十九角形
両辺の立方根を取ると
2 cos  2 π 19 + ω ⋅ 2 cos  16 π 19 + ω 2 ⋅ 2 cos  14 π 19 = 3 α + 7 β + 12 − 6 ω ( β + 1 ) + 3 ω 2 ( α − 1 ) 3 2 cos  2 π 19 + ω 2 ⋅ 2 cos  16 π 19 + ω ⋅ 2 cos  14 π 19 = 3 α + 7 β + 12 − 6 ω 2 ( β + 1 ) + 3 ω ( α − 1 ) 3 {\displaystyle {\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}}}