{sin(2π/13)sin(5π/13)sin(6π/13)}/{sin(π/13)sin(3π/13)sin(4π/13)}=?

[2013.07.13]{sin(2π/13)sin(5π/13)sin(6π/13)}/{sin(π/13)sin(3π/13)sin(4π/13)}=?

■問題

       {sin(2π/13)sin(5π/13)sin(6π/13)}/{sin(π/13)sin(3π/13)sin(4π/13)}を求めよ。

直ぐに答えを知りたい方は、こちらを参照。

■予想を立てる。
       a={sin(2π/13)sin(5π/13)sin(6π/13)}/{sin(π/13)sin(3π/13)sin(4π/13)}
とする。
aは正の実数であるので、aを300桁程度の精度で計算して、連分数で表示してみると、
       a≒[3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4]
となるので、
       a=[3^・]
と予想できる。

[3^・]は、2次方程式 x=3+(1/x)、つまり、x^2-3x-1=0の正根であるので、
       a=(3+sqrt(13))/2 ---------(*)
と予想できる。

[pari/gp-2.6.0による計算]

gp> \p 300
   realprecision = 308 significant digits (300 digits displayed)
gp> f(x)=sin(2*x)*sin(5*x)*sin(6*x)/(sin(x)*sin(3*x)*sin(4*x))
time = 22 ms.
%1 = (x)->sin(2*x)*sin(5*x)*sin(6*x)/(sin(x)*sin(3*x)*sin(4*x))
gp> a=f(Pi/13)
time = 33 ms.
%2 = 3.30277563773199464655961063373524797312564828692262310635522652811358347414650522260230954100924535883675709101203177018801533913234890385258150858344635487887134528213707631661691519748117347239813661499814400163442821360653605636658453610264875089277941922240732693446053769769624128165512234141832
gp> contfrac(a)
time = 9 ms.
%3 = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4]
gp> b=(3+sqrt(13))/2
time = 10 ms.
%4 = 3.30277563773199464655961063373524797312564828692262310635522652811358347414650522260230954100924535883675709101203177018801533913234890385258150858344635487887134528213707631661691519748117347239813661499814400163442821360653605636658453610264875089277941922240732693446053769769624128165512234141832
gp> a-b
time = 8 ms.
%5 = 2.225073859 E-308

■予想(*)を証明する。
1の原始26乗根
    ζ = ζ₂₆ = e^2πi/26 = e^πi/13
を使って、aをQ(ζ)上で計算する。

ζの最小多項式は、(ζ¹³+1)/(ζ+1)、つまり、
    ζ¹² - ζ¹¹ + ζ¹⁰ - ζ⁹ + ζ⁸ - ζ⁷ + ζ⁶ - ζ⁵ + ζ⁴ - ζ³ + ζ² - ζ + 1
である。

    a = {sin(2π/13)sin(8π/13)sin(6π/13)}/{sin(π/13)sin(3π/13)sin(4π/13)}
    = {sin(2π/13)/sin(π/13)}{sin(6π/13)/sin(3π/13)}{sin(8π/13)/sin(4π/13)}
sinの2倍角公式より、
    = {2cos(π/13)}{2cos(3π/13)}{2cos(4π/13)}
    = {ζ+ζ^-1}{ζ³+ζ^-3}{ζ⁴+ζ^-4}
    = - ζ¹¹ + ζ⁸ - ζ⁷ + ζ⁶ - ζ⁵ + ζ² + 2

ここで、σ:ζ→ζ⁵∈Gal(Q(ζ)/Q)をa∈Q(ζ)に作用させると、
    a^σ = {ζ⁵+ζ^-5}{ζ¹⁵+ζ^-15}{ζ²⁰+ζ^-20}
    = {ζ⁵+ζ^-5}{ζ²+ζ^-2}{ζ⁶+ζ^-6}
    = ζ¹¹ - ζ⁸ + ζ⁷ - ζ⁶ + ζ⁵ - ζ² + 1
となる。
よって、
    a+a^σ = (- ζ¹¹ + ζ⁸ - ζ⁷ + ζ⁶ - ζ⁵ + ζ² + 2)+(ζ¹¹ - ζ⁸ + ζ⁷ - ζ⁶ + ζ⁵ - ζ² + 1)
    = 3

    a・a^σ = (- ζ¹¹ + ζ⁸ - ζ⁷ + ζ⁶ - ζ⁵ + ζ² + 2)(ζ¹¹ - ζ⁸ + ζ⁷ - ζ⁶ + ζ⁵ - ζ² + 1)
    = -1
となるので、aとa^σは、２次方程式
    x²- 3x - 1 = 0
の2根であり、a > 0なので、
    a = (3+sqrt(13))/2
    a^σ = (3-sqrt(13))/2
である。

[pari/gp-2.6.0による計算]

gp> u=Mod(x,(x^13+1)/(x+1))
time = 15 ms.
%1 = Mod(x, x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp> g(x)=(x+(1/x))*(x^3+(1/x^3))*(x^4+(1/x^4))
time = 2 ms.
%2 = (x)->(x+(1/x))*(x^3+(1/x^3))*(x^4+(1/x^4))
gp> g(u)
time = 15 ms.
%3 = Mod(-x^11 + x^8 - x^7 + x^6 - x^5 + x^2 + 2, x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp> g(u^5)
time = 3 ms.
%4 = Mod(-x^11 + x^8 - x^7 + x^6 - x^5 + x^2 + 2, x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp> g(u^2)
time = 1 ms.
%5 = Mod(x^11 - x^8 + x^7 - x^6 + x^5 - x^2 + 1, x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp> g(u)+g(u^5)
time = 32 ms.
%6 = Mod(3, x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp> g(u)*g(u^5)
time = 6 ms.
%7 = Mod(-1, x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)

■aとa^σは、Q(ζ)の単数(unit)である。
[証明]
a・a^σ=-1より、直ちに分かる。

■参考文献[1]によると、aはQ(sqrt(13))の基本単数(fundamental unit)である。

[pari/gp-2.6.0による計算]

gp> F=bnfinit(x^2-13)
time = 315 ms.
%1 = [[;], matrix(0,7), [-1.194763217287109304111930829 + 3.141592653589793238462643383*I; 1.194763217287109304111930829 + 6.283185307179586476925286767*I], [0.2848090500183463082388395777, -0.4243834417614225495940935198, 0.6945940812433962163187951216, 0.E-38, -0.2848090500183463082388395777, 0.4243834417614225495940935198, -0.6945940812433962163187951216 + 3.141592653589793238462643383*I; -0.2848090500183463082388395777 + 3.141592653589793238462643383*I, 0.4243834417614225495940935198, -0.6945940812433962163187951216, 5.87747176 E-39 + 3.141592653589793238462643383*I, 0.2848090500183463082388395777 + 3.141592653589793238462643383*I, -0.4243834417614225495940935198, 0.6945940812433962163187951216 + 3.141592653589793238462643383*I], [[3, [0, 2]~, 1, 1, [-1, -3; -1, 0]], [17, [-7, 2]~, 1, 1, [-8, 6; 2, -10]], [23, [-5, 2]~, 1, 1, [7, 6; 2, 5]], [13, [1, 2]~, 2, 1, [1, 6; 2, -1]], [3, [2, 2]~, 1, 1, [0, -3; -1, 1]], [17, [9, 2]~, 1, 1, [-7, 6; 2, -9]], [23, [7, 2]~, 1, 1, [-5, 6; 2, -7]]], 0, [x^2 - 13, [2, 0], 13, 2, [[1, -2.302775637731994646559610634; 1, 1.302775637731994646559610634], [1, -2.302775637731994646559610634; 1, 1.302775637731994646559610634], [1, -2; 1, 1], [2, -1; -1, 7], [13, 7; 0, 1], [7, 1; 1, 2], [13, [7, 3; 1, 6]]], [-3.605551275463989293119221268, 3.605551275463989293119221268], [1, 1/2*x - 1/2], [1, 1; 0, 2], [1, 0, 0, 3; 0, 1, 1, -1]], [[1, [], []], 1.194763217287109304111930829, 1, [2, -1], [1/2*x + 3/2]], [[;], [], []], 0]
gp> F.fu
time = 10 ms.
%2 = [Mod(1/2*x + 3/2, x^2 - 13)]

[参考文献]

[1]B. Mazur, "Modular Curves and Arithmetic", Proc. of the Inetnational Congress of Math., Aug 16-24, 1983, Warszawa.

Last Update: 2013.07.27

H.Nakao