cos(2*n*pi/47)的舍命求解。这次内容高能之至，可能引起思绪错乱。

一．23次单位根

k1=exp(2*pi*j/23); k2=exp(4*pi*j/23); k3=exp(6*pi*j/23); k4=exp(8*pi*j/23); k5=exp(10*pi*j/23); k6=exp(12*pi*j/23);

(资料图片仅供参考)

k7=exp(14*pi*j/23); k8=exp(16*pi*j/23); k9=exp(18*pi*j/23); k10=exp(20*pi*j/23); k11=exp(22*pi*j/23); k12=1/k11; k13=1/k10;

k14=1/k9; k15=1/k8; k16=1/k7; k17=1/k6; k18=1/k5; k19=1/k4; k20=1/k3; k21=1/k2; k22=1/k1;

k的表达式全部于《23边形》一文给出。

二．用22个含有23次单位根的23次根式线性表示cos(2*n*pi/47)。以2为底数。

E0=2*(cos(2*pi/47)+cos(4*pi/47)+cos(8*pi/47)+cos(16*pi/47)+cos(32*pi/47)+cos(64*pi/47)+cos(34*pi/47)+cos(68*pi/47)+cos(42*pi/47)+cos(84*pi/47)+cos(74*pi/47)+cos(54*pi/47)+cos(14*pi/47)+cos(28*pi/47)+cos(56*pi/47)+cos(18*pi/47)+cos(36*pi/47)+cos(72*pi/47)+cos(50*pi/47)+cos(6*pi/47)+cos(12*pi/47)+cos(24*pi/47)+cos(48*pi/47));

E1=2*(cos(2*pi/47)+k1*cos(4*pi/47)+k2*cos(8*pi/47)+k3*cos(16*pi/47)+k4*cos(32*pi/47)+k5*cos(64*pi/47)+k6*cos(34*pi/47)+k7*cos(68*pi/47)+k8*cos(42*pi/47)+k9*cos(84*pi/47)+k10*cos(74*pi/47)+k11*cos(54*pi/47)+k12*cos(14*pi/47)+k13*cos(28*pi/47)+k14*cos(56*pi/47)+k15*cos(18*pi/47)+k16*cos(36*pi/47)+k17*cos(72*pi/47)+k18*cos(50*pi/47)+k19*cos(6*pi/47)+k20*cos(12*pi/47)+k21*cos(24*pi/47)+k22*cos(48*pi/47));

E2=2*(cos(2*pi/47)+k2*cos(4*pi/47)+k4*cos(8*pi/47)+k6*cos(16*pi/47)+k8*cos(32*pi/47)+k10*cos(64*pi/47)+k12*cos(34*pi/47)+k14*cos(68*pi/47)+k16*cos(42*pi/47)+k18*cos(84*pi/47)+k20*cos(74*pi/47)+k22*cos(54*pi/47)+k1*cos(14*pi/47)+k3*cos(28*pi/47)+k5*cos(56*pi/47)+k7*cos(18*pi/47)+k9*cos(36*pi/47)+k11*cos(72*pi/47)+k13*cos(50*pi/47)+k15*cos(6*pi/47)+k17*cos(12*pi/47)+k19*cos(24*pi/47)+k21*cos(48*pi/47));

E3=2*(cos(2*pi/47)+k3*cos(4*pi/47)+k6*cos(8*pi/47)+k9*cos(16*pi/47)+k12*cos(32*pi/47)+k15*cos(64*pi/47)+k18*cos(34*pi/47)+k21*cos(68*pi/47)+k1*cos(42*pi/47)+k4*cos(84*pi/47)+k7*cos(74*pi/47)+k10*cos(54*pi/47)+k13*cos(14*pi/47)+k16*cos(28*pi/47)+k19*cos(56*pi/47)+k22*cos(18*pi/47)+k2*cos(36*pi/47)+k5*cos(72*pi/47)+k8*cos(50*pi/47)+k11*cos(6*pi/47)+k14*cos(12*pi/47)+k17*cos(24*pi/47)+k20*cos(48*pi/47));

E4=2*(cos(2*pi/47)+k4*cos(4*pi/47)+k8*cos(8*pi/47)+k12*cos(16*pi/47)+k16*cos(32*pi/47)+k20*cos(64*pi/47)+k1*cos(34*pi/47)+k5*cos(68*pi/47)+k9*cos(42*pi/47)+k13*cos(84*pi/47)+k17*cos(74*pi/47)+k21*cos(54*pi/47)+k2*cos(14*pi/47)+k6*cos(28*pi/47)+k10*cos(56*pi/47)+k14*cos(18*pi/47)+k18*cos(36*pi/47)+k22*cos(72*pi/47)+k3*cos(50*pi/47)+k7*cos(6*pi/47)+k11*cos(12*pi/47)+k15*cos(24*pi/47)+k19*cos(48*pi/47));

E5=2*(cos(2*pi/47)+k5*cos(4*pi/47)+k10*cos(8*pi/47)+k15*cos(16*pi/47)+k20*cos(32*pi/47)+k2*cos(64*pi/47)+k7*cos(34*pi/47)+k12*cos(68*pi/47)+k17*cos(42*pi/47)+k22*cos(84*pi/47)+k4*cos(74*pi/47)+k9*cos(54*pi/47)+k14*cos(14*pi/47)+k19*cos(28*pi/47)+k1*cos(56*pi/47)+k6*cos(18*pi/47)+k11*cos(36*pi/47)+k16*cos(72*pi/47)+k21*cos(50*pi/47)+k3*cos(6*pi/47)+k8*cos(12*pi/47)+k13*cos(24*pi/47)+k18*cos(48*pi/47));

E6=2*(cos(2*pi/47)+k6*cos(4*pi/47)+k12*cos(8*pi/47)+k18*cos(16*pi/47)+k1*cos(32*pi/47)+k7*cos(64*pi/47)+k13*cos(34*pi/47)+k19*cos(68*pi/47)+k2*cos(42*pi/47)+k8*cos(84*pi/47)+k14*cos(74*pi/47)+k20*cos(54*pi/47)+k3*cos(14*pi/47)+k9*cos(28*pi/47)+k15*cos(56*pi/47)+k21*cos(18*pi/47)+k4*cos(36*pi/47)+k10*cos(72*pi/47)+k16*cos(50*pi/47)+k22*cos(6*pi/47)+k5*cos(12*pi/47)+k11*cos(24*pi/47)+k17*cos(48*pi/47));

E7=2*(cos(2*pi/47)+k7*cos(4*pi/47)+k14*cos(8*pi/47)+k21*cos(16*pi/47)+k5*cos(32*pi/47)+k12*cos(64*pi/47)+k19*cos(34*pi/47)+k3*cos(68*pi/47)+k10*cos(42*pi/47)+k17*cos(84*pi/47)+k1*cos(74*pi/47)+k8*cos(54*pi/47)+k15*cos(14*pi/47)+k22*cos(28*pi/47)+k6*cos(56*pi/47)+k13*cos(18*pi/47)+k20*cos(36*pi/47)+k4*cos(72*pi/47)+k11*cos(50*pi/47)+k18*cos(6*pi/47)+k2*cos(12*pi/47)+k9*cos(24*pi/47)+k16*cos(48*pi/47));

E8=2*(cos(2*pi/47)+k8*cos(4*pi/47)+k16*cos(8*pi/47)+k1*cos(16*pi/47)+k9*cos(32*pi/47)+k17*cos(64*pi/47)+k2*cos(34*pi/47)+k10*cos(68*pi/47)+k18*cos(42*pi/47)+k3*cos(84*pi/47)+k11*cos(74*pi/47)+k19*cos(54*pi/47)+k4*cos(14*pi/47)+k12*cos(28*pi/47)+k20*cos(56*pi/47)+k5*cos(18*pi/47)+k13*cos(36*pi/47)+k21*cos(72*pi/47)+k6*cos(50*pi/47)+k14*cos(6*pi/47)+k22*cos(12*pi/47)+k7*cos(24*pi/47)+k15*cos(48*pi/47));

E9=2*(cos(2*pi/47)+k9*cos(4*pi/47)+k18*cos(8*pi/47)+k4*cos(16*pi/47)+k13*cos(32*pi/47)+k22*cos(64*pi/47)+k8*cos(34*pi/47)+k17*cos(68*pi/47)+k3*cos(42*pi/47)+k12*cos(84*pi/47)+k21*cos(74*pi/47)+k7*cos(54*pi/47)+k16*cos(14*pi/47)+k2*cos(28*pi/47)+k11*cos(56*pi/47)+k20*cos(18*pi/47)+k6*cos(36*pi/47)+k15*cos(72*pi/47)+k1*cos(50*pi/47)+k10*cos(6*pi/47)+k19*cos(12*pi/47)+k5*cos(24*pi/47)+k14*cos(48*pi/47));

E10=2*(cos(2*pi/47)+k10*cos(4*pi/47)+k20*cos(8*pi/47)+k7*cos(16*pi/47)+k17*cos(32*pi/47)+k4*cos(64*pi/47)+k14*cos(34*pi/47)+k1*cos(68*pi/47)+k11*cos(42*pi/47)+k21*cos(84*pi/47)+k8*cos(74*pi/47)+k18*cos(54*pi/47)+k5*cos(14*pi/47)+k15*cos(28*pi/47)+k2*cos(56*pi/47)+k12*cos(18*pi/47)+k22*cos(36*pi/47)+k9*cos(72*pi/47)+k19*cos(50*pi/47)+k6*cos(6*pi/47)+k16*cos(12*pi/47)+k3*cos(24*pi/47)+k13*cos(48*pi/47));

E11=2*(cos(2*pi/47)+k11*cos(4*pi/47)+k22*cos(8*pi/47)+k10*cos(16*pi/47)+k21*cos(32*pi/47)+k9*cos(64*pi/47)+k20*cos(34*pi/47)+k8*cos(68*pi/47)+k19*cos(42*pi/47)+k7*cos(84*pi/47)+k18*cos(74*pi/47)+k6*cos(54*pi/47)+k17*cos(14*pi/47)+k5*cos(28*pi/47)+k16*cos(56*pi/47)+k4*cos(18*pi/47)+k15*cos(36*pi/47)+k3*cos(72*pi/47)+k14*cos(50*pi/47)+k2*cos(6*pi/47)+k13*cos(12*pi/47)+k1*cos(24*pi/47)+k12*cos(48*pi/47));

E12=47/E11; E13=47/E10; E14=47/E9; E15=47/E8; E16=47/E7; E17=47/E6; E18=47/E5; E19=47/E4;

E20=47/E3; E21=47/E2; E22=47/E1;

X1=E1^23; X2=E2^23; X3=E3^23; X4=E4^23; X5=E5^23; X6=E6^23; X7=E7^23; X8=E8^23;

X9=E9^23; X10=E10^23; X11=E11^23; X12=E12^23; X13=E13^23; X14=E14^23; X15=E15^23; X16=E16^23;

X17=E17^23; X18=E18^23; X19=E19^23; X20=E20^23; X21=E21^23; X22=E22^23;

假设X1~X22能用23次单位根线性表示，表达式为Xn=C0+C1*k1^n+……+C22*k22^n，C0~C22都是庞大的整数！

C0=(X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+X12+X13+X14+X15+X16+X17+X18+X19+X20+X21+X22+X0)/23=-21419266697599461;

C1=(X1/k1^1+X2/k2^1+X3/k3^1+X4/k4^1+X5/k5^1+X6/k6^1+X7/k7^1+X8/k8^1+X9/k9^1+X10/k10^1+X11/k11^1+X12/k12^1+X13/k13^1+X14/k14^1+X15/k15^1+X16/k16^1+X17/k17^1+X18/k18^1+X19/k19^1+X20/k20^1+X21/k21^1+X22/k22^1+X0)/23=2407563234519470365;

C2=(X1/k1^2+X2/k2^2+X3/k3^2+X4/k4^2+X5/k5^2+X6/k6^2+X7/k7^2+X8/k8^2+X9/k9^2+X10/k10^2+X11/k11^2+X12/k12^2+X13/k13^2+X14/k14^2+X15/k15^2+X16/k16^2+X17/k17^2+X18/k18^2+X19/k19^2+X20/k20^2+X21/k21^2+X22/k22^2+X0)/23=-1362260327737730672;

C3=(X1/k1^3+X2/k2^3+X3/k3^3+X4/k4^3+X5/k5^3+X6/k6^3+X7/k7^3+X8/k8^3+X9/k9^3+X10/k10^3+X11/k11^3+X12/k12^3+X13/k13^3+X14/k14^3+X15/k15^3+X16/k16^3+X17/k17^3+X18/k18^3+X19/k19^3+X20/k20^3+X21/k21^3+X22/k22^3+X0)/23=2395617115052993394;

C4=(X1/k1^4+X2/k2^4+X3/k3^4+X4/k4^4+X5/k5^4+X6/k6^4+X7/k7^4+X8/k8^4+X9/k9^4+X10/k10^4+X11/k11^4+X12/k12^4+X13/k13^4+X14/k14^4+X15/k15^4+X16/k16^4+X17/k17^4+X18/k18^4+X19/k19^4+X20/k20^4+X21/k21^4+X22/k22^4+X0)/23=-393717458288946471;

C5=(X1/k1^5+X2/k2^5+X3/k3^5+X4/k4^5+X5/k5^5+X6/k6^5+X7/k7^5+X8/k8^5+X9/k9^5+X10/k10^5+X11/k11^5+X12/k12^5+X13/k13^5+X14/k14^5+X15/k15^5+X16/k16^5+X17/k17^5+X18/k18^5+X19/k19^5+X20/k20^5+X21/k21^5+X22/k22^5+X0)/23=1511346796802899014;

C6=(X1/k1^6+X2/k2^6+X3/k3^6+X4/k4^6+X5/k5^6+X6/k6^6+X7/k7^6+X8/k8^6+X9/k9^6+X10/k10^6+X11/k11^6+X12/k12^6+X13/k13^6+X14/k14^6+X15/k15^6+X16/k16^6+X17/k17^6+X18/k18^6+X19/k19^6+X20/k20^6+X21/k21^6+X22/k22^6+X0)/23=-9407573159118751021;

C7=(X1/k1^7+X2/k2^7+X3/k3^7+X4/k4^7+X5/k5^7+X6/k6^7+X7/k7^7+X8/k8^7+X9/k9^7+X10/k10^7+X11/k11^7+X12/k12^7+X13/k13^7+X14/k14^7+X15/k15^7+X16/k16^7+X17/k17^7+X18/k18^7+X19/k19^7+X20/k20^7+X21/k21^7+X22/k22^7+X0)/23=-1692273137144054721;

C8=(X1/k1^8+X2/k2^8+X3/k3^8+X4/k4^8+X5/k5^8+X6/k6^8+X7/k7^8+X8/k8^8+X9/k9^8+X10/k10^8+X11/k11^8+X12/k12^8+X13/k13^8+X14/k14^8+X15/k15^8+X16/k16^8+X17/k17^8+X18/k18^8+X19/k19^8+X20/k20^8+X21/k21^8+X22/k22^8+X0)/23=741328698064911855;

C9=(X1/k1^9+X2/k2^9+X3/k3^9+X4/k4^9+X5/k5^9+X6/k6^9+X7/k7^9+X8/k8^9+X9/k9^9+X10/k10^9+X11/k11^9+X12/k12^9+X13/k13^9+X14/k14^9+X15/k15^9+X16/k16^9+X17/k17^9+X18/k18^9+X19/k19^9+X20/k20^9+X21/k21^9+X22/k22^9+X0)/23=-1602526444387269086;

C10=(X1/k1^10+X2/k2^10+X3/k3^10+X4/k4^10+X5/k5^10+X6/k6^10+X7/k7^10+X8/k8^10+X9/k9^10+X10/k10^10+X11/k11^10+X12/k12^10+X13/k13^10+X14/k14^10+X15/k15^10+X16/k16^10+X17/k17^10+X18/k18^10+X19/k19^10+X20/k20^10+X21/k21^10+X22/k22^10+X0)/23=-6606192542568108857;

C11=(X1/k1^11+X2/k2^11+X3/k3^11+X4/k4^11+X5/k5^11+X6/k6^11+X7/k7^11+X8/k8^11+X9/k9^11+X10/k10^11+X11/k11^11+X12/k12^11+X13/k13^11+X14/k14^11+X15/k15^11+X16/k16^11+X17/k17^11+X18/k18^11+X19/k19^11+X20/k20^11+X21/k21^11+X22/k22^11+X0)/23=3930485083326055184;

C12=(X1/k1^12+X2/k2^12+X3/k3^12+X4/k4^12+X5/k5^12+X6/k6^12+X7/k7^12+X8/k8^12+X9/k9^12+X10/k10^12+X11/k11^12+X12/k12^12+X13/k13^12+X14/k14^12+X15/k15^12+X16/k16^12+X17/k17^12+X18/k18^12+X19/k19^12+X20/k20^12+X21/k21^12+X22/k22^12+X0)/23=1576906243279346707;

C13=(X1/k1^13+X2/k2^13+X3/k3^13+X4/k4^13+X5/k5^13+X6/k6^13+X7/k7^13+X8/k8^13+X9/k9^13+X10/k10^13+X11/k11^13+X12/k12^13+X13/k13^13+X14/k14^13+X15/k15^13+X16/k16^13+X17/k17^13+X18/k18^13+X19/k19^13+X20/k20^13+X21/k21^13+X22/k22^13+X0)/23=1842535238960285195;

C14=(X1/k1^14+X2/k2^14+X3/k3^14+X4/k4^14+X5/k5^14+X6/k6^14+X7/k7^14+X8/k8^14+X9/k9^14+X10/k10^14+X11/k11^14+X12/k12^14+X13/k13^14+X14/k14^14+X15/k15^14+X16/k16^14+X17/k17^14+X18/k18^14+X19/k19^14+X20/k20^14+X21/k21^14+X22/k22^14+X0)/23=5133268272465685189;

C15=(X1/k1^15+X2/k2^15+X3/k3^15+X4/k4^15+X5/k5^15+X6/k6^15+X7/k7^15+X8/k8^15+X9/k9^15+X10/k10^15+X11/k11^15+X12/k12^15+X13/k13^15+X14/k14^15+X15/k15^15+X16/k16^15+X17/k17^15+X18/k18^15+X19/k19^15+X20/k20^15+X21/k21^15+X22/k22^15+X0)/23=-3922714564881345220;

C16=(X1/k1^16+X2/k2^16+X3/k3^16+X4/k4^16+X5/k5^16+X6/k6^16+X7/k7^16+X8/k8^16+X9/k9^16+X10/k10^16+X11/k11^16+X12/k12^16+X13/k13^16+X14/k14^16+X15/k15^16+X16/k16^16+X17/k17^16+X18/k18^16+X19/k19^16+X20/k20^16+X21/k21^16+X22/k22^16+X0)/23=-1735763312965995866;

C17=(X1/k1^17+X2/k2^17+X3/k3^17+X4/k4^17+X5/k5^17+X6/k6^17+X7/k7^17+X8/k8^17+X9/k9^17+X10/k10^17+X11/k11^17+X12/k12^17+X13/k13^17+X14/k14^17+X15/k15^17+X16/k16^17+X17/k17^17+X18/k18^17+X19/k19^17+X20/k20^17+X21/k21^17+X22/k22^17+X0)/23=-2924746611514392042;

C18=(X1/k1^18+X2/k2^18+X3/k3^18+X4/k4^18+X5/k5^18+X6/k6^18+X7/k7^18+X8/k8^18+X9/k9^18+X10/k10^18+X11/k11^18+X12/k12^18+X13/k13^18+X14/k14^18+X15/k15^18+X16/k16^18+X17/k17^18+X18/k18^18+X19/k19^18+X20/k20^18+X21/k21^18+X22/k22^18+X0)/23=1236790755952073690;

C19=(X1/k1^19+X2/k2^19+X3/k3^19+X4/k4^19+X5/k5^19+X6/k6^19+X7/k7^19+X8/k8^19+X9/k9^19+X10/k10^19+X11/k11^19+X12/k12^19+X13/k13^19+X14/k14^19+X15/k15^19+X16/k16^19+X17/k17^19+X18/k18^19+X19/k19^19+X20/k20^19+X21/k21^19+X22/k22^19+X0)/23=6188690192745096588;

C20=(X1/k1^20+X2/k2^20+X3/k3^20+X4/k4^20+X5/k5^20+X6/k6^20+X7/k7^20+X8/k8^20+X9/k9^20+X10/k10^20+X11/k11^20+X12/k12^20+X13/k13^20+X14/k14^20+X15/k15^20+X16/k16^20+X17/k17^20+X18/k18^20+X19/k19^20+X20/k20^20+X21/k21^20+X22/k22^20+X0)/23=-700321639254472937;

C21=(X1/k1^21+X2/k2^21+X3/k3^21+X4/k4^21+X5/k5^21+X6/k6^21+X7/k7^21+X8/k8^21+X9/k9^21+X10/k10^21+X11/k11^21+X12/k12^21+X13/k13^21+X14/k14^21+X15/k15^21+X16/k16^21+X17/k17^21+X18/k18^21+X19/k19^21+X20/k20^21+X21/k21^21+X22/k22^21+X0)/23=1550448814188407053;

C22=(X1/k1^22+X2/k2^22+X3/k3^22+X4/k4^22+X5/k5^22+X6/k6^22+X7/k7^22+X8/k8^22+X9/k9^22+X10/k10^22+X11/k11^22+X12/k12^22+X13/k13^22+X14/k14^22+X15/k15^22+X16/k16^22+X17/k17^22+X18/k18^22+X19/k19^22+X20/k20^22+X21/k21^22+X22/k22^22+X0)/23=1854528019201442119;

继而根据1+k1+……+k22=0整理得到：

X1=47*(-39913772040405140+11766706708894218*k1-68442305254024953*k2+11512533954288325*k3-47835010159369970*k4-7301728136139215*k5-239619174006812620*k6-75463854390329720*k7-23685091939075112*k8-73554350289121515*k9-180015331101479808*k10+44169299236693895*k11-5906846296214796*k12-255165537045892*k13+69760430920515810*k14-122920054980484837*k15-76389177280158255*k16-101686694270549663*k17-13143346026582307*k18+92216216458375627*k19-54358503371402448*k20-6469770319426278*k21);

X2=47*(-39913772040405140+11766706708894218*k2-68442305254024953*k4+11512533954288325*k6-47835010159369970*k8-7301728136139215*k10-239619174006812620*k12-75463854390329720*k14-23685091939075112*k16-73554350289121515*k18-180015331101479808*k20+44169299236693895*k22-5906846296214796*k1-255165537045892*k3+69760430920515810*k5-122920054980484837*k7-76389177280158255*k9-101686694270549663*k11-13143346026582307*k13+92216216458375627*k15-54358503371402448*k17-6469770319426278*k19);

X3=47*(-39913772040405140+11766706708894218*k3-68442305254024953*k6+11512533954288325*k9-47835010159369970*k12-7301728136139215*k15-239619174006812620*k18-75463854390329720*k21-23685091939075112*k1-73554350289121515*k4-180015331101479808*k7+44169299236693895*k10-5906846296214796*k13-255165537045892*k16+69760430920515810*k19-122920054980484837*k22-76389177280158255*k2-101686694270549663*k5-13143346026582307*k8+92216216458375627*k11-54358503371402448*k14-6469770319426278*k17);

X4=47*(-39913772040405140+11766706708894218*k4-68442305254024953*k8+11512533954288325*k12-47835010159369970*k16-7301728136139215*k20-239619174006812620*k1-75463854390329720*k5-23685091939075112*k9-73554350289121515*k13-180015331101479808*k17+44169299236693895*k21-5906846296214796*k2-255165537045892*k6+69760430920515810*k10-122920054980484837*k14-76389177280158255*k18-101686694270549663*k22-13143346026582307*k3+92216216458375627*k7-54358503371402448*k11-6469770319426278*k15);

X5=47*(-39913772040405140+11766706708894218*k5-68442305254024953*k10+11512533954288325*k15-47835010159369970*k20-7301728136139215*k2-239619174006812620*k7-75463854390329720*k12-23685091939075112*k17-73554350289121515*k22-180015331101479808*k4+44169299236693895*k9-5906846296214796*k14-255165537045892*k19+69760430920515810*k1-122920054980484837*k6-76389177280158255*k11-101686694270549663*k16-13143346026582307*k21+92216216458375627*k3-54358503371402448*k8-6469770319426278*k13);

X6=47*(-39913772040405140+11766706708894218*k6-68442305254024953*k12+11512533954288325*k18-47835010159369970*k1-7301728136139215*k7-239619174006812620*k13-75463854390329720*k19-23685091939075112*k2-73554350289121515*k8-180015331101479808*k14+44169299236693895*k20-5906846296214796*k3-255165537045892*k9+69760430920515810*k15-122920054980484837*k21-76389177280158255*k4-101686694270549663*k10-13143346026582307*k16+92216216458375627*k22-54358503371402448*k5-6469770319426278*k11);

X7=47*(-39913772040405140+11766706708894218*k7-68442305254024953*k14+11512533954288325*k21-47835010159369970*k5-7301728136139215*k12-239619174006812620*k19-75463854390329720*k3-23685091939075112*k10-73554350289121515*k17-180015331101479808*k1+44169299236693895*k8-5906846296214796*k15-255165537045892*k22+69760430920515810*k6-122920054980484837*k13-76389177280158255*k20-101686694270549663*k4-13143346026582307*k11+92216216458375627*k18-54358503371402448*k2-6469770319426278*k9);

X8=47*(-39913772040405140+11766706708894218*k8-68442305254024953*k16+11512533954288325*k1-47835010159369970*k9-7301728136139215*k17-239619174006812620*k2-75463854390329720*k10-23685091939075112*k18-73554350289121515*k3-180015331101479808*k11+44169299236693895*k19-5906846296214796*k4-255165537045892*k12+69760430920515810*k20-122920054980484837*k5-76389177280158255*k13-101686694270549663*k21-13143346026582307*k6+92216216458375627*k14-54358503371402448*k22-6469770319426278*k7);

X9=47*(-39913772040405140+11766706708894218*k9-68442305254024953*k18+11512533954288325*k4-47835010159369970*k13-7301728136139215*k22-239619174006812620*k8-75463854390329720*k17-23685091939075112*k3-73554350289121515*k12-180015331101479808*k21+44169299236693895*k7-5906846296214796*k16-255165537045892*k2+69760430920515810*k11-122920054980484837*k20-76389177280158255*k6-101686694270549663*k15-13143346026582307*k1+92216216458375627*k10-54358503371402448*k19-6469770319426278*k5);

X10=47*(-39913772040405140+11766706708894218*k10-68442305254024953*k20+11512533954288325*k7-47835010159369970*k17-7301728136139215*k4-239619174006812620*k14-75463854390329720*k1-23685091939075112*k11-73554350289121515*k21-180015331101479808*k8+44169299236693895*k18-5906846296214796*k5-255165537045892*k15+69760430920515810*k2-122920054980484837*k12-76389177280158255*k22-101686694270549663*k9-13143346026582307*k19+92216216458375627*k6-54358503371402448*k16-6469770319426278*k3);

X11=47*(-39913772040405140+11766706708894218*k11-68442305254024953*k22+11512533954288325*k10-47835010159369970*k21-7301728136139215*k9-239619174006812620*k20-75463854390329720*k8-23685091939075112*k19-73554350289121515*k7-180015331101479808*k18+44169299236693895*k6-5906846296214796*k17-255165537045892*k5+69760430920515810*k16-122920054980484837*k4-76389177280158255*k15-101686694270549663*k3-13143346026582307*k14+92216216458375627*k2-54358503371402448*k13-6469770319426278*k1);

X12=47*(-39913772040405140+11766706708894218/k11-68442305254024953/k22+11512533954288325/k10-47835010159369970/k21-7301728136139215/k9-239619174006812620/k20-75463854390329720/k8-23685091939075112/k19-73554350289121515/k7-180015331101479808/k18+44169299236693895/k6-5906846296214796/k17-255165537045892/k5+69760430920515810/k16-122920054980484837/k4-76389177280158255/k15-101686694270549663/k3-13143346026582307/k14+92216216458375627/k2-54358503371402448/k13-6469770319426278/k1);

X13=47*(-39913772040405140+11766706708894218/k10-68442305254024953/k20+11512533954288325/k7-47835010159369970/k17-7301728136139215/k4-239619174006812620/k14-75463854390329720/k1-23685091939075112/k11-73554350289121515/k21-180015331101479808/k8+44169299236693895/k18-5906846296214796/k5-255165537045892/k15+69760430920515810/k2-122920054980484837/k12-76389177280158255/k22-101686694270549663/k9-13143346026582307/k19+92216216458375627/k6-54358503371402448/k16-6469770319426278/k3);

X14=47*(-39913772040405140+11766706708894218/k9-68442305254024953/k18+11512533954288325/k4-47835010159369970/k13-7301728136139215/k22-239619174006812620/k8-75463854390329720/k17-23685091939075112/k3-73554350289121515/k12-180015331101479808/k21+44169299236693895/k7-5906846296214796/k16-255165537045892/k2+69760430920515810/k11-122920054980484837/k20-76389177280158255/k6-101686694270549663/k15-13143346026582307/k1+92216216458375627/k10-54358503371402448/k19-6469770319426278/k5);

X15=47*(-39913772040405140+11766706708894218/k8-68442305254024953/k16+11512533954288325/k1-47835010159369970/k9-7301728136139215/k17-239619174006812620/k2-75463854390329720/k10-23685091939075112/k18-73554350289121515/k3-180015331101479808/k11+44169299236693895/k19-5906846296214796/k4-255165537045892/k12+69760430920515810/k20-122920054980484837/k5-76389177280158255/k13-101686694270549663/k21-13143346026582307/k6+92216216458375627/k14-54358503371402448/k22-6469770319426278/k7);

X16=47*(-39913772040405140+11766706708894218/k7-68442305254024953/k14+11512533954288325/k21-47835010159369970/k5-7301728136139215/k12-239619174006812620/k19-75463854390329720/k3-23685091939075112/k10-73554350289121515/k17-180015331101479808/k1+44169299236693895/k8-5906846296214796/k15-255165537045892/k22+69760430920515810/k6-122920054980484837/k13-76389177280158255/k20-101686694270549663/k4-13143346026582307/k11+92216216458375627/k18-54358503371402448/k2-6469770319426278/k9);

X17=47*(-39913772040405140+11766706708894218/k6-68442305254024953/k12+11512533954288325/k18-47835010159369970/k1-7301728136139215/k7-239619174006812620/k13-75463854390329720/k19-23685091939075112/k2-73554350289121515/k8-180015331101479808/k14+44169299236693895/k20-5906846296214796/k3-255165537045892/k9+69760430920515810/k15-122920054980484837/k21-76389177280158255/k4-101686694270549663/k10-13143346026582307/k16+92216216458375627/k22-54358503371402448/k5-6469770319426278/k11);

X18=47*(-39913772040405140+11766706708894218/k5-68442305254024953/k10+11512533954288325/k15-47835010159369970/k20-7301728136139215/k2-239619174006812620/k7-75463854390329720/k12-23685091939075112/k17-73554350289121515/k22-180015331101479808/k4+44169299236693895/k9-5906846296214796/k14-255165537045892/k19+69760430920515810/k1-122920054980484837/k6-76389177280158255/k11-101686694270549663/k16-13143346026582307/k21+92216216458375627/k3-54358503371402448/k8-6469770319426278/k13);

X19=47*(-39913772040405140+11766706708894218/k4-68442305254024953/k8+11512533954288325/k12-47835010159369970/k16-7301728136139215/k20-239619174006812620/k1-75463854390329720/k5-23685091939075112/k9-73554350289121515/k13-180015331101479808/k17+44169299236693895/k21-5906846296214796/k2-255165537045892/k6+69760430920515810/k10-122920054980484837/k14-76389177280158255/k18-101686694270549663/k22-13143346026582307/k3+92216216458375627/k7-54358503371402448/k11-6469770319426278/k15);

X20=47*(-39913772040405140+11766706708894218/k3-68442305254024953/k6+11512533954288325/k9-47835010159369970/k12-7301728136139215/k15-239619174006812620/k18-75463854390329720/k21-23685091939075112/k1-73554350289121515/k4-180015331101479808/k7+44169299236693895/k10-5906846296214796/k13-255165537045892/k16+69760430920515810/k19-122920054980484837/k22-76389177280158255/k2-101686694270549663/k5-13143346026582307/k8+92216216458375627/k11-54358503371402448/k14-6469770319426278/k17);

X21=47*(-39913772040405140+11766706708894218/k2-68442305254024953/k4+11512533954288325/k6-47835010159369970/k8-7301728136139215/k10-239619174006812620/k12-75463854390329720/k14-23685091939075112/k16-73554350289121515/k18-180015331101479808/k20+44169299236693895/k22-5906846296214796/k1-255165537045892/k3+69760430920515810/k5-122920054980484837/k7-76389177280158255/k9-101686694270549663/k11-13143346026582307/k13+92216216458375627/k15-54358503371402448/k17-6469770319426278/k19);

X22=47*(-39913772040405140+11766706708894218/k1-68442305254024953/k2+11512533954288325/k3-47835010159369970/k4-7301728136139215/k5-239619174006812620/k6-75463854390329720/k7-23685091939075112/k8-73554350289121515/k9-180015331101479808/k10+44169299236693895/k11-5906846296214796/k12-255165537045892/k13+69760430920515810/k14-122920054980484837/k15-76389177280158255/k16-101686694270549663/k17-13143346026582307/k18+92216216458375627/k19-54358503371402448/k20-6469770319426278/k21);

切勿忽略以下关系式

E1/X1^(1/23)=1; E2/X2^(1/23)=1; E3/X3^(1/23)=k7; E4/X4^(1/23)=k8; E5/X5^(1/23)=k6;

E6/X6^(1/23)=k5; E7/X7^(1/23)=k20; E8/X8^(1/23)=k4; E9/X9^(1/23)=k17; E10/X10^(1/23)=k5; E11/X11^(1/23)=k3;

最终得出：cos(2^u*pi/47) (u=1, 2, ……23)

· u=1

cos(2*pi/47)=(-1+X1^(1/23)+X22^(1/23)+X2^(1/23)+X21^(1/23)+k7*X3^(1/23)+k16*X20^(1/23)+k8*X4^(1/23)+k15*X19^(1/23)+k6*X5^(1/23)+k17*X18^(1/23)+k5*X6^(1/23)+k18*X17^(1/23)+k20*X7^(1/23)+k3*X16^(1/23)+k4*X8^(1/23)+k19*X15^(1/23)+k17*X9^(1/23)+k6*X14^(1/23)+k5*X10^(1/23)+k18*X13^(1/23)+k3*X11^(1/23)+k20*X12^(1/23))/46;

· u=2

cos(4*pi/47)=(-1+k22*X1^(1/23)+k1*X22^(1/23)+k21*X2^(1/23)+k2*X21^(1/23)+k4*X3^(1/23)+k19*X20^(1/23)+k4*X4^(1/23)+k19*X19^(1/23)+k1*X5^(1/23)+k22*X18^(1/23)+k22*X6^(1/23)+k1*X17^(1/23)+k13*X7^(1/23)+k10*X16^(1/23)+k19*X8^(1/23)+k4*X15^(1/23)+k8*X9^(1/23)+k15*X14^(1/23)+k18*X10^(1/23)+k5*X13^(1/23)+k15*X11^(1/23)+k8*X12^(1/23))/46;

……

一个神奇的发现：u每加上1，X1^(1/23)前面的系数要除以k1，X2^(1/23)前面的系数要除以k2，X3^(1/23)前面的系数要除以k3，X4^(1/23)前面的系数要除以k4，……X22^(1/23)前面的系数要除以k22。后续的式子不列出，大家可以很容易根据这个规律推导出来。

下回书——sin(2*k*pi/47)的解析式！

关键词： C-17