$

$u_{5}=6\sin {\frac {32\pi }{19}}+6\sin {\frac {34\pi }{19}}+6\sin {\frac {10\pi }{19}}-{\sqrt {19}}$

$u_{6}=6\sin {\frac {26\pi }{19}}+6\sin {\frac {30\pi }{19}}+6\sin {\frac {20\pi }{19}}+{\sqrt {19}}$

$u_{1}+u_{3}+u_{5}=0$

$u_{1}u_{3}+u_{3}u_{5}+u_{5}c_{1}=-57=p$

$u_{1}u_{3}u_{5}=11{\sqrt {19}}=-q$

$\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}={\frac {-25137}{4}}={\frac {-3^{3}\cdot 7^{2}\cdot 19}{4}}$

$u_{1}=\omega ^{2}{\sqrt[{3}]{\tfrac {11{\sqrt {19}}+3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}+\omega {\sqrt[{3}]{\tfrac {11{\sqrt {19}}-3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}=\omega ^{2}{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11+21{\sqrt {3}}i}{38}}}+\omega {\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11-21{\sqrt {3}}i}{38}}}$

$u_{3}={\sqrt[{3}]{\tfrac {11{\sqrt {19}}+3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}+{\sqrt[{3}]{\tfrac {11{\sqrt {19}}-3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}={\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11+21{\sqrt {3}}i}{38}}}+{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11-21{\sqrt {3}}i}{38}}}$

$u_{5}=\omega {\sqrt[{3}]{\tfrac {11{\sqrt {19}}+3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}+\omega ^{2}{\sqrt[{3}]{\tfrac {11{\sqrt {19}}-3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}=\omega {\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11+21{\sqrt {3}}i}{38}}}+\omega ^{2}{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11-21{\sqrt {3}}i}{38}}}$

$u_{2}={\sqrt[{3}]{\tfrac {-11{\sqrt {19}}+3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}+{\sqrt[{3}]{\tfrac {-11{\sqrt {19}}-3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}={\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {-11+21{\sqrt {3}}i}{38}}}+{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {-11-21{\sqrt {3}}i}{38}}}=-\omega {\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11+21{\sqrt {3}}i}{38}}}-\omega ^{2}{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11-21{\sqrt {3}}i}{38}}}$

$u_{4}=\omega ^{2}{\sqrt[{3}]{\tfrac {-11{\sqrt {19}}+3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}+\omega {\sqrt[{3}]{\tfrac {-11{\sqrt {19}}-3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}=\omega ^{2}{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {-11+21{\sqrt {3}}i}{38}}}+\omega {\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {-11-21{\sqrt {3}}i}{38}}}=-\omega ^{2}{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11+21{\sqrt {3}}i}{38}}}-\omega {\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11-21{\sqrt {3}}i}{38}}}$

$u_{6}=\omega {\sqrt[{3}]{\tfrac {-11{\sqrt {19}}+3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}+\omega ^{2}{\sqrt[{3}]{\tfrac {-11{\sqrt {19}}-3{\sqrt {3}}i\cdot 7{\sqrt {19}}}{2}}}=\omega {\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {-11+21{\sqrt {3}}i}{38}}}+\omega ^{2}{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {-11-21{\sqrt {3}}i}{38}}}=-{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11+21{\sqrt {3}}i}{38}}}-{\sqrt {19}}\cdot {\sqrt[{3}]{\tfrac {11-21{\sqrt {3}}i}{38}}}$

$2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+2\sin {\frac {22\pi }{19}}={\frac {u_{1}+{\sqrt {19}}}{3}}$

$\left(\omega \cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega ^{2}\cdot 2\sin {\frac {22\pi }{19}}\right)^{3}$

$\left(\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}\right)^{3}$

${\begin{aligned}(a+b+c)^{3}=&a^{3}+b^{3}+c^{3}+3a^{2}b+3ab^{2}+3b^{2}c+3bc^{2}+3c^{2}a+3ca^{2}+6abc\\(a+\omega b+\omega ^{2}c)^{3}=&a^{3}+b^{3}+c^{3}+6abc+3\omega (a^{2}b+b^{2}c+c^{2}a)+3\omega ^{2}(ab^{2}+bc^{2}+ca^{2})\\(a+\omega ^{2}b+\omega c)^{3}=&a^{3}+b^{3}+c^{3}+6abc+3\omega ^{2}(a^{2}b+b^{2}c+c^{2}a)+3\omega (ab^{2}+bc^{2}+ca^{2})\\\end{aligned}}$

$8\sin \alpha \sin \beta \sin \gamma =2\sin(\alpha -\beta +\gamma )-2\sin(\alpha -\beta -\gamma )-2\sin(\alpha +\beta +\gamma )+2\sin(\alpha +\beta -\gamma )$

$8\sin \alpha \sin \beta \sin \gamma =-2\left(\sin(\alpha +\beta +\gamma )+\sin(\alpha -\beta -\gamma )+\sin(\beta -\gamma -\alpha )+\sin(\gamma -\alpha -\beta )\right)$

$8\sin ^{2}\alpha \sin \beta =-2\left(\sin(2\alpha +\beta )-2\sin \beta -\sin(2\alpha -\beta )\right)$

$8\sin ^{3}\alpha =6\sin \alpha -2\sin 3\alpha$

$\left(\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}\right)^{3}={\frac {3u_{1}-7u_{2}+10{\sqrt {19}}}{3}}+\omega (2u_{1}+u_{2}-u_{3})+\omega ^{2}(2u_{1}+2{\sqrt {19}})$

$\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}={\sqrt[{3}]{{\tfrac {3u_{1}-7u_{2}+10{\sqrt {19}}}{3}}+\omega (2u_{1}+u_{2}-u_{3})+\omega ^{2}(2u_{1}+2{\sqrt {19}})}}$

$\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}={\sqrt[{3}]{{\tfrac {3u_{1}+7u_{5}+10{\sqrt {19}}}{3}}+\omega (3u_{1})+\omega ^{2}(2u_{1}+2{\sqrt {19}})}}$

$\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}={\sqrt[{3}]{{\tfrac {7u_{5}+10{\sqrt {19}}}{3}}+(1+3\omega +2\omega ^{2})u_{1}+\omega ^{2}(2{\sqrt {19}})}}$

$\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}={\sqrt[{3}]{{\tfrac {7u_{5}+10{\sqrt {19}}}{3}}+(\omega -1)u_{1}+\omega ^{2}(2{\sqrt {19}})}}$

$\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}={\frac {1}{3}}{\sqrt[{3}]{63u_{5}+90{\sqrt {19}}+27\omega u_{1}-27u_{1}+54\omega ^{2}{\sqrt {19}}}}$

$\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}={\frac {1}{3}}{\sqrt[{3}]{63(\omega A+\omega ^{2}B)+90{\sqrt {19}}+27\omega (\omega ^{2}A+\omega B)-27(\omega ^{2}A+\omega B)+54\omega ^{2}{\sqrt {19}}}}$

$\omega ^{2}\cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega \cdot 2\sin {\frac {22\pi }{19}}={\frac {1}{3}}{\sqrt[{3}]{(90\omega +54)A+(90\omega ^{2}-27\omega )B+(90+54\omega ^{2}){\sqrt {19}}}}$

同様に式変形して

$\omega \cdot 2\sin {\frac {2\pi }{19}}+2\sin {\frac {14\pi }{19}}+\omega ^{2}\cdot 2\sin {\frac {22\pi }{19}}={\frac {1}{3}}{\sqrt[{3}]{(90\omega -27\omega ^{2})A+(90\omega ^{2}+54)B+(90+54\omega ){\sqrt {19}}}}$

$6\sin {\frac {2\pi }{19}}={\tfrac {\omega ^{2}A+\omega B+{\sqrt {19}}}{3}}+{\tfrac {\omega {\sqrt[{3}]{(90\omega +54)A+(90\omega ^{2}-27\omega )B+(90+54\omega ^{2}){\sqrt {19}}}}}{3}}+{\tfrac {\omega ^{2}{\sqrt[{3}]{(90\omega -27\omega ^{2})A+(90\omega ^{2}+54)B+(90+54\omega ){\sqrt {19}}}}}{3}}$

$\sin {\frac {2\pi }{19}}={\tfrac {\omega ^{2}A+\omega B+{\sqrt {19}}}{18}}+{\tfrac {\omega {\sqrt[{3}]{(90\omega +54)A+(90\omega ^{2}-27\omega )B+(90+54\omega ^{2}){\sqrt {19}}}}}{18}}+{\tfrac {\omega ^{2}{\sqrt[{3}]{(90\omega -27\omega ^{2})A+(90\omega ^{2}+54)B+(90+54\omega ){\sqrt {19}}}}}{18}}$

正三十七角形

${\begin{aligned}u_{1}&=6\sin {\frac {2\pi }{37}}+6\sin {\frac {52\pi }{37}}+6\sin {\frac {20\pi }{37}}-{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\\u_{2}&=6\sin {\frac {4\pi }{37}}+6\sin {\frac {30\pi }{37}}+6\sin {\frac {40\pi }{37}}+{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\\u_{3}&=6\sin {\frac {8\pi }{37}}+6\sin {\frac {60\pi }{37}}+6\sin {\frac {6\pi }{37}}+{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\\u_{4}&=6\sin {\frac {16\pi }{37}}+6\sin {\frac {46\pi }{37}}+6\sin {\frac {12\pi }{37}}-{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\\u_{5}&=6\sin {\frac {32\pi }{37}}+6\sin {\frac {18\pi }{37}}+6\sin {\frac {24\pi }{37}}-{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\\u_{6}&=6\sin {\frac {64\pi }{37}}+6\sin {\frac {36\pi }{37}}+6\sin {\frac {48\pi }{37}}+{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\\u_{7}&=6\sin {\frac {54\pi }{37}}+6\sin {\frac {72\pi }{37}}+6\sin {\frac {22\pi }{37}}+{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\\u_{8}&=6\sin {\frac {34\pi }{37}}+6\sin {\frac {70\pi }{37}}+6\sin {\frac {44\pi }{37}}-{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\\u_{9}&=6\sin {\frac {68\pi }{37}}+6\sin {\frac {66\pi }{37}}+6\sin {\frac {14\pi }{37}}-{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\\u_{10}&=6\sin {\frac {62\pi }{37}}+6\sin {\frac {58\pi }{37}}+6\sin {\frac {28\pi }{37}}+{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\\u_{11}&=6\sin {\frac {50\pi }{37}}+6\sin {\frac {42\pi }{37}}+6\sin {\frac {56\pi }{37}}+{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\\u_{12}&=6\sin {\frac {26\pi }{37}}+6\sin {\frac {10\pi }{37}}+6\sin {\frac {38\pi }{37}}-{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\\\end{aligned}}$

${\begin{aligned}&u_{1}+u_{5}+u_{9}=0\\&u_{1}u_{5}+u_{5}u_{9}+u_{9}u_{1}={\frac {-111-3{\sqrt {37}}}{2}}={\frac {-3(37+{\sqrt {37}})}{2}}=p\\&u_{1}u_{5}u_{9}=11{\sqrt {185+14{\sqrt {37}}}}=-q\\&\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}={\frac {-27(185+14{\sqrt {37}})}{4}}\\\end{aligned}}$

${\begin{aligned}&u_{1}=\omega ^{2}{\sqrt[{3}]{\tfrac {11{\sqrt {185+14{\sqrt {37}}}}+3{\sqrt {3}}i\cdot {\sqrt {185+14{\sqrt {37}}}}}{2}}}+\omega {\sqrt[{3}]{\tfrac {11{\sqrt {185+14{\sqrt {37}}}}-3{\sqrt {3}}i\cdot {\sqrt {185+14{\sqrt {37}}}}}{2}}}=\omega ^{2}{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}+\omega {\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\\&u_{5}={\sqrt[{3}]{\tfrac {11{\sqrt {185+14{\sqrt {37}}}}+3{\sqrt {3}}i\cdot {\sqrt {185+14{\sqrt {37}}}}}{2}}}+{\sqrt[{3}]{\tfrac {11{\sqrt {185+14{\sqrt {37}}}}-3{\sqrt {3}}i\cdot {\sqrt {185+14{\sqrt {37}}}}}{2}}}={\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}+{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\\&u_{9}=\omega {\sqrt[{3}]{\tfrac {11{\sqrt {185+14{\sqrt {37}}}}+3{\sqrt {3}}i\cdot {\sqrt {185+14{\sqrt {37}}}}}{2}}}+\omega ^{2}{\sqrt[{3}]{\tfrac {11{\sqrt {185+14{\sqrt {37}}}}-3{\sqrt {3}}i\cdot {\sqrt {185+14{\sqrt {37}}}}}{2}}}=\omega {\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}+\omega ^{2}{\sqrt {\tfrac {37+{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\\\end{aligned}}$

${\begin{aligned}&u_{2}={\sqrt[{3}]{\tfrac {11{\sqrt {185-14{\sqrt {37}}}}+3{\sqrt {3}}i\cdot {\sqrt {185-14{\sqrt {37}}}}}{2}}}+{\sqrt[{3}]{\tfrac {11{\sqrt {185-14{\sqrt {37}}}}-3{\sqrt {3}}i\cdot {\sqrt {185-14{\sqrt {37}}}}}{2}}}={\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}+{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\\&u_{6}=\omega {\sqrt[{3}]{\tfrac {11{\sqrt {185-14{\sqrt {37}}}}+3{\sqrt {3}}i\cdot {\sqrt {185-14{\sqrt {37}}}}}{2}}}+\omega ^{2}{\sqrt[{3}]{\tfrac {11{\sqrt {185-14{\sqrt {37}}}}-3{\sqrt {3}}i\cdot {\sqrt {185-14{\sqrt {37}}}}}{2}}}=\omega {\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}+\omega ^{2}{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\\&u_{10}=\omega ^{2}{\sqrt[{3}]{\tfrac {11{\sqrt {185-14{\sqrt {37}}}}+3{\sqrt {3}}i\cdot {\sqrt {185-14{\sqrt {37}}}}}{2}}}+\omega {\sqrt[{3}]{\tfrac {11{\sqrt {185-14{\sqrt {37}}}}-3{\sqrt {3}}i\cdot {\sqrt {185-14{\sqrt {37}}}}}{2}}}=\omega ^{2}{\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}+\omega {\sqrt {\tfrac {37-{\sqrt {37}}}{2}}}\cdot {\sqrt[{3}]{\tfrac {11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\\\end{aligned}}$

${\begin{aligned}\left(\omega \cdot 2\sin {\frac {2\pi }{37}}+\omega ^{2}\cdot 2\sin {\frac {52\pi }{37}}+2\sin {\frac {20\pi }{37}}\right)^{3}&=2(u_{2}-\beta )+(u_{1}+\alpha )+{\frac {(u_{9}+\alpha )}{3}}+\omega (2(u_{1}+\alpha )+(u_{5}+\alpha )-(u_{6}-\beta ))+\omega ^{2}(3(u_{1}+\alpha )+(u_{10}-\beta ))\\\left(\omega \cdot 6\sin {\frac {2\pi }{37}}+\omega ^{2}\cdot 6\sin {\frac {52\pi }{37}}+6\sin {\frac {20\pi }{37}}\right)^{3}&=54(C+D-\beta )+27(\omega ^{2}A+\omega B+\alpha )+9(\omega A+\omega ^{2}B+\alpha )+\omega (54(\omega ^{2}A+\omega B+\alpha )+27(A+B+\alpha )-27(\omega C+\omega ^{2}D-\beta ))+\omega ^{2}(81(\omega ^{2}A+\omega B+\alpha )+27(\omega ^{2}C+\omega D-\beta ))\\&=(27\omega ^{2}+9\omega +54+27\omega +81\omega )A+(27\omega +9\omega ^{2}+54\omega ^{2}+27\omega +81)B+(54-27\omega ^{2}+27\omega )C+(54-27+27)D+(27+9+54\omega +27\omega +81\omega ^{2})\alpha +(-54+27\omega -27\omega ^{2})\beta \\&=(90\omega +27)A+(9\omega ^{2}+27)B+(54\omega +81)C+54D-45\alpha +(54\omega -27)\beta \\\end{aligned}}$

ピアポント素数

• 正七角形

${\sqrt[{3}]{\frac {1+3{\sqrt {3}}\cdot i}{2{\sqrt {7}}}}}={\sqrt[{3}]{\frac {3\omega +2}{\sqrt {7}}}}$

${\sqrt[{3}]{\frac {1-3{\sqrt {3}}\cdot i}{2{\sqrt {7}}}}}={\sqrt[{3}]{\frac {3\omega ^{2}+2}{\sqrt {7}}}}$

• 高次方程式 z^n-1=0 の代数的解 ///計算ミス？

${\begin{aligned}\sin {\frac {2\pi }{7}}=&{\frac {1}{6}}{\sqrt {21+{\sqrt[{3}]{28(1+3{\sqrt {3}}i)}}+{\sqrt[{3}]{28(1-3{\sqrt {3}}i)}}-{\sqrt[{3}]{{\frac {49}{4}}(-13+3{\sqrt {3}}i)}}-{\sqrt[{3}]{{\frac {49}{4}}(-13-3{\sqrt {3}}i)}}}}\\=&{\frac {1}{6}}{\sqrt {21+2{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {1+3{\sqrt {3}}i}{2{\sqrt {7}}}}}+2{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {1-3{\sqrt {3}}i}{2{\sqrt {7}}}}}-{\frac {7}{\sqrt[{3}]{2}}}\cdot {\sqrt[{3}]{\frac {-13+3{\sqrt {3}}i}{14}}}-{\frac {7}{\sqrt[{3}]{2}}}\cdot {\sqrt[{3}]{\frac {-13-3{\sqrt {3}}i}{14}}}}}\\=&{\frac {1}{6}}{\sqrt {21+2{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {3\omega +2}{\sqrt {7}}}}+2{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {3\omega ^{2}+2}{\sqrt {7}}}}-{\frac {7}{\sqrt[{3}]{2}}}\cdot {\sqrt[{3}]{\frac {3\omega -5}{7}}}-{\frac {7}{\sqrt[{3}]{2}}}\cdot {\sqrt[{3}]{\frac {3\omega ^{2}-5}{7}}}}}\end{aligned}}$

• 正十三角形

${\sqrt[{3}]{\frac {-5+3{\sqrt {3}}i}{2{\sqrt {13}}}}}={\sqrt[{3}]{\frac {3\omega -1}{\sqrt {13}}}}$

${\sqrt[{3}]{\frac {-5-3{\sqrt {3}}i}{2{\sqrt {13}}}}}={\sqrt[{3}]{\frac {3\omega ^{2}-1}{\sqrt {13}}}}$

• 正十九角形

${\sqrt[{3}]{\frac {7+3{\sqrt {3}}i}{2{\sqrt {19}}}}}={\sqrt[{3}]{\frac {3\omega +5}{\sqrt {19}}}}$

${\sqrt[{3}]{\frac {7-3{\sqrt {3}}i}{2{\sqrt {19}}}}}={\sqrt[{3}]{\frac {3\omega ^{2}+5}{\sqrt {19}}}}$

• 正三十七角形

${\sqrt[{3}]{\frac {-11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}={\sqrt[{3}]{\frac {3\omega -4}{\sqrt {37}}}}$

${\sqrt[{3}]{\frac {-11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}={\sqrt[{3}]{\frac {3\omega ^{2}-4}{\sqrt {37}}}}$

• 正七十三角形

${\sqrt[{3}]{\frac {7+9{\sqrt {3}}i}{2{\sqrt {73}}}}}={\sqrt[{3}]{\frac {9\omega +8}{\sqrt {73}}}}$

${\sqrt[{3}]{\frac {7-9{\sqrt {3}}i}{2{\sqrt {73}}}}}={\sqrt[{3}]{\frac {9\omega ^{2}+8}{\sqrt {73}}}}$

• 正九十七角形

${\sqrt[{3}]{\frac {19+3{\sqrt {3}}i}{2{\sqrt {97}}}}}={\sqrt[{3}]{\frac {3\omega +11}{\sqrt {97}}}}$

${\sqrt[{3}]{\frac {19-3{\sqrt {3}}i}{2{\sqrt {97}}}}}={\sqrt[{3}]{\frac {3\omega ^{2}+11}{\sqrt {97}}}}$

• 正百九角形

${\sqrt[{3}]{\frac {-2+12{\sqrt {3}}i}{2{\sqrt {109}}}}}={\sqrt[{3}]{\frac {12\omega +5}{\sqrt {109}}}}$

${\sqrt[{3}]{\frac {-2-12{\sqrt {3}}i}{2{\sqrt {109}}}}}={\sqrt[{3}]{\frac {12\omega ^{2}+5}{\sqrt {109}}}}$

• 正百六十三角形

${\sqrt[{3}]{\frac {25+3{\sqrt {3}}i}{2{\sqrt {163}}}}}={\sqrt[{3}]{\frac {3\omega +14}{\sqrt {163}}}}$

${\sqrt[{3}]{\frac {25-3{\sqrt {3}}i}{2{\sqrt {163}}}}}={\sqrt[{3}]{\frac {3\omega ^{2}+14}{\sqrt {163}}}}$

• 正百九十三角形

${\sqrt[{3}]{\frac {-23+9{\sqrt {3}}i}{2{\sqrt {193}}}}}={\sqrt[{3}]{\frac {9\omega -7}{\sqrt {193}}}}$

${\sqrt[{3}]{\frac {-23-9{\sqrt {3}}i}{2{\sqrt {193}}}}}={\sqrt[{3}]{\frac {9\omega ^{2}-7}{\sqrt {193}}}}$

• 正四百三十三角形

${\sqrt[{3}]{\frac {-2+24{\sqrt {3}}i}{2{\sqrt {433}}}}}={\sqrt[{3}]{\frac {24\omega +11}{\sqrt {433}}}}$

${\sqrt[{3}]{\frac {-2-24{\sqrt {3}}i}{2{\sqrt {433}}}}}={\sqrt[{3}]{\frac {24\omega ^{2}+11}{\sqrt {433}}}}$

• 正四百八十七角形

${\sqrt[{3}]{\frac {25+21{\sqrt {3}}i}{2{\sqrt {487}}}}}={\sqrt[{3}]{\frac {21\omega +23}{\sqrt {487}}}}$

${\sqrt[{3}]{\frac {25-21{\sqrt {3}}i}{2{\sqrt {487}}}}}={\sqrt[{3}]{\frac {21\omega ^{2}+23}{\sqrt {487}}}}$

• 正五百七十七角形

${\sqrt[{3}]{\frac {-11+27{\sqrt {3}}i}{2{\sqrt {577}}}}}={\sqrt[{3}]{\frac {27\omega +8}{\sqrt {577}}}}$

${\sqrt[{3}]{\frac {-11-27{\sqrt {3}}i}{2{\sqrt {577}}}}}={\sqrt[{3}]{\frac {27\omega ^{2}+8}{\sqrt {577}}}}$

• 正七百六十九角形

${\sqrt[{3}]{\frac {49+15{\sqrt {3}}i}{2{\sqrt {769}}}}}={\sqrt[{3}]{\frac {15\omega +32}{\sqrt {769}}}}$

${\sqrt[{3}]{\frac {49-15{\sqrt {3}}i}{2{\sqrt {769}}}}}={\sqrt[{3}]{\frac {15\omega ^{2}+32}{\sqrt {769}}}}$

• 正1153角形

${\sqrt[{3}]{\frac {7+39{\sqrt {3}}i}{2{\sqrt {1153}}}}}={\sqrt[{3}]{\frac {39\omega +23}{\sqrt {1153}}}}$

${\sqrt[{3}]{\frac {7-39{\sqrt {3}}i}{2{\sqrt {1153}}}}}={\sqrt[{3}]{\frac {39\omega ^{2}+23}{\sqrt {1153}}}}$

• 正1297角形

${\sqrt[{3}]{\frac {25+39{\sqrt {3}}i}{2{\sqrt {1297}}}}}={\sqrt[{3}]{\frac {39\omega +32}{\sqrt {1297}}}}$

${\sqrt[{3}]{\frac {25-39{\sqrt {3}}i}{2{\sqrt {1297}}}}}={\sqrt[{3}]{\frac {39\omega ^{2}+32}{\sqrt {1297}}}}$

• 正1459角形

${\sqrt[{3}]{\frac {-56+30{\sqrt {3}}i}{2{\sqrt {1459}}}}}={\sqrt[{3}]{\frac {30\omega -13}{\sqrt {1459}}}}$

${\sqrt[{3}]{\frac {-56-30{\sqrt {3}}i}{2{\sqrt {1459}}}}}={\sqrt[{3}]{\frac {30\omega ^{2}-13}{\sqrt {1459}}}}$

• 正2593角形

${\sqrt[{3}]{\frac {25+57{\sqrt {3}}i}{2{\sqrt {2593}}}}}={\sqrt[{3}]{\frac {57\omega +41}{\sqrt {2593}}}}$

${\sqrt[{3}]{\frac {25-57{\sqrt {3}}i}{2{\sqrt {2593}}}}}={\sqrt[{3}]{\frac {57\omega ^{2}+41}{\sqrt {2593}}}}$

• 正2917角形

${\sqrt[{3}]{\frac {106+12{\sqrt {3}}i}{2{\sqrt {2917}}}}}={\sqrt[{3}]{\frac {12\omega +59}{\sqrt {2917}}}}$

${\sqrt[{3}]{\frac {106-12{\sqrt {3}}i}{2{\sqrt {2917}}}}}={\sqrt[{3}]{\frac {12\omega ^{2}+59}{\sqrt {2917}}}}$

• 正3457角形

${\sqrt[{3}]{\frac {-110+24{\sqrt {3}}i}{2{\sqrt {3457}}}}}={\sqrt[{3}]{\frac {24\omega -43}{\sqrt {3457}}}}$

${\sqrt[{3}]{\frac {-110-24{\sqrt {3}}i}{2{\sqrt {3457}}}}}={\sqrt[{3}]{\frac {24\omega ^{2}-43}{\sqrt {3457}}}}$

• 正3889角形

${\sqrt[{3}]{\frac {-2+72{\sqrt {3}}i}{2{\sqrt {3889}}}}}={\sqrt[{3}]{\frac {72\omega +35}{\sqrt {3889}}}}$

${\sqrt[{3}]{\frac {-2-72{\sqrt {3}}i}{2{\sqrt {3889}}}}}={\sqrt[{3}]{\frac {72\omega ^{2}+35}{\sqrt {3889}}}}$

• 正10369角形

${\sqrt[{3}]{\frac {-137+87{\sqrt {3}}i}{2{\sqrt {10369}}}}}={\sqrt[{3}]{\frac {87\omega -25}{\sqrt {10369}}}}$

${\sqrt[{3}]{\frac {-137-87{\sqrt {3}}i}{2{\sqrt {10369}}}}}={\sqrt[{3}]{\frac {87\omega ^{2}-25}{\sqrt {10369}}}}$

• 正12289角形

${\sqrt[{3}]{\frac {193+63{\sqrt {3}}i}{2{\sqrt {12289}}}}}={\sqrt[{3}]{\frac {63\omega +128}{\sqrt {12289}}}}$

${\sqrt[{3}]{\frac {193-63{\sqrt {3}}i}{2{\sqrt {12289}}}}}={\sqrt[{3}]{\frac {63\omega ^{2}+128}{\sqrt {12289}}}}$

• 正17497角形

${\sqrt[{3}]{\frac {241+63{\sqrt {3}}i}{2{\sqrt {17497}}}}}={\sqrt[{3}]{\frac {63\omega +152}{\sqrt {17497}}}}$

${\sqrt[{3}]{\frac {241-63{\sqrt {3}}i}{2{\sqrt {17497}}}}}={\sqrt[{3}]{\frac {63\omega ^{2}+152}{\sqrt {17497}}}}$

• 正18433角形

${\sqrt[{3}]{\frac {-263+39{\sqrt {3}}i}{2{\sqrt {18433}}}}}={\sqrt[{3}]{\frac {39\omega -112}{\sqrt {18433}}}}$

${\sqrt[{3}]{\frac {-263-39{\sqrt {3}}i}{2{\sqrt {18433}}}}}={\sqrt[{3}]{\frac {39\omega ^{2}-112}{\sqrt {18433}}}}$

• 正39367角形

${\sqrt[{3}]{\frac {-380+66{\sqrt {3}}i}{2{\sqrt {39367}}}}}={\sqrt[{3}]{\frac {66\omega -157}{\sqrt {39367}}}}$

${\sqrt[{3}]{\frac {-380-66{\sqrt {3}}i}{2{\sqrt {39367}}}}}={\sqrt[{3}]{\frac {66\omega ^{2}-157}{\sqrt {39367}}}}$

• 正52489角形

${\sqrt[{3}]{\frac {241+225{\sqrt {3}}i}{2{\sqrt {52489}}}}}={\sqrt[{3}]{\frac {225\omega +233}{\sqrt {52489}}}}$

${\sqrt[{3}]{\frac {241-225{\sqrt {3}}i}{2{\sqrt {52489}}}}}={\sqrt[{3}]{\frac {225\omega ^{2}+233}{\sqrt {52489}}}}$

話者数と記事数

話者数と記事数

順位	言語名	話者数	WP	純記事数
1	中国語	約13億7000万人	zh	1,183,145
2	英語	5億3000万人	en	6,271,094
3	ヒンディー語	4億9000万人	hi	146,376
4	スペイン語	4億2000万人	es	1,666,952
5	アラビア語	2億3000万人	ar	1,106,896
6	ベンガル語	2億2000万人	bn	104,960
7	ポルトガル語	2億1500万人	pt	1,060,018
8	ロシア語	1億8000万人	ru	1,706,413
9	日本語	1億3400万人	ja	1,258,125
10	ドイツ語	1億3000万人	de	2,548,932
11	フランス語	1億2300万人	fr	2,308,622
12	パンジャーブ語（インド、パキスタン）	9000万人	pa、pnb	96,600
13	ジャワ語	7500万人	jv	62,652
14	朝鮮語	7500万人	ko	535,452
15	ベトナム語	7000万人	vi	1,262,347
16	テルグ語	7000万人	te	70,796
17	マラーティー語	6800万人	mr	70,323
18	タミル語	7400万人	ta	135,326
19	ペルシア語	4600万人	fa	773,101
20	ウルドゥー語	6100万人	ur	161,620
21	イタリア語	6100万人	it	1,679,734
22	トルコ語	6000万人	tr	394,837
23	グジャラート語	4600万人	gu	29,500
24	ポーランド語	5000万人	pl	1,462,628
25	ウクライナ語	4500万人	uk	1,079,067
26	マラヤーラム語	3600万人	ml	72,342
27	カンナダ語	3500万人	kn	26,811
28	アゼルバイジャン語（アゼルバイジャン、イラン）	3300万人	az、azb	419,513
29	オリヤー語	3200万人	or	15,639
30	ビルマ語	3200万人	my	100,200
31	タイ語	4600万人	th	139,634
32	スンダ語	2700万人	su	60,645
33	クルド語（クルマンジー、ソラニー）	2600万人	ku、ckb	67,439
34	パシュトー語	2700万人	ps	12,054
35	ハウサ語	2400万人	ha	8,164
36	ルーマニア語	2400万人	ro	417,552
37	インドネシア語	2300万人	id	563,639
38	ウズベク語	2000万人	uz	139,849
39	シンド語	2000万人	sd	14,062
40	セブアノ語	2000万人	ceb	5,596,787
41	ヨルバ語	1900万人	yo	33,334
42	ソマリ語	1300〜2000万人	so	5,960
43	ラーオ語	1900万人以上	lo	3,623
44	オロモ語	1800万人	om	1,054
45	マレー語	1800万人	ms	347,181
46	イボ語	1800万人	ig	2,088
47	オランダ語	1700万人	nl	2,048,262
48	アムハラ語	1700万人	am	14,911
49	マダガスカル語	1700万人	mg	93,763
50	タガログ語	1700万人	tl	58,326
51	ネパール語	1700万人	ne	31,779
52	アッサム語	1500万人	as	8,225
53	ハンガリー語	1500万人	hu	485,105
54	ショナ語	1500万人	sn	6,752
55	クメール語	1400万人	km	8,238
56	チワン語	1400万人	za	1,960
57	マドゥラ語	1400万人	mad	714
58	シンハラ語	1300万人	si	16,849
59	フラニ語	1300万人以上	ff	278
60	ベルベル語 （カビル語など）	1300万人以上	kab	6,079
61	チェコ語	1200万人	cs	476,275
62	ギリシア語	1200万人	el	189,551
63	セルビア語	1100万人	sr	643,792
64	ケチュア語	1040万人	qu	22,825

• ネイティブスピーカーの数が多い言語の一覧
• Wikipedia:全言語版の統計

キガリから

1. キガリ ～ アレクサンドリア 5,379km [2]
2. キガリ ～ タンジェ 8,042km [3]
3. キガリ ～ ダカール 7,394km [4]
4. キガリ ～ アビジャン 5,509km [5]
5. キガリ ～ ドゥアラ 3,682km [6]
6. キガリ ～ ルアンダ 3,010km [7]
7. キガリ ～ ナミベ 3,400km [8]

   キガリ ～ ルブンバシ ～ ナミベ 3,845km [9]

8. キガリ ～ ケープタウン 5,075km [10]
9. キガリ ～ ダーバン 4,277km [11]
10. キガリ ～ ナカラ 2,769km [12]
11. キガリ ～ モンバサ 1,473km [13]
12. キガリ ～ ジブチ市 3,282km [14]

アフリカ以外へ

• キガリ ～ ローマ 8,916km [15]
• キガリ ～ タリン 10,315km [16]
• キガリ ～ イスタンブール 7,152km [17]
• キガリ ～ モスクワ 9,100km [18]
• キガリ ～ テヘラン 7,196km [19]
• キガリ ～ ドバイ 7,696km [20]

テンプレート

pt:Predefinição:Línguas da África

アフリカの鉄道

アフリカの鉄道路線図

ケープ・カイロ鉄道

100万都市の一覧（アフリカ）

• 
  カサブランカ
• 
  フェズ
• 
  アルジェ
• 
  トリポリ
• 
  アレクサンドリア
• 
  ショブラ・エル・ケイマ
• 
  カイロ
• 
  ギーザ
• 
  オムドゥルマン
• 
  ハルツーム
• 
  アディスアベバ
• 
  モガディシュ
• 
  ナイロビ
• 
  カンパラ
• 
  キガリ
• 
  ダルエスサラーム
• 
  ルサカ
• 
  ハラレ
• 
  マプト
• 
  アンタナナリボ
• 
  プレトリア
• 
  ヨハネスブルグ
• 
  ソウェト
• 
  ダーバン
• 
  ポート・エリザベス
• 
  ケープタウン
• 
  ルアンダ
• 
  ルブンバシ
• 
  ムブジマイ
• 
  カナンガ
• 
  キンシャサ
• 
  ヤウンデ
• 
  ドゥアラ
• 
  マイドゥグリ [1]
• 
  カノ
• 
  アブジャ
• 
  カドゥナ
• 
  アバ
• 
  ポートハーコート
• 
  オニチャ
• 
  ベニンシティ
• 
  イロリン
• 
  イバダン
• 
  ラゴス
• 
  アクラ
• 
  クマシ
• 
  ワガドゥグー
• 
  アビジャン
• 
  バマコ
• 
  コナクリ
• 
  ダカール

統計

https://www.citypopulation.de/world/Agglomerations.html

2020年アフリカ100万都市

カイロ

ラゴス

ヨハネスブルグ

キンシャサ

ルアンダ

ダルエスサラーム

ハルツーム

ナイロビ

アレクサンドリア

アビジャン

アクラ

カノ

カサブランカ

ケープタウン

アディスアベバ

ダカール

イバダン

アブジャ

バマコ

カンパラ

ヤウンデ

ダーバン

ドゥアラ

クマシ

ルサカ

チュニス

ワガドゥグー

コナクリ

P

マプト

ブラザヴィル

ルブンバシ

ロメ

ムブジマイ

ハラレ

ラバト

モガディシュ

カドゥナ

コトヌー

フリータウン

B

モンロビア

ンジャメナ

ポート・エリザベス

アンタナナリボ

ニアメ

フェズ

ヌアクショット

ベンゲラ

トリポリ

カナンガ

モンバサ

マイドゥグリ

O

アガディール

バンギ

キガリ

ポワントノワール

キサンガニ

A

スース

マラケシュ

タンジェ

トゥーバ

イロリン

リロングウェ

ジョス

バンジュール

E

W

ザリア

アルジェ

オラン

アフリカの100万都市

エジプトのOpenWeatherMapにある都市

El Gouna

Makadi Bay

Port el Ghalib

Naama Bay

Ain Sukhna

Muḩāfaz̧at al Uqşur

Beheira

Masākin al Akhshāb wa ash Shāy

Al ‘Atabah

Zefta

Wasit

Toukh

Tanda

Ţāmiyah

Ţalkhā

Talā

Ţahţā

Sumusţā as Sulţānī

Muḩāfaz̧at Sūhāj

Sohag

Siwa Oasis

Sīdī Sālim

Shirbīn

Shibīn al Qanāṭir

Shibīn al Kawm

Sharm el-Sheikh

Muḩāfaz̧at Shamāl Sīnā

Sarābiyūm

Samannūd

Samālūţ

Sāḩil Ţahţā

Rosetta

Quwaysinā

Quţūr

Kousa

Muḩāfaz̧at Qinā

Qinā

Qaşr al Farāfirah

Qalyūb

Nuwaybi‘a

Nişf Atrīb

Naj‘ Tinjār

Naja' Ḥammādī

Naj‘ al Khuţbah

Minyat an Naşr

Munūf

Muḩāfaz̧at Maţrūḩ

Maţāy

Mashtūl as Sūq

Mersa Matruh

Marsa Alam

Manfalūţ

Mallawī

Madīnat Sittah Uktūbar

Madīnat as Sādāt

Kawm Umbū

Kawm Ḩamādah

Kafr Shukr

Kafr Şaqr

Kafr az Zayyāt

Kafr ash Shaykh

Kafr ash Shaykh

Kafr ad Dawwār

Juhaynah

Al Jubayl

エジプトのOpenWeatherMapにある都市

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