三十九角形
両辺の立方根を取ると
2 cos  2 π 39 + 2 cos  32 π 39 + 2 cos  34 π 39 = 1 4 ( 1 − 13 − 6 ( 13 − 3 13 ) ) = x 1 2 cos  4 π 39 + 2 cos  14 π 39 + 2 cos  10 π 39 = 1 4 ( 1 + 13 + 6 ( 13 + 3 13 ) ) = x 2 2 cos  8 π 39 + 2 cos  28 π 39 + 2 cos  20 π 39 = 1 4 ( 1 − 13 + 6 ( 13 − 3 13 ) ) = x 3 2 cos  16 π 39 + 2 cos  22 π 39 + 2 cos  38 π 39 = 1 4 ( 1 + 13 − 6 ( 13 + 3 13 ) ) = x 4 {\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{39}}+2\cos {\frac {32\pi }{39}}+2\cos {\frac {34\pi }{39}}={\frac {1}{4}}\left(1-{\sqrt {13}}-{\sqrt {6\left(13-3{\sqrt {13}}\right)}}\right)=x_{1}\\2\cos {\frac {4\pi }{39}}+2\cos {\frac {14\pi }{39}}+2\cos {\frac {10\pi }{39}}={\frac {1}{4}}\left(1+{\sqrt {13}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)=x_{2}\\2\cos {\frac {8\pi }{39}}+2\cos {\frac {28\pi }{39}}+2\cos {\frac {20\pi }{39}}={\frac {1}{4}}\left(1-{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}\right)=x_{3}\\2\cos {\frac {16\pi }{39}}+2\cos {\frac {22\pi }{39}}+2\cos {\frac {38\pi }{39}}={\frac {1}{4}}\left(1+{\sqrt {13}}-{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)=x_{4}\\\end{aligned}}}