六十五角形
2 cos  2 π 65 ⋅ 2 cos  32 π 65 + 2 cos  32 π 65 ⋅ 2 cos  8 π 65 + 2 cos  8 π 65 ⋅ 2 cos  2 π 65 = x 3 + 2 cos  2 π 13 + 2 cos  6 π 13 + 2 cos  8 π 13 = x 3 + − 1 + 13 2 2 cos  2 π 65 ⋅ 2 cos  32 π 65 ⋅ 2 cos  8 π 65 = x 8 + 2 cos  2 π 5 = x 8 + − 1 + 5 2 {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{65}}\cdot 2\cos {\frac {32\pi }{65}}+2\cos {\frac {32\pi }{65}}\cdot 2\cos {\frac {8\pi }{65}}+2\cos {\frac {8\pi }{65}}\cdot 2\cos {\frac {2\pi }{65}}\\&=x_{3}+2\cos {\frac {2\pi }{13}}+2\cos {\frac {6\pi }{13}}+2\cos {\frac {8\pi }{13}}\\&=x_{3}+{\frac {-1+{\sqrt {13}}}{2}}\\&2\cos {\frac {2\pi }{65}}\cdot 2\cos {\frac {32\pi }{65}}\cdot 2\cos {\frac {8\pi }{65}}=x_{8}+2\cos {\frac {2\pi }{5}}=x_{8}+{\frac {-1+{\sqrt {5}}}{2}}\\\end{aligned}}}