注：這一次的運(yùn)算比31邊形的情形翻了近一倍。

一．給出三次單位根和七次單位根

1. 三次

w1=(-1+sqrt(-3))/2; w2=(-1-sqrt(-3))/2;

2. 七次

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．求解cos(2*k*pi/43)

思路：先3次后7次，以-2和6為底數(shù)。

1.?14*cos(2*k*pi/43)的非7次根式實(shí)數(shù)項(xiàng)

A10=2*(cos(2*pi/43)+cos(82*pi/43)+cos(8*pi/43)+cos(70*pi/43)+cos(32*pi/43)+cos(22*pi/43)+cos(42*pi/43));

A20=2*(cos(12*pi/43)+cos(62*pi/43)+cos(48*pi/43)+cos(76*pi/43)+cos(20*pi/43)+cos(46*pi/43)+cos(80*pi/43));

A30=2*(cos(72*pi/43)+cos(28*pi/43)+cos(30*pi/43)+cos(26*pi/43)+cos(34*pi/43)+cos(18*pi/43)+cos(50*pi/43));

A11=2*(cos(2*pi/43)+v1*cos(82*pi/43)+v2*cos(8*pi/43)+v3*cos(70*pi/43)+v4*cos(32*pi/43)+v5*cos(22*pi/43)+v6*cos(42*pi/43));

A21=2*(cos(12*pi/43)+v1*cos(62*pi/43)+v2*cos(48*pi/43)+v3*cos(76*pi/43)+v4*cos(20*pi/43)+v5*cos(46*pi/43)+v6*cos(80*pi/43));

A31=2*(cos(72*pi/43)+v1*cos(28*pi/43)+v2*cos(30*pi/43)+v3*cos(26*pi/43)+v4*cos(34*pi/43)+v5*cos(18*pi/43)+v6*cos(50*pi/43));

A12=2*(cos(2*pi/43)+v2*cos(82*pi/43)+v4*cos(8*pi/43)+v6*cos(70*pi/43)+v1*cos(32*pi/43)+v3*cos(22*pi/43)+v5*cos(42*pi/43));

A22=2*(cos(12*pi/43)+v2*cos(62*pi/43)+v4*cos(48*pi/43)+v6*cos(76*pi/43)+v1*cos(20*pi/43)+v3*cos(46*pi/43)+v5*cos(80*pi/43));

A32=2*(cos(72*pi/43)+v2*cos(28*pi/43)+v4*cos(30*pi/43)+v6*cos(26*pi/43)+v1*cos(34*pi/43)+v3*cos(18*pi/43)+v5*cos(50*pi/43));

A13=2*(cos(2*pi/43)+v3*cos(82*pi/43)+v6*cos(8*pi/43)+v2*cos(70*pi/43)+v5*cos(32*pi/43)+v1*cos(22*pi/43)+v4*cos(42*pi/43));

A23=2*(cos(12*pi/43)+v3*cos(58*pi/43)+v6*cos(48*pi/43)+v2*cos(76*pi/43)+v5*cos(20*pi/43)+v1*cos(46*pi/43)+v4*cos(80*pi/43));

A33=2*(cos(72*pi/43)+v3*cos(28*pi/43)+v6*cos(30*pi/43)+v2*cos(26*pi/43)+v5*cos(34*pi/43)+v1*cos(18*pi/43)+v4*cos(50*pi/43));

A14=2*(cos(2*pi/43)+v4*cos(82*pi/43)+v1*cos(8*pi/43)+v5*cos(70*pi/43)+v2*cos(32*pi/43)+v6*cos(22*pi/43)+v3*cos(42*pi/43));

A24=2*(cos(12*pi/43)+v4*cos(62*pi/43)+v1*cos(48*pi/43)+v5*cos(76*pi/43)+v2*cos(20*pi/43)+v6*cos(46*pi/43)+v3*cos(80*pi/43));

A34=2*(cos(72*pi/43)+v4*cos(28*pi/43)+v1*cos(30*pi/43)+v5*cos(26*pi/43)+v2*cos(34*pi/43)+v6*cos(18*pi/43)+v3*cos(50*pi/43));

A15=2*(cos(2*pi/43)+v5*cos(82*pi/43)+v3*cos(8*pi/43)+v1*cos(70*pi/43)+v6*cos(32*pi/43)+v4*cos(22*pi/43)+v2*cos(42*pi/43));

A25=2*(cos(12*pi/43)+v5*cos(62*pi/43)+v3*cos(48*pi/43)+v1*cos(76*pi/43)+v6*cos(20*pi/43)+v4*cos(46*pi/43)+v2*cos(80*pi/43));

A35=2*(cos(72*pi/43)+v5*cos(28*pi/43)+v3*cos(30*pi/43)+v1*cos(26*pi/43)+v6*cos(34*pi/43)+v4*cos(18*pi/43)+v2*cos(50*pi/43));

A16=2*(cos(2*pi/43)+v6*cos(82*pi/43)+v5*cos(8*pi/43)+v4*cos(70*pi/43)+v3*cos(32*pi/43)+v2*cos(22*pi/43)+v1*cos(42*pi/43));

A26=2*(cos(12*pi/43)+v6*cos(62*pi/43)+v5*cos(48*pi/43)+v4*cos(76*pi/43)+v3*cos(20*pi/43)+v2*cos(46*pi/43)+v1*cos(80*pi/43));

A36=2*(cos(72*pi/43)+v6*cos(28*pi/43)+v5*cos(30*pi/43)+v4*cos(26*pi/43)+v3*cos(34*pi/43)+v2*cos(18*pi/43)+v1*cos(50*pi/43));

A10=(-1+(-172-129*sqrt(-3))^(1/3)+(-172+129*sqrt(-3))^(1/3))/3;

A20=(-1+w1*(-172-129*sqrt(-3))^(1/3)+w2*(-172+129*sqrt(-3))^(1/3))/3;

A30=(-1+w2*(-172-129*sqrt(-3))^(1/3)+w1*(-172+129*sqrt(-3))^(1/3))/3;

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2.?14*cos(2*k*pi/43)的6個(gè)7次復(fù)根式項(xiàng)，設(shè)其內(nèi)表達(dá)式寫(xiě)成Pyz=Xy1+Xy2*v1^z+Xy3*v3^z+Xy4*v4^z+Xy5*v5^z+Xy6*v6^z。

P11=A11^7; P12=A12^7; P13=A13^7; P14=A14^7; P15=A15^7; P16=A16^7; P10=A10^7;

P21=A21^7; P22=A22^7; P23=A23^7; P24=A24^7; P25=A25^7; P26=A26^7; P20=A20^7;

P31=A31^7; P32=A32^7; P33=A33^7; P34=A34^7; P35=A35^7; P36=A36^7; P30=A30^7;

依次求解Xy0, Xy1......, Xy6。

I.?X10=(P11+P12+P13+P14+P15+P16+P10)/7;

X20=(P21+P22+P23+P24+P25+P26+P20)/7;

X30=(P31+P32+P33+P34+P35+P36+P30)/7;

三者滿足同一個(gè)整系數(shù)一元三次方程，是三個(gè)不同實(shí)數(shù)根。后續(xù)X1y、X2y、X3y（y不等于0）也有這樣的性質(zhì)。

X10=(-159748+(-8096+5127*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-8096-5127*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))/3;

X20=(-159748+(-8096+5127*sqrt(-3))*w1*(-172-129*sqrt(-3))^(1/3)+(-8096-5127*sqrt(-3))*w2*(-172+129*sqrt(-3))^(1/3))/3;

X30=(-159748+(-8096+5127*sqrt(-3))*w2*(-172-129*sqrt(-3))^(1/3)+(-8096-5127*sqrt(-3))*w1*(-172+129*sqrt(-3))^(1/3))/3;

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II. X11=(P11/v1+P12/v2+P13/v3+P14/v4+P15/v5+P16/v6+P10)/7;

X21=(P21/v1+P22/v2+P23/v3+P24/v4+P25/v5+P26/v6+P20)/7;

X31=(P31/v1+P32/v2+P33/v3+P34/v4+P35/v5+P36/v6+P30)/7;

X11=(90727+(5820.5-5596.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(5820.5+5596.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))/3;

X21=(90727+(5820.5-5596.5*sqrt(-3))*w1*(-172-129*sqrt(-3))^(1/3)+(5820.5+5596.5*sqrt(-3))*w2*(-172+129*sqrt(-3))^(1/3))/3;

X31=(90727+(5820.5-5596.5*sqrt(-3))*w2*(-172-129*sqrt(-3))^(1/3)+(5820.5+5596.5*sqrt(-3))*w1*(-172+129*sqrt(-3))^(1/3))/3;

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III. X12=(P11/v2+P12/v4+P13/v6+P14/v1+P15/v3+P16/v5+P10)/7;

X22=(P21/v2+P22/v4+P23/v6+P24/v1+P25/v3+P26/v5+P20)/7;

X32=(P31/v2+P32/v4+P33/v6+P34/v1+P35/v3+P36/v5+P30)/7;

X12=(147014+(11557-546*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(11557+546*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))/3;

X22=(147014+(11557-546*sqrt(-3))*w1*(-172-129*sqrt(-3))^(1/3)+(11557+546*sqrt(-3))*w2*(-172+129*sqrt(-3))^(1/3))/3;

X32=(147014+(11557-546*sqrt(-3))*w2*(-172-129*sqrt(-3))^(1/3)+(11557+546*sqrt(-3))*w1*(-172+129*sqrt(-3))^(1/3))/3;

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IV. X13=(P11/v3+P12/v6+P13/v2+P14/v5+P15/v1+P16/v4+P10)/7;

X23=(P21/v3+P22/v6+P23/v2+P24/v5+P25/v1+P26/v4+P20)/7;

X33=(P31/v3+P32/v6+P33/v2+P34/v5+P35/v1+P36/v4+P30)/7;

X13=(-44422+(6664+13069*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(6664-13069*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))/3;

X23=(-44422+(6664+13069*sqrt(-3))*w1*(-172-129*sqrt(-3))^(1/3)+(6664-13069*sqrt(-3))*w2*(-172+129*sqrt(-3))^(1/3))/3;

X33=(-44422+(6664+13069*sqrt(-3))*w2*(-172-129*sqrt(-3))^(1/3)+(6664-13069*sqrt(-3))*w1*(-172+129*sqrt(-3))^(1/3))/3;

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V. X14=(P11/v4+P12/v1+P13/v5+P14/v2+P15/v6+P16/v3+P10)/7;

X24=(P21/v4+P22/v1+P23/v5+P24/v2+P25/v6+P26/v3+P20)/7;

X34=(P31/v4+P32/v1+P33/v5+P34/v2+P35/v6+P36/v3+P30)/7;

X14=(7952+(1445.5+2922.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(1445.5-2922.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))/3;

X24=(7952+(1445.5+2922.5*sqrt(-3))*w1*(-172-129*sqrt(-3))^(1/3)+(1445.5-2922.5*sqrt(-3))*w2*(-172+129*sqrt(-3))^(1/3))/3;

X34=(7952+(1445.5+2922.5*sqrt(-3))*w2*(-172-129*sqrt(-3))^(1/3)+(1445.5-2922.5*sqrt(-3))*w1*(-172+129*sqrt(-3))^(1/3))/3;

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VI. X15=(P11/v5+P12/v3+P13/v1+P14/v6+P15/v4+P16/v2+P10)/7;

X25=(P21/v5+P22/v3+P23/v1+P24/v6+P25/v4+P26/v2+P20)/7;

X35=(P31/v5+P32/v3+P33/v1+P34/v6+P35/v4+P36/v2+P30)/7;

X15=(39557+(-1074.5-13653.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-1074.5+13653.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))/3;

X25=(39557+(-1074.5-13653.5*sqrt(-3))*w1*(-172-129*sqrt(-3))^(1/3)+(-1074.5+13653.5*sqrt(-3))*w2*(-172+129*sqrt(-3))^(1/3))/3;

X35=(39557+(-1074.5-13653.5*sqrt(-3))*w2*(-172-129*sqrt(-3))^(1/3)+(-1074.5+13653.5*sqrt(-3))*w1*(-172+129*sqrt(-3))^(1/3))/3;

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VII. X16=(P11/v6+P12/v5+P13/v4+P14/v3+P15/v2+P16/v1+P10)/7;

X26=(P21/v6+P22/v5+P23/v4+P24/v3+P25/v2+P26/v1+P20)/7;

X36=(P31/v6+P32/v5+P33/v4+P34/v3+P35/v2+P36/v1+P30)/7;

X16=(-116963+(-10664.5-2215.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-10664.5+2215.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))/3;

X26=(-116963+(-10664.5-2215.5*sqrt(-3))*w1*(-172-129*sqrt(-3))^(1/3)+(-10664.5+2215.5*sqrt(-3))*w2*(-172+129*sqrt(-3))^(1/3))/3;

X36=(-116963+(-10664.5-2215.5*sqrt(-3))*w2*(-172-129*sqrt(-3))^(1/3)+(-10664.5+2215.5*sqrt(-3))*w1*(-172+129*sqrt(-3))^(1/3))/3;

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巧用1+v1+......+v6=0消去含Xy6的項(xiàng)，最終得到：

P11=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5)/3;

P12=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3)/3;

P13=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1)/3;

P14=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6)/3;

P15=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4)/3;

P16=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2)/3;

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P21=(-42785+(-12298-2387*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-12298+2387*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-3171+9933*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-3171-9933*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(263977+(-13615+10276*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13615-10276*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(72541+(-31591+1022*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-31591-1022*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(124915+(-13762+3486*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13762-3486*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(156520+(12362+10514*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12362-10514*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5)/3;

P22=(-42785+(-12298-2387*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-12298+2387*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-3171+9933*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-3171-9933*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(263977+(-13615+10276*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13615-10276*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(72541+(-31591+1022*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-31591-1022*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(124915+(-13762+3486*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13762-3486*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(156520+(12362+10514*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12362-10514*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3)/3;

P23=(-42785+(-12298-2387*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-12298+2387*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-3171+9933*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-3171-9933*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(263977+(-13615+10276*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13615-10276*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(72541+(-31591+1022*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-31591-1022*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(124915+(-13762+3486*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13762-3486*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(156520+(12362+10514*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12362-10514*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1)/3;

P24=(-42785+(-12298-2387*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-12298+2387*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-3171+9933*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-3171-9933*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(263977+(-13615+10276*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13615-10276*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(72541+(-31591+1022*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-31591-1022*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(124915+(-13762+3486*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13762-3486*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(156520+(12362+10514*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12362-10514*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6)/3;

P25=(-42785+(-12298-2387*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-12298+2387*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-3171+9933*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-3171-9933*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(263977+(-13615+10276*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13615-10276*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(72541+(-31591+1022*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-31591-1022*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(124915+(-13762+3486*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13762-3486*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(156520+(12362+10514*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12362-10514*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4)/3;

P26=(-42785+(-12298-2387*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-12298+2387*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-3171+9933*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-3171-9933*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(263977+(-13615+10276*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13615-10276*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(72541+(-31591+1022*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-31591-1022*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(124915+(-13762+3486*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13762-3486*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(156520+(12362+10514*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12362-10514*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2)/3;

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P31=(-42785+(9729.5-4955.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9729.5+4955.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-13314-6552*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13314+6552*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(263977+(-8606.5-11945.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-8606.5+11945.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(72541+(14262.5-16306.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(14262.5+16306.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(124915+(1652-8624*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(1652+8624*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(156520+(-21952+924*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-21952-924*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5)/3;

P32=(-42785+(9729.5-4955.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9729.5+4955.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-13314-6552*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13314+6552*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(263977+(-8606.5-11945.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-8606.5+11945.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(72541+(14262.5-16306.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(14262.5+16306.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(124915+(1652-8624*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(1652+8624*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(156520+(-21952+924*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-21952-924*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3)/3;

P33=(-42785+(9729.5-4955.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9729.5+4955.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-13314-6552*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13314+6552*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(263977+(-8606.5-11945.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-8606.5+11945.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(72541+(14262.5-16306.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(14262.5+16306.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(124915+(1652-8624*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(1652+8624*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(156520+(-21952+924*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-21952-924*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1)/3;

P34=(-42785+(9729.5-4955.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9729.5+4955.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-13314-6552*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13314+6552*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(263977+(-8606.5-11945.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-8606.5+11945.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(72541+(14262.5-16306.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(14262.5+16306.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(124915+(1652-8624*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(1652+8624*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(156520+(-21952+924*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-21952-924*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6)/3;

P35=(-42785+(9729.5-4955.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9729.5+4955.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-13314-6552*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13314+6552*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(263977+(-8606.5-11945.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-8606.5+11945.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(72541+(14262.5-16306.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(14262.5+16306.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(124915+(1652-8624*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(1652+8624*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(156520+(-21952+924*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-21952-924*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4)/3;

P36=(-42785+(9729.5-4955.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9729.5+4955.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(-13314-6552*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-13314+6552*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(263977+(-8606.5-11945.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-8606.5+11945.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(72541+(14262.5-16306.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(14262.5+16306.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(124915+(1652-8624*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(1652+8624*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(156520+(-21952+924*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(-21952-924*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2)/3;

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3.?用Ay0, Py1^(1/7), ......Py6^(1/7)線性表示cos(2*k*pi/43)。會(huì)不可避免用到7次單位根系數(shù)，每7個(gè)以02為底數(shù)的k值對(duì)應(yīng)同一個(gè)A、P值，而每個(gè)k值對(duì)應(yīng)唯一一套6個(gè)7次單位根系數(shù)的組合。列舉全部的cos(2*k*pi/43)值如下：

I.以k=2為開(kāi)頭

cos(2*pi/43)=(A10+v1*P11^(1/7)+v6*P16^(1/7)+v2*P12^(1/7)+v5*P15^(1/7)+v1*P13^(1/7)+v6*P14^(1/7))/14;

cos(82*pi/43)=(A10+P11^(1/7)+P16^(1/7)+P12^(1/7)+P15^(1/7)+v5*P13^(1/7)+v2*P14^(1/7))/14;

cos(8*pi/43)=(A10+v6*P11^(1/7)+v1*P16^(1/7)+v5*P12^(1/7)+v2*P15^(1/7)+v2*P13^(1/7)+v5*P14^(1/7))/14;

cos(70*pi/43)=(A10+v5*P11^(1/7)+v2*P16^(1/7)+v3*P12^(1/7)+v4*P15^(1/7)+v6*P13^(1/7)+v1*P14^(1/7))/14;

cos(32*pi/43)=(A10+v4*P11^(1/7)+v3*P16^(1/7)+v1*P12^(1/7)+v6*P15^(1/7)+v3*P13^(1/7)+v4*P14^(1/7))/14;

cos(22*pi/43)=(A10+v3*P11^(1/7)+v4*P16^(1/7)+v6*P12^(1/7)+v1*P15^(1/7)+P13^(1/7)+P14^(1/7))/14;

cos(42*pi/43)=(A10+v2*P11^(1/7)+v5*P16^(1/7)+v4*P12^(1/7)+v3*P15^(1/7)+v4*P13^(1/7)+v3*P14^(1/7))/14;

II.以k=12為開(kāi)頭

cos(12*pi/43)=(A20+v6*P21^(1/7)+v1*P26^(1/7)+v6*P22^(1/7)+v1*P25^(1/7)+v3*P23^(1/7)+v4*P24^(1/7))/14;

cos(62*pi/43)=(A20+v5*P21^(1/7)+v2*P26^(1/7)+v4*P22^(1/7)+v3*P25^(1/7)+P23^(1/7)+P24^(1/7))/14;

cos(48*pi/43)=(A20+v4*P21^(1/7)+v3*P26^(1/7)+v2*P22^(1/7)+v5*P25^(1/7)+v4*P23^(1/7)+v3*P24^(1/7))/14;

cos(76*pi/43)=(A20+v3*P21^(1/7)+v4*P26^(1/7)+P22^(1/7)+P25^(1/7)+v1*P23^(1/7)+v6*P24^(1/7))/14;

cos(20*pi/43)=(A20+v2*P21^(1/7)+v5*P26^(1/7)+v5*P22^(1/7)+v2*P25^(1/7)+v5*P23^(1/7)+v2*P24^(1/7))/14;

cos(46*pi/43)=(A20+v1*P21^(1/7)+v6*P26^(1/7)+v3*P22^(1/7)+v4*P25^(1/7)+v2*P23^(1/7)+v5*P24^(1/7))/14;

cos(80*pi/43)=(A20+P21^(1/7)+P26^(1/7)+v1*P22^(1/7)+v6*P25^(1/7)+v6*P23^(1/7)+v1*P24^(1/7))/14;

III.以k=72為開(kāi)頭

cos(72*pi/43)=(A30+P31^(1/7)+P36^(1/7)+v1*P32^(1/7)+v6*P35^(1/7)+v1*P33^(1/7)+v6*P34^(1/7))/14;

cos(28*pi/43)=(A30+v6*P31^(1/7)+v1*P36^(1/7)+v6*P32^(1/7)+v1*P35^(1/7)+v5*P33^(1/7)+v2*P34^(1/7))/14;

cos(30*pi/43)=(A30+v5*P31^(1/7)+v2*P36^(1/7)+v4*P32^(1/7)+v3*P35^(1/7)+v2*P33^(1/7)+v5*P34^(1/7))/14;

cos(26*pi/43)=(A30+v4*P31^(1/7)+v3*P36^(1/7)+v2*P32^(1/7)+v5*P35^(1/7)+v6*P33^(1/7)+v1*P34^(1/7))/14;

cos(34*pi/43)=(A30+v3*P31^(1/7)+v4*P36^(1/7)+P32^(1/7)+P35^(1/7)+v3*P33^(1/7)+v4*P34^(1/7))/14;

cos(18*pi/43)=(A30+v2*P31^(1/7)+v5*P36^(1/7)+v5*P32^(1/7)+v2*P35^(1/7)+P33^(1/7)+P34^(1/7))/14;

cos(50*pi/43)=(A30+v1*P31^(1/7)+v6*P36^(1/7)+v3*P32^(1/7)+v4*P35^(1/7)+v4*P33^(1/7)+v3*P34^(1/7))/14.

未完待續(xù)，下篇，揭秘sin(2*k*pi/43)！