■原始根とガウス和(その49)

2cos(2π/7)-2cos(π/7)+2cos(4π/7)=-1

2sin(2π/7)-2sin(π/7)+2sin(4π/7)=√7

4cos(2π/13)-4cos(5π/13)+4cos(6π/13)=√13-1

と計算された。

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 なお,

  [参]栗原将人「ガウスの数論世界をゆく」数学書房によると

[1]sin(2π/7)+sin(4π/7)+sin(8π/7)=(√7)/2

[2]sin(2π/13)+sin(6π/13)+sin(18π/13)=(26−6√13)^1/2/4

[3]sin(2π/41)+sin(20π/41)+sin(32π/41)+sin(36π/41)+sin(49π/41)=1/8(A+B−C)^1/2

  A=164+12√41

  B=(14+2√41)(82−10√41)^1/2

  C=16{82+10√41)^1/2

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cos(2π/7)+cos(4π/7)+cos(8π/7)=-1/2

cos(2π/13)+cos(6π/13)+cos(18π/13)=(-1+√13)/4

cos(2π/17)+cos(8π/17)=(-1+√17)/8+{(34-2√17)^1/2}/8

cos(2π/17)=(-1+√17)/16+{(34-2√17)^1/2}/16+1/16・C^1/2

C=68+12root17+2(-1+root17)・(34-2root17)^1/2-16・(34+2root17)

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