(Murty, 1982 June)证明或否证
[ an frac{3pi}{11}+4sin frac{2pi}{11}=sqrt{11}.]
令$x=e^{2pi i/11}$,则
[2left( 2isin frac{2pi}{11}
ight) =2left( x-x^{10}
ight),]
而
egin{align*}
i an frac{3pi}{11}&=frac{x^3-1}{x^3+1}=frac{x^3-x^{33}}{1+x^3}
\
&=x^3-x^6+x^9-x+x^4-x^7+x^{10}-x^2+x^5-x^8.
end{align*}
相加可知
[i an frac{3pi}{11}+i ext{4}sin frac{2pi}{11}=S-S',]
其中$S=x+x^3+x^4+x^5+x^9$且
[S'=x^{10}+x^8+x^7+x^6+x^2=x^{-1}+x^{-3}+x^{-4}+x^{-5}+x^{-9}.]
由于[1+S+S'=frac{x^{11}-1}{x-1}=0,qquad S+S'=-1,]
相乘有$SS'=5+2(S+S')=3$.因此$S$和$S'$是$u^2+u+3=0$的根$frac{-1pm isqrt{11}}{2}$,则$S-S'=pm isqrt{11}$.因为$ an frac{3pi}{11}$和$sin frac{2pi}{11}$均为正,我们可知
[ an frac{3pi}{11}+4sin frac{2pi}{11}=sqrt{11}.]