一．7次單位根

v1=exp(2*pi*j/7)=(-2+(28+84*sqrt(-3))^(1/3)+(28-84*sqrt(-3))^(1/3)+j*(2*sqrt(7)-w2*(52*sqrt(7)+12*sqrt(-21))^(1/3)-w1*(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

v2=exp(4*pi*j/7)=(-2+w2*(28+84*sqrt(-3))^(1/3)+w1*(28-84*sqrt(-3))^(1/3)+j*(2*sqrt(7)-w1*(52*sqrt(7)+12*sqrt(-21))^(1/3)-w2*(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

v3=exp(6*pi*j/7)=(-2+w1*(28+84*sqrt(-3))^(1/3)+w2*(28-84*sqrt(-3))^(1/3)-j*(2*sqrt(7)-(52*sqrt(7)+12*sqrt(-21))^(1/3)-(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

v4=exp(8*pi*j/7)=(-2+w1*(28+84*sqrt(-3))^(1/3)+w2*(28-84*sqrt(-3))^(1/3)+j*(2*sqrt(7)-(52*sqrt(7)+12*sqrt(-21))^(1/3)-(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

v5=exp(10*pi*j/7)=(-2+w2*(28+84*sqrt(-3))^(1/3)+w1*(28-84*sqrt(-3))^(1/3)-j*(2*sqrt(7)-w1*(52*sqrt(7)+12*sqrt(-21))^(1/3)-w2*(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

v6=exp(12*pi*j/7)=(-2+(28+84*sqrt(-3))^(1/3)+(28-84*sqrt(-3))^(1/3)-j*(2*sqrt(7)-w2*(52*sqrt(7)+12*sqrt(-21))^(1/3)-w1*(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

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二．cos(2*k*pi/29)的最簡(jiǎn)解析式求解

A0=2*(cos(2*pi/29)+cos(14*pi/29)+cos(40*pi/29)+cos(48*pi/29)+cos(46*pi/29)+cos(32*pi/29)+cos(50*pi/29))=(-1+sqrt(29))/2;

A1=2*(cos(2*pi/29)+v1*cos(14*pi/29)+v2*cos(40*pi/29)+v3*cos(48*pi/29)+v4*cos(46*pi/29)+v5*cos(32*pi/29)+v6*cos(50*pi/29));

A2=2*(cos(2*pi/29)+v2*cos(14*pi/29)+v4*cos(40*pi/29)+v6*cos(48*pi/29)+v1*cos(46*pi/29)+v3*cos(32*pi/29)+v5*cos(50*pi/29));

A3=2*(cos(2*pi/29)+v3*cos(14*pi/29)+v6*cos(40*pi/29)+v2*cos(48*pi/29)+v5*cos(46*pi/29)+v1*cos(32*pi/29)+v4*cos(50*pi/29));

A4=2*(cos(2*pi/29)+v4*cos(14*pi/29)+v1*cos(40*pi/29)+v5*cos(48*pi/29)+v2*cos(46*pi/29)+v6*cos(32*pi/29)+v3*cos(50*pi/29));

A5=2*(cos(2*pi/29)+v5*cos(14*pi/29)+v3*cos(40*pi/29)+v1*cos(48*pi/29)+v6*cos(46*pi/29)+v4*cos(32*pi/29)+v2*cos(50*pi/29));

A6=2*(cos(2*pi/29)+v6*cos(14*pi/29)+v5*cos(40*pi/29)+v4*cos(48*pi/29)+v3*cos(46*pi/29)+v2*cos(32*pi/29)+v1*cos(50*pi/29));

B0=2*(cos(34*pi/29)+cos(6*pi/29)+cos(42*pi/29)+cos(4*pi/29)+cos(28*pi/29)+cos(22*pi/29)+cos(38*pi/29))=(-1-sqrt(29))/2;

B1=2*(cos(34*pi/29)+v1*cos(6*pi/29)+v2*cos(42*pi/29)+v3*cos(4*pi/29)+v4*cos(28*pi/29)+v5*cos(22*pi/29)+v6*cos(38*pi/29));

B2=2*(cos(34*pi/29)+v2*cos(6*pi/29)+v4*cos(42*pi/29)+v6*cos(4*pi/29)+v1*cos(28*pi/29)+v3*cos(22*pi/29)+v5*cos(38*pi/29));

B3=2*(cos(34*pi/29)+v3*cos(6*pi/29)+v6*cos(42*pi/29)+v2*cos(4*pi/29)+v5*cos(28*pi/29)+v1*cos(22*pi/29)+v4*cos(38*pi/29));

B4=2*(cos(34*pi/29)+v4*cos(6*pi/29)+v1*cos(42*pi/29)+v5*cos(4*pi/29)+v2*cos(28*pi/29)+v6*cos(22*pi/29)+v3*cos(38*pi/29));

B5=2*(cos(34*pi/29)+v5*cos(6*pi/29)+v3*cos(42*pi/29)+v1*cos(4*pi/29)+v6*cos(28*pi/29)+v4*cos(22*pi/29)+v2*cos(38*pi/29));

B6=2*(cos(34*pi/29)+v6*cos(6*pi/29)+v5*cos(42*pi/29)+v4*cos(4*pi/29)+v3*cos(28*pi/29)+v2*cos(22*pi/29)+v1*cos(38*pi/29));

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X1=A1^7; X2=A2^7; X3=A3^7; X4=A4^7; X5=A5^7; X6=A6^7; X0=A0^7;

Y1=B1^7; Y2=B2^7; Y3=B3^7; Y4=B4^7; Y5=B5^7; Y6=B6^7; Y0=B0^7;

(X1+X2+X3+X4+X5+X6+X0)/7=-18869-2110*sqrt(29);

(Y1+Y2+Y3+Y4+Y5+Y6+Y0)/7=-18869+2110*sqrt(29);

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(X1/v1+X2/v2+X3/v3+X4/v4+X5/v5+X6/v6+X0)/49=1555-515*sqrt(29);

(Y1/v1+Y2/v2+Y3/v3+Y4/v4+Y5/v5+Y6/v6+Y0)/49=1555+515*sqrt(29);

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(X1/v2+X2/v4+X3/v6+X4/v1+X5/v3+X6/v5+X0)/49=453+823*sqrt(29);

(Y1/v2+Y2/v4+Y3/v6+Y4/v1+Y5/v3+Y6/v5+Y0)/49=453-823*sqrt(29);

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(X1/v3+X2/v6+X3/v2+X4/v5+X5/v1+X6/v4+X0)/49=-1577+120*sqrt(29);

(Y1/v3+Y2/v6+Y3/v2+Y4/v5+Y5/v1+Y6/v4+Y0)/49=-1577-120*sqrt(29);

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(X1/v4+X2/v1+X3/v5+X4/v2+X5/v6+X6/v3+X0)/49=(-5735-591*sqrt(29))/2;

(Y1/v4+Y2/v1+Y3/v5+Y4/v2+Y5/v6+Y6/v3+Y0)/49=(-5735+591*sqrt(29))/2;

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(X1/v5+X2/v3+X3/v1+X4/v6+X5/v4+X6/v2+X0)/49=(761+943*sqrt(29))/2;

(Y1/v5+Y2/v3+Y3/v1+Y4/v6+Y5/v4+Y6/v2+Y0)/49=(761-943*sqrt(29))/2;

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(X1/v6+X2/v5+X3/v4+X4/v3+X5/v2+X6/v1+X0)/49=(9055-509*sqrt(29))/2;

(Y1/v6+Y2/v5+Y3/v4+Y4/v3+Y5/v2+Y6/v1+Y0)/49=(9055+509*sqrt(29))/2;

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那么

X1=-(101123+657*sqrt(29))/2-7*((5945+521*sqrt(29))/2*v1+(8149-2155*sqrt(29))/2*v2+(12209-749*sqrt(29))/2*v3+(7395+41*sqrt(29))*v4+(4147-726*sqrt(29))*v5);

X2=-(101123+657*sqrt(29))/2-7*((5945+521*sqrt(29))/2*v2+(8149-2155*sqrt(29))/2*v4+(12209-749*sqrt(29))/2*v6+(7395+41*sqrt(29))*v1+(4147-726*sqrt(29))*v3);

X3=-(101123+657*sqrt(29))/2-7*((5945+521*sqrt(29))/2*v3+(8149-2155*sqrt(29))/2*v6+(12209-749*sqrt(29))/2*v2+(7395+41*sqrt(29))*v5+(4147-726*sqrt(29))*v1);

X4=-(101123+657*sqrt(29))/2-7*((5945+521*sqrt(29))/2*v4+(8149-2155*sqrt(29))/2*v1+(12209-749*sqrt(29))/2*v5+(7395+41*sqrt(29))*v2+(4147-726*sqrt(29))*v6);

X5=-(101123+657*sqrt(29))/2-7*((5945+521*sqrt(29))/2*v5+(8149-2155*sqrt(29))/2*v3+(12209-749*sqrt(29))/2*v1+(7395+41*sqrt(29))*v6+(4147-726*sqrt(29))*v4);

X6=-(101123+657*sqrt(29))/2-7*((5945+521*sqrt(29))/2*v6+(8149-2155*sqrt(29))/2*v5+(12209-749*sqrt(29))/2*v4+(7395+41*sqrt(29))*v3+(4147-726*sqrt(29))*v2);

P1=2*X1^(1/7); P2=2*X2^(1/7); P3=2*X3^(1/7);

P4=2*X4^(1/7); P5=2*X5^(1/7); P6=2*X6^(1/7);

A1=X1^(1/7); A2=X2^(1/7); A3=X3^(1/7)*v4;

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Y1=-(101123-657*sqrt(29))/2-7*((5945-521*sqrt(29))/2*v1+(8149+2155*sqrt(29))/2*v2+(12209+749*sqrt(29))/2*v3+(7395-41*sqrt(29))*v4+(4147+726*sqrt(29))*v5);

Y2=-(101123-657*sqrt(29))/2-7*((5945-521*sqrt(29))/2*v2+(8149+2155*sqrt(29))/2*v4+(12209+749*sqrt(29))/2*v6+(7395-41*sqrt(29))*v1+(4147+726*sqrt(29))*v3);

Y3=-(101123-657*sqrt(29))/2-7*((5945-521*sqrt(29))/2*v3+(8149+2155*sqrt(29))/2*v6+(12209+749*sqrt(29))/2*v2+(7395-41*sqrt(29))*v5+(4147+726*sqrt(29))*v1);

Y4=-(101123-657*sqrt(29))/2-7*((5945-521*sqrt(29))/2*v4+(8149+2155*sqrt(29))/2*v1+(12209+749*sqrt(29))/2*v5+(7395-41*sqrt(29))*v2+(4147+726*sqrt(29))*v6);

Y5=-(101123-657*sqrt(29))/2-7*((5945-521*sqrt(29))/2*v5+(8149+2155*sqrt(29))/2*v3+(12209+749*sqrt(29))/2*v1+(7395-41*sqrt(29))*v6+(4147+726*sqrt(29))*v4);

Y6=-(101123-657*sqrt(29))/2-7*((5945-521*sqrt(29))/2*v6+(8149+2155*sqrt(29))/2*v5+(12209+749*sqrt(29))/2*v4+(7395-41*sqrt(29))*v3+(4147+726*sqrt(29))*v2);

Q1=2*Y1^(1/7); Q2=2*Y2^(1/7); Q3=2*Y3^(1/7);

Q4=2*Y4^(1/7); Q5=2*Y5^(1/7); Q6=2*Y6^(1/7);

B1=Y1^(1/7)*v2; B2=Y2^(1/7)*v5; B3=Y3^(1/7)*v3;

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最終結(jié)果：

cos(2*pi/29)=(-1+sqrt(29)+P1+P6+P2+P5+v4*P3+v3*P4)/28;

cos(14*pi/29)=(-1+sqrt(29)+v6*P1+v1*P6+v5*P2+v2*P5+v1*P3+v6*P4)/28;

cos(40*pi/29)=(-1+sqrt(29)+v5*P1+v2*P6+v3*P2+v4*P5+v5*P3+v2*P4)/28;

cos(48*pi/29)=(-1+sqrt(29)+v4*P1+v3*P6+v1*P2+v6*P5+v2*P3+v5*P4)/28;

cos(46*pi/29)=(-1+sqrt(29)+v3*P1+v4*P6+v6*P2+v1*P5+v6*P3+v1*P4)/28;

cos(32*pi/29)=(-1+sqrt(29)+v2*P1+v5*P6+v4*P2+v3*P5+v3*P3+v4*P4)/28;

cos(50*pi/29)=(-1+sqrt(29)+v1*P1+v6*P6+v2*P2+v5*P5+P3+P4)/28;

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cos(34*pi/29)=(-1-sqrt(29)+v2*Q1+v5*Q6+v5*Q2+v2*Q5+v3*Q3+v4*Q4)/28;

cos(6*pi/29)=(-1-sqrt(29)+v1*Q1+v6*Q6+v3*Q2+v4*Q5+Q3+Q4)/28;

cos(42*pi/29)=(-1-sqrt(29)+Q1+Q6+v1*Q2+v6*Q5+v4*Q3+v3*Q4)/28;

cos(4*pi/29)=(-1-sqrt(29)+v6*Q1+v1*Q6+v6*Q2+v1*Q5+v1*Q3+v6*Q4)/28;

cos(28*pi/29)=(-1-sqrt(29)+v5*Q1+v2*Q6+v4*Q2+v3*Q5+v5*Q3+v2*Q4)/28;

cos(22*pi/29)=(-1-sqrt(29)+v4*Q1+v3*Q6+v2*Q2+v5*Q5+v2*Q3+v5*Q4)/28;

cos(38*pi/29)=(-1-sqrt(29)+v3*Q1+v4*Q6+Q2+Q5+v6*Q3+v1*Q4)/28;

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三．sin(2*k*pi/29)的最簡(jiǎn)解析式求解

C0=2*(sin(2*pi/29)+sin(14*pi/29)+sin(40*pi/29)+sin(48*pi/29)+sin(46*pi/29)+sin(32*pi/29)+sin(50*pi/29))=-sqrt(58+10*sqrt(29))/2;

C1=2*(sin(2*pi/29)+v1*sin(14*pi/29)+v2*sin(40*pi/29)+v3*sin(48*pi/29)+v4*sin(46*pi/29)+v5*sin(32*pi/29)+v6*sin(50*pi/29));

C2=2*(sin(2*pi/29)+v2*sin(14*pi/29)+v4*sin(40*pi/29)+v6*sin(48*pi/29)+v1*sin(46*pi/29)+v3*sin(32*pi/29)+v5*sin(50*pi/29));

C3=2*(sin(2*pi/29)+v3*sin(14*pi/29)+v6*sin(40*pi/29)+v2*sin(48*pi/29)+v5*sin(46*pi/29)+v1*sin(32*pi/29)+v4*sin(50*pi/29));

C4=2*(sin(2*pi/29)+v4*sin(14*pi/29)+v1*sin(40*pi/29)+v5*sin(48*pi/29)+v2*sin(46*pi/29)+v6*sin(32*pi/29)+v3*sin(50*pi/29));

C5=2*(sin(2*pi/29)+v5*sin(14*pi/29)+v3*sin(40*pi/29)+v1*sin(48*pi/29)+v6*sin(46*pi/29)+v4*sin(32*pi/29)+v2*sin(50*pi/29));

C6=2*(sin(2*pi/29)+v6*sin(14*pi/29)+v5*sin(40*pi/29)+v4*sin(48*pi/29)+v3*sin(46*pi/29)+v2*sin(32*pi/29)+v1*sin(50*pi/29));

D0=2*(sin(34*pi/29)+sin(6*pi/29)+sin(42*pi/29)+sin(4*pi/29)+sin(28*pi/29)+sin(22*pi/29)+sin(38*pi/29))=-sqrt(58-10*sqrt(29))/2;

D1=2*(sin(34*pi/29)+v1*sin(6*pi/29)+v2*sin(42*pi/29)+v3*sin(4*pi/29)+v4*sin(28*pi/29)+v5*sin(22*pi/29)+v6*sin(38*pi/29));

D2=2*(sin(34*pi/29)+v2*sin(6*pi/29)+v4*sin(42*pi/29)+v6*sin(4*pi/29)+v1*sin(28*pi/29)+v3*sin(22*pi/29)+v5*sin(38*pi/29));

D3=2*(sin(34*pi/29)+v3*sin(6*pi/29)+v6*sin(42*pi/29)+v2*sin(4*pi/29)+v5*sin(28*pi/29)+v1*sin(22*pi/29)+v4*sin(38*pi/29));

D4=2*(sin(34*pi/29)+v4*sin(6*pi/29)+v1*sin(42*pi/29)+v5*sin(4*pi/29)+v2*sin(28*pi/29)+v6*sin(22*pi/29)+v3*sin(38*pi/29));

D5=2*(sin(34*pi/29)+v5*sin(6*pi/29)+v3*sin(42*pi/29)+v1*sin(4*pi/29)+v6*sin(28*pi/29)+v4*sin(22*pi/29)+v2*sin(38*pi/29));

D6=2*(sin(34*pi/29)+v6*sin(6*pi/29)+v5*sin(42*pi/29)+v4*sin(4*pi/29)+v3*sin(28*pi/29)+v2*sin(22*pi/29)+v1*sin(38*pi/29));

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M1=(C1^7)/sqrt(377+70*sqrt(29)); M2=(C2^7)/sqrt(377+70*sqrt(29));

M3=(C3^7)/sqrt(377+70*sqrt(29)); M4=(C4^7)/sqrt(377+70*sqrt(29));

M5=(C5^7)/sqrt(377+70*sqrt(29)); M6=(C6^7)/sqrt(377+70*sqrt(29));

M0=(C0^7)/sqrt(377+70*sqrt(29));

N1=(D1^7)/sqrt(377-70*sqrt(29)); N2=(D2^7)/sqrt(377-70*sqrt(29));

N3=(D3^7)/sqrt(377-70*sqrt(29)); N4=(D4^7)/sqrt(377-70*sqrt(29));

N5=(D5^7)/sqrt(377-70*sqrt(29)); N6=(D6^7)/sqrt(377-70*sqrt(29));

N0=(D0^7)/sqrt(377-70*sqrt(29));

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(M1+M2+M3+M4+M5+M6+M0)/7=-28923+5156*sqrt(29);

(N1+N2+N3+N4+N5+N6+N0)/7=28923+5156*sqrt(29);

(M1/v1+M2/v2+M3/v3+M4/v4+M5/v5+M6/v6+M0)/49=4253-807*sqrt(29);

(N1/v1+N2/v2+N3/v3+N4/v4+N5/v5+N6/v6+N0)/49=-4253-807*sqrt(29);

(M1/v2+M2/v4+M3/v6+M4/v1+M5/v3+M6/v5+M0)/49=-2535+473*sqrt(29);

(N1/v2+N2/v4+N3/v6+N4/v1+N5/v3+N6/v5+N0)/49=2535+473*sqrt(29);

(M1/v3+M2/v6+M3/v2+M4/v5+M5/v1+M6/v4+M0)/49=-867+164*sqrt(29);

(N1/v3+N2/v6+N3/v2+N4/v5+N5/v1+N6/v4+N0)/49=867+164*sqrt(29);

(M1/v4+M2/v1+M3/v5+M4/v2+M5/v6+M6/v3+M0)/49=(7147-1353*sqrt(29))/2;

(N1/v4+N2/v1+N3/v5+N4/v2+N5/v6+N6/v3+N0)/49=(-7147-1353*sqrt(29))/2;

(M1/v5+M2/v3+M3/v1+M4/v6+M5/v4+M6/v2+M0)/49=(-8319+1495*sqrt(29))/2;

(N1/v5+N2/v3+N3/v1+N4/v6+N5/v4+N6/v2+N0)/49=(8319+1495*sqrt(29))/2;

(M1/v6+M2/v5+M3/v4+M4/v3+M5/v2+M6/v1+M0)/49=(7133-1387*sqrt(29))/2;

(N1/v6+N2/v5+N3/v4+N4/v3+N5/v2+N6/v1+N0)/49=(-7133-1387*sqrt(29))/2;

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那么設(shè)R=-M*sqrt(377+70*sqrt(29))，S=N*sqrt(377-70*sqrt(29))

R1=sqrt(377+70*sqrt(29))*((107777-20021*sqrt(29))/2+7*(-(1373-227*sqrt(29))/2*v1+(12203-2333*sqrt(29))/2*v2+(8867-1715*sqrt(29))/2*v3-(7+17*sqrt(29))*v4+(7726-1441*sqrt(29))*v5));

R2=sqrt(377+70*sqrt(29))*((107777-20021*sqrt(29))/2+7*(-(1373-227*sqrt(29))/2*v2+(12203-2333*sqrt(29))/2*v4+(8867-1715*sqrt(29))/2*v6-(7+17*sqrt(29))*v1+(7726-1441*sqrt(29))*v3));

R3=sqrt(377+70*sqrt(29))*((107777-20021*sqrt(29))/2+7*(-(1373-227*sqrt(29))/2*v3+(12203-2333*sqrt(29))/2*v6+(8867-1715*sqrt(29))/2*v2-(7+17*sqrt(29))*v5+(7726-1441*sqrt(29))*v1));

R4=sqrt(377+70*sqrt(29))*((107777-20021*sqrt(29))/2+7*(-(1373-227*sqrt(29))/2*v4+(12203-2333*sqrt(29))/2*v1+(8867-1715*sqrt(29))/2*v5-(7+17*sqrt(29))*v2+(7726-1441*sqrt(29))*v6));

R5=sqrt(377+70*sqrt(29))*((107777-20021*sqrt(29))/2+7*(-(1373-227*sqrt(29))/2*v5+(12203-2333*sqrt(29))/2*v3+(8867-1715*sqrt(29))/2*v1-(7+17*sqrt(29))*v6+(7726-1441*sqrt(29))*v4));

R6=sqrt(377+70*sqrt(29))*((107777-20021*sqrt(29))/2+7*(-(1373-227*sqrt(29))/2*v6+(12203-2333*sqrt(29))/2*v5+(8867-1715*sqrt(29))/2*v4-(7+17*sqrt(29))*v3+(7726-1441*sqrt(29))*v2));

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S1=sqrt(377-70*sqrt(29))*((107777+20021*sqrt(29))/2+7*(-(1373+227*sqrt(29))/2*v1+(12203+2333*sqrt(29))/2*v2+(8867+1715*sqrt(29))/2*v3-(7-17*sqrt(29))*v4+(7726+1441*sqrt(29))*v5));

S2=sqrt(377-70*sqrt(29))*((107777+20021*sqrt(29))/2+7*(-(1373+227*sqrt(29))/2*v2+(12203+2333*sqrt(29))/2*v4+(8867+1715*sqrt(29))/2*v6-(7-17*sqrt(29))*v1+(7726+1441*sqrt(29))*v3));

S3=sqrt(377-70*sqrt(29))*((107777+20021*sqrt(29))/2+7*(-(1373+227*sqrt(29))/2*v3+(12203+2333*sqrt(29))/2*v6+(8867+1715*sqrt(29))/2*v2-(7-17*sqrt(29))*v5+(7726+1441*sqrt(29))*v1));

S4=sqrt(377-70*sqrt(29))*((107777+20021*sqrt(29))/2+7*(-(1373+227*sqrt(29))/2*v4+(12203+2333*sqrt(29))/2*v1+(8867+1715*sqrt(29))/2*v5-(7-17*sqrt(29))*v2+(7726+1441*sqrt(29))*v6));

S5=sqrt(377-70*sqrt(29))*((107777+20021*sqrt(29))/2+7*(-(1373+227*sqrt(29))/2*v5+(12203+2333*sqrt(29))/2*v3+(8867+1715*sqrt(29))/2*v1-(7-17*sqrt(29))*v6+(7726+1441*sqrt(29))*v4));

S6=sqrt(377-70*sqrt(29))*((107777+20021*sqrt(29))/2+7*(-(1373+227*sqrt(29))/2*v6+(12203+2333*sqrt(29))/2*v5+(8867+1715*sqrt(29))/2*v4-(7-17*sqrt(29))*v3+(7726+1441*sqrt(29))*v2));

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設(shè)K1=2*R1^(1/7); K2=2*R2^(1/7); K3=2*R3^(1/7);

K4=2*R4^(1/7); K5=2*R5^(1/7); K6=2*R6^(1/7);

L1=2*S1^(1/7); L2=2*S2^(1/7); L3=2*S3^(1/7);

L4=2*S4^(1/7); L5=2*S5^(1/7); L6=2*S6^(1/7);

則-v4*K1=2*C1;-v5*K2=2*C2; -v6*K3=2*C3; v4*L1=2*D1; v2*L2=2*D2; v2*L3=2*D3;

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最終結(jié)果：

sin(2*pi/29)=-(sqrt(58+10*sqrt(29))+v4*K1+v3*K6+v5*K2+v2*K5+v6*K3+v1*K4)/28;

sin(14*pi/29)=-(sqrt(58+10*sqrt(29))+v3*K1+v4*K6+v3*K2+v4*K5+v3*K3+v4*K4)/28;

sin(40*pi/29)=-(sqrt(58+10*sqrt(29))+v2*K1+v5*K6+v1*K2+v6*K5+K3+K4)/28;

sin(48*pi/29)=-(sqrt(58+10*sqrt(29))+v1*K1+v6*K6+v6*K2+v1*K5+v4*K3+v3*K4)/28;

sin(46*pi/29)=-(sqrt(58+10*sqrt(29))+K1+K6+v4*K2+v3*K5+v1*K3+v6*K4)/28;

sin(32*pi/29)=-(sqrt(58+10*sqrt(29))+v6*K1+v1*K6+v2*K2+v5*K5+v5*K3+v2*K4)/28;

sin(50*pi/29)=-(sqrt(58+10*sqrt(29))+v5*K1+v4*K6+K2+K5+v2*K3+v5*K4)/28;

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sin(34*pi/29)=(-sqrt(58-10*sqrt(29))+v4*L1+v3*L6+v2*L2+v5*L5+v2*L3+v5*L4)/28;

sin(6*pi/29)=(-sqrt(58-10*sqrt(29))+v3*L1+v4*L6+L2+L5+v6*L3+v1*L4)/28;

sin(42*pi/29)=(-sqrt(58-10*sqrt(29))+v2*L1+v5*L6+v5*L2+v2*L5+v3*L3+v4*L4)/28;

sin(4*pi/29)=(-sqrt(58-10*sqrt(29))+v1*L1+v6*L6+v3*L2+v4*L5+L3+L4)/28;

sin(28*pi/29)=(-sqrt(58-10*sqrt(29))+L1+L6+v1*L2+v6*L5+v4*L3+v3*L4)/28;

sin(22*pi/29)=(-sqrt(58-10*sqrt(29))+v6*L1+v1*L6+v6*L2+v1*L5+v1*L3+v6*L4)/28;

sin(38*pi/29)=(-sqrt(58-10*sqrt(29))+v5*L1+v2*L6+v4*L2+v3*L5+v5*L3+v2*L4)/28;

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四．29次單位根

例如：

exp(2*pi*j/29)=(-1+sqrt(29)+P1+P6+P2+P5+v4*P3+v3*P4-j*(sqrt(58+10*sqrt(29))+v4*K1+v3*K6+v5*K2+v2*K5+v6*K3+v1*K4))/28;

exp(34*pi*j/29)=(-1-sqrt(29)+v2*Q1+v5*Q6+v5*Q2+v2*Q5+v3*Q3+v4*Q4+j*(-sqrt(58-10*sqrt(29))+v4*L1+v3*L6+v2*L2+v5*L5+v2*L3+v5*L4))/28.