73邊形-(一)
注：w1=(-1+sqrt(-3))/2; w2=(-1-sqrt(-3))/2, j=sqrt(-1)
計算工具：Magma與Matlab。
第一節(jié).求解sin(2*k*pi/73)
I.分組
A111=2*(sin(2*pi/73)+sin(16*pi/73)+sin(128*pi/73));
A112=2*(sin(4*pi/73)+sin(32*pi/73)+sin(110*pi/73));
A113=2*(sin(8*pi/73)+sin(64*pi/73)+sin(74*pi/73));
A121=2*(sin(92*pi/73)+sin(6*pi/73)+sin(48*pi/73));
A122=2*(sin(38*pi/73)+sin(12*pi/73)+sin(96*pi/73));
A123=2*(sin(76*pi/73)+sin(24*pi/73)+sin(46*pi/73));
A211=2*(sin(20*pi/73)+sin(14*pi/73)+sin(112*pi/73));
A212=2*(sin(40*pi/73)+sin(28*pi/73)+sin(78*pi/73));
A213=2*(sin(80*pi/73)+sin(56*pi/73)+sin(10*pi/73));
A221=2*(sin(44*pi/73)+sin(60*pi/73)+sin(42*pi/73));
A222=-2*(sin(58*pi/73)+sin(26*pi/73)+sin(62*pi/73));
A223=2*(sin(30*pi/73)+sin(94*pi/73)+sin(22*pi/73));
?
B111=4*(sin(2*pi/73)*sin(16*pi/73)+sin(16*pi/73)*sin(128*pi/73)+sin(128*pi/73)*sin(2*pi/73));
B112=4*(sin(4*pi/73)*sin(32*pi/73)+sin(32*pi/73)*sin(110*pi/73)+sin(110*pi/73)*sin(4*pi/73));
B113=4*(sin(8*pi/73)*sin(64*pi/73)+sin(64*pi/73)*sin(74*pi/73)+sin(74*pi/73)*sin(8*pi/73));
B121=4*(sin(92*pi/73)*sin(6*pi/73)+sin(6*pi/73)*sin(48*pi/73)+sin(48*pi/73)*sin(92*pi/73));
B122=4*(sin(38*pi/73)*sin(12*pi/73)+sin(12*pi/73)*sin(96*pi/73)+sin(96*pi/73)*sin(38*pi/73));
B123=4*(sin(76*pi/73)*sin(24*pi/73)+sin(24*pi/73)*sin(46*pi/73)+sin(46*pi/73)*sin(76*pi/73));
B211=4*(sin(20*pi/73)*sin(14*pi/73)+sin(14*pi/73)*sin(112*pi/73)+sin(112*pi/73)*sin(20*pi/73));
B212=4*(sin(40*pi/73)*sin(28*pi/73)+sin(28*pi/73)*sin(78*pi/73)+sin(78*pi/73)*sin(40*pi/73));
B213=4*(sin(80*pi/73)*sin(56*pi/73)+sin(56*pi/73)*sin(10*pi/73)+sin(10*pi/73)*sin(80*pi/73));
B221=4*(sin(44*pi/73)*sin(60*pi/73)+sin(60*pi/73)*sin(42*pi/73)+sin(42*pi/73)*sin(44*pi/73));
B222=4*(sin(58*pi/73)*sin(26*pi/73)+sin(26*pi/73)*sin(62*pi/73)+sin(62*pi/73)*sin(58*pi/73));
B223=4*(sin(30*pi/73)*sin(94*pi/73)+sin(94*pi/73)*sin(22*pi/73)+sin(22*pi/73)*sin(30*pi/73));
?
C111=8*(sin(2*pi/73)*sin(16*pi/73)*sin(128*pi/73));
C112=8*(sin(4*pi/73)*sin(32*pi/73)*sin(110*pi/73));
C113=8*(sin(8*pi/73)*sin(64*pi/73)*sin(74*pi/73));
C121=8*(sin(92*pi/73)*sin(6*pi/73)*sin(48*pi/73));
C122=8*(sin(38*pi/73)*sin(12*pi/73)*sin(96*pi/73));
C123=8*(sin(76*pi/73)*sin(24*pi/73)*sin(46*pi/73));
C211=8*(sin(20*pi/73)*sin(14*pi/73)*sin(112*pi/73));
C212=8*(sin(40*pi/73)*sin(28*pi/73)*sin(78*pi/73));
C213=8*(sin(80*pi/73)*sin(56*pi/73)*sin(10*pi/73));
C221=8*(sin(44*pi/73)*sin(60*pi/73)*sin(42*pi/73));
C222=-8*(sin(58*pi/73)*sin(26*pi/73)*sin(62*pi/73));
C223=8*(sin(30*pi/73)*sin(94*pi/73)*sin(22*pi/73));
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II.A B C系列的數(shù)量關系
II.1.A
(A111+A112+A113)^2 + (A121+A122+A123)^2 = (73-sqrt(73))/2;
(A211+A212+A113)^2 + (A221+A222+A223)^2 = (73+sqrt(73))/2;
((A111+A112+A113)^2 - (A121+A122+A123)^2)^2 + ((A211+A212+A213)^2 - (A221+A222+A223)^2)^2
=1241;
((A111+A112+A113)^2 - (A121+A122+A123)^2)^2 - ((A211+A212+A213)^2 - (A221+A222+A223)^2)^2
=19*sqrt(73);
X11=(A111+A112+A113) = sqrt(73-sqrt(73)-sqrt(2482+38*sqrt(73)))/2;
X12=(A121+A122+A123) = sqrt(73-sqrt(73)+sqrt(2482+38*sqrt(73)))/2;
X21=(A211+A212+A213) = sqrt(73+sqrt(73)+sqrt(2482-38*sqrt(73)))/2;
X22=(A221+A222+A223) = sqrt(73+sqrt(73)-sqrt(2482-38*sqrt(73)))/2;
?
A111*A112+A112*A113+A113*A111 + A121*A122+A122*A123+A123*A121=sqrt(73);
A211*A212+A212*A213+A213*A211 + A221*A222+A222*A223+A223*A221=-sqrt(73);
(A111*A112+A112*A113+A113*A111 -(A121*A122+A122*A123+A123*A121))^2+(A211*A212+A212*A213+A213*A211 -( A221*A222+A222*A223+A223*A221))^2=730;
(A111*A112+A112*A113+A113*A111 -(A121*A122+A122*A123+A123*A121))^2-(A211*A212+A212*A213+A213*A211 -( A221*A222+A222*A223+A223*A221))^2=-72*sqrt(73);
Y11=(A111*A112+A112*A113+A113*A111)=(sqrt(73)-sqrt(365-36*sqrt(73)))/2;
Y12=(A121*A122+A122*A123+A123*A121)=(sqrt(73)+sqrt(365-36*sqrt(73)))/2;
Y21=(A211*A212+A212*A213+A213*A211)=(-sqrt(73)+sqrt(365+36*sqrt(73)))/2;
Y22=(A221*A222+A222*A223+A223*A221)=(-sqrt(73)-sqrt(365+36*sqrt(73)))/2;
?
(A111*A112*A113)^2+(A121*A122*A123)^2=730-84*sqrt(73);
(A211*A212*A213)^2+(A221*A222*A223)^2=730+84*sqrt(73);
((A111*A112*A113)^2-(A121*A122*A123)^2)^2+((A211*A212*A213)^2-(A221*A222*A223)^2)^2=2022976;
((A111*A112*A113)^2-(A121*A122*A123)^2)^2-((A211*A212*A213)^2-(A221*A222*A223)^2)^2=236736*sqrt(73);
Z11=(A111*A112*A113)=sqrt(365-42*sqrt(73)-2*sqrt(63218-7398*sqrt(73)));
Z12=(A121*A122*A123)=sqrt(365-42*sqrt(73)+2*sqrt(63218-7398*sqrt(73)));
Z21=(A211*A212*A213)=sqrt(365+42*sqrt(73)-2*sqrt(63218+7398*sqrt(73)));
Z22=(A211*A212*A213)=-sqrt(365+42*sqrt(73)+2*sqrt(63218+7398*sqrt(73)));
?
P11=2*X11^3-9*X11*Y11+27*Z11; K11=P11^2;
P12=2*X12^3-9*X12*Y12+27*Z12; K12=P12^2;
P21=2*X21^3-9*X21*Y21+27*Z21; K21=P21^2;
P22=2*X22^3-9*X22*Y22+27*Z22; K22=P22^2;
K11+K12+K21+K22=261997;
K11+K12-K21-K22=-30571*sqrt(73);
(K11-K12)^2+(K21-K22)^2=68178480305;
(K11-K12)^2-(K21-K22)^2=7979653211*sqrt(73).
P11=sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))/2;
P12=sqrt(261997-30571*sqrt(73)+(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))/2;
P21=sqrt(261997+30571*sqrt(73)+(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))/2;
P22=-sqrt(261997+30571*sqrt(73)-(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))/2;
?
Q11=sqrt(X11^2*Y11^2-27*Z11^2+18*X11*Y11*Z11-4*X11^3*Z11-4*Y11^3);
Q12=sqrt(X12^2*Y12^2-27*Z12^2+18*X12*Y12*Z12-4*X12^3*Z12-4*Y12^3);
Q21=sqrt(X21^2*Y21^2-27*Z21^2+18*X21*Y21*Z21-4*X21^3*Z21-4*Y21^3);
Q22=sqrt(X22^2*Y22^2-27*Z22^2+18*X22*Y22*Z22-4*X22^3*Z22-4*Y22^3);
Q11+Q12+Q21+Q22=23141;
Q11+Q12-Q21-Q22=-2707*sqrt(73);
(Q11-Q12)^2+(Q21-Q22)^2=533224193;
(Q11-Q12)^2-(Q21-Q22)^2=-62409163*sqrt(73);
Q11=sqrt(23141-2707*sqrt(73)+(77-31*sqrt(73))*sqrt(9125-1068*sqrt(73)))/2;
Q12=sqrt(23141-2707*sqrt(73)-(77-31*sqrt(73))*sqrt(9125-1068*sqrt(73)))/2;
Q21=sqrt(23141+2707*sqrt(73)-(77+31*sqrt(73))*sqrt(9125+1068*sqrt(73)))/2;
Q22=sqrt(23141+2707*sqrt(73)+(77+31*sqrt(73))*sqrt(9125+1068*sqrt(73)))/2;
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根據(jù)數(shù)值驗證，得到12個A的值
A111=(X11+w1*(P11+Q11*sqrt(-27))^(1/3)+w2*(P11-Q11*sqrt(-27))^(1/3))/3=(sqrt(73-sqrt(73)-sqrt(2482+38*sqrt(73)))+w1*(2*sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))+6*sqrt(-69423+8121*sqrt(73)-(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w2*(2*sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))-6*sqrt(-69423+8121*sqrt(73)-(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/6;
?
A112=(sqrt(73-sqrt(73)-sqrt(2482+38*sqrt(73)))+w2*(2*sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))+6*sqrt(-69423+8121*sqrt(73)-(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w1*(2*sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))-6*sqrt(-69423+8121*sqrt(73)-(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/6;
?
A113=(sqrt(73-sqrt(73)-sqrt(2482+38*sqrt(73)))+(2*sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))+6*sqrt(-69423+8121*sqrt(73)-(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+(2*sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))-6*sqrt(-69423+8121*sqrt(73)-(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/6;
?
A121=(sqrt(73-sqrt(73)+sqrt(2482+38*sqrt(73)))+w1*(2*sqrt(261997-30571*sqrt(73)+(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))+6*sqrt(-69423+8121*sqrt(73)+(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w2*(2*sqrt(261997-30571*sqrt(73)+(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))-6*sqrt(-69423+8121*sqrt(73)+(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/6;
?
A122=(sqrt(73-sqrt(73)+sqrt(2482+38*sqrt(73)))+w2*(2*sqrt(261997-30571*sqrt(73)+(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))+6*sqrt(-69423+8121*sqrt(73)+(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w1*(2*sqrt(261997-30571*sqrt(73)+(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))-6*sqrt(-69423+8121*sqrt(73)+(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/6;
?
A123=(sqrt(73-sqrt(73)+sqrt(2482+38*sqrt(73)))+(2*sqrt(261997-30571*sqrt(73)+(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))+6*sqrt(-69423+8121*sqrt(73)+(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+(2*sqrt(261997-30571*sqrt(73)+(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))-6*sqrt(-69423+8121*sqrt(73)+(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/6;
?
A211=(X21+w1*(P21+Q21*sqrt(-27))^(1/3)+w2*(P21-Q21*sqrt(-27))^(1/3))/3=(sqrt(73+sqrt(73)+sqrt(2482-38*sqrt(73)))+w1*(2*sqrt(261997+30571*sqrt(73)+(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))+6*sqrt(-69423-8121*sqrt(73)+(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w2*(2*sqrt(261997+30571*sqrt(73)+(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))-6*sqrt(-69423-8121*sqrt(73)+(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/6;
?
A212=(sqrt(73+sqrt(73)+sqrt(2482-38*sqrt(73)))+(2*sqrt(261997+30571*sqrt(73)+(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))+6*sqrt(-69423-8121*sqrt(73)+(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+(2*sqrt(261997+30571*sqrt(73)+(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))-6*sqrt(-69423-8121*sqrt(73)+(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/6;
?
A213=(sqrt(73+sqrt(73)+sqrt(2482-38*sqrt(73)))+w2*(2*sqrt(261997+30571*sqrt(73)+(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))+6*sqrt(-69423-8121*sqrt(73)+(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w1*(2*sqrt(261997+30571*sqrt(73)+(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))-6*sqrt(-69423-8121*sqrt(73)+(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/6;
?
A221=(sqrt(73+sqrt(73)-sqrt(2482-38*sqrt(73)))-w1*(2*sqrt(261997+30571*sqrt(73)-(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))+6*sqrt(-69423-8121*sqrt(73)-(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)-w2*(2*sqrt(261997+30571*sqrt(73)-(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))-6*sqrt(-69423-8121*sqrt(73)-(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/6;
?
A222=(sqrt(73+sqrt(73)-sqrt(2482-38*sqrt(73)))-(2*sqrt(261997+30571*sqrt(73)-(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))+6*sqrt(-69423-8121*sqrt(73)-(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)-(2*sqrt(261997+30571*sqrt(73)-(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))-6*sqrt(-69423-8121*sqrt(73)-(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/6;
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A223=(sqrt(73+sqrt(73)-sqrt(2482-38*sqrt(73)))-w2*(2*sqrt(261997+30571*sqrt(73)-(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))+6*sqrt(-69423-8121*sqrt(73)-(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)-w1*(2*sqrt(261997+30571*sqrt(73)-(4123-935*sqrt(73))*sqrt(9125+1068*sqrt(73)))-6*sqrt(-69423-8121*sqrt(73)-(231+93*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/6;
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II.2.C
C111=-A112, C112=-A113, C113=-A111;
C121=-A122, C122=-A123, C123=-A121;
C211=-A212, C212=-A213, C213=-A211;
C221=-A222, C222=-A223, C223=-A221;
II.3.B
B111+B112+B113+B121+B122+B123 + B211+B212+B213+B221+B222+B223 =0;
B111+B112+B113+B121+B122+B123 - (B211+B212+B213+B221+B222+B223) =-2*sqrt(73);
(B111+B112+B113-(B121+B122+B123))^2+(B211+B212+B213-(B221+B222+B223))^2=146;
(B111+B112+B113-(B121+B122+B123))^2-(B211+B212+B213-(B221+B222+B223))^2=-16*sqrt(73);
R11=B111+B112+B113=(-sqrt(73)-sqrt(73-8*sqrt(73)))/2;
R12=B121+B122+B123=(-sqrt(73)+sqrt(73-8*sqrt(73)))/2;
R21=B211+B212+B213=(sqrt(73)-sqrt(73+8*sqrt(73)))/2;
R22=B221+B222+B223=(sqrt(73)+sqrt(73+8*sqrt(73)))/2;
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B111*B112+B112*B113+B113*B111 + B121*B122+B122*B123+B123*B121 = 0;
B211*B212+B212*B213+B213*B211 + B221*B222+B222*B223+B223*B221 = 0;
S11=B111*B112+B112*B113+B113*B111=sqrt(146-6*sqrt(73))/2;
S12=B121*B122+B122*B123+B123*B121=-sqrt(146-6*sqrt(73))/2;
S21=B211*B212+B212*B213+B213*B211=-sqrt(146+6*sqrt(73))/2;
S22=B221*B222+B222*B223+B223*B221=sqrt(146+6*sqrt(73))/2;
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B111*B112*B113 + B121*B122*B123 + B211*B212*B213 + B221*B222*B223=-73;
B111*B112*B113 + B121*B122*B123 -(B211*B212*B213 + B221*B222*B223)=13*sqrt(73);
(B111*B112*B113 -(B121*B122*B123))^2 + (B211*B212*B213 -(B221*B222*B223))^2=9125;
(B111*B112*B113 -(B121*B122*B123))^2 - (B211*B212*B213 -(B221*B222*B223))^2=-1025*sqrt(73);
T11=B111*B112*B113=(-73+13*sqrt(73)-5*sqrt(730-82*sqrt(73)))/4;
T12=B111*B112*B113=(-73+13*sqrt(73)+5*sqrt(730-82*sqrt(73)))/4;
T21=B211*B212*B213=(-73-13*sqrt(73)+5*sqrt(730+82*sqrt(73)))/4;
T22=B221*B222*B223=(-73-13*sqrt(73)-5*sqrt(730+82*sqrt(73)))/4;
?
K11=2*R11^3-9*R11*S11+27*T11;
K12=2*R12^3-9*R12*S12+27*T12;
K21=2*R21^3-9*R21*S21+27*T21;
K22=2*R22^3-9*R22*S22+27*T22;
K11+K12 + K21+K22=-876; K11+K12 - K21-K22=158*sqrt(73);
(K11-K12)^2 + (K21-K22)^2 = 958490;
(K11-K12)^2 - (K21-K22)^2 = -99496*sqrt(73);
K11=(-438+79*sqrt(73)-sqrt(479245-49748*sqrt(73)))/2;
K12=(-438+79*sqrt(73)+sqrt(479245-49748*sqrt(73)))/2;
K21=(-438-79*sqrt(73)+sqrt(479245+49748*sqrt(73)))/2;
K22=(-438-79*sqrt(73)-sqrt(479245+49748*sqrt(73)))/2;
?注：到這一步，原先求得的K值被覆蓋。
L11=sqrt(R11^2*S11^2-27*T11^2-4*R11^3*T11-4*S11^3+18*R11*S11*T11);
L12=sqrt(R12^2*S12^2-27*T12^2-4*R12^3*T12-4*S12^3+18*R12*S12*T12);
L21=sqrt(R21^2*S21^2-27*T21^2-4*R21^3*T21-4*S21^3+18*R21*S21*T21);
L22=sqrt(R22^2*S22^2-27*T22^2-4*R22^3*T22-4*S22^3+18*R22*S22*T22);
L11+L12+L21+L22=292;
L11+L12-L21-L22=-26*sqrt(73);
(L11-L12)^2+(L21-L22)^2=29930;
(L11-L12)^2-(L21-L22)^2=-3496*sqrt(73);
L11=(146-13*sqrt(73)+sqrt(14965-1748*sqrt(73)))/2;
L12=(146-13*sqrt(73)-sqrt(14965-1748*sqrt(73)))/2;
L21=(146+13*sqrt(73)-sqrt(14965+1748*sqrt(73)))/2;
L22=(146+13*sqrt(73)+sqrt(14965+1748*sqrt(73)))/2;
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B111=(-sqrt(73)-sqrt(73-8*sqrt(73))+w2*(-876+158*sqrt(73)-2*sqrt(479245-49748*sqrt(73))+6*sqrt(3)*j*(146-13*sqrt(73)+sqrt(14965-1748*sqrt(73))))^(1/3)+w1*(-876+158*sqrt(73)-2*sqrt(479245-49748*sqrt(73))-6*sqrt(3)*j*(146-13*sqrt(73)+sqrt(14965-1748*sqrt(73))))^(1/3))/6;
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B112=(-sqrt(73)-sqrt(73-8*sqrt(73))+w1*(-876+158*sqrt(73)-2*sqrt(479245-49748*sqrt(73))+6*sqrt(3)*j*(146-13*sqrt(73)+sqrt(14965-1748*sqrt(73))))^(1/3)+w2*(-876+158*sqrt(73)-2*sqrt(479245-49748*sqrt(73))-6*sqrt(3)*j*(146-13*sqrt(73)+sqrt(14965-1748*sqrt(73))))^(1/3))/6;
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B113=(-sqrt(73)-sqrt(73-8*sqrt(73))+(-876+158*sqrt(73)-2*sqrt(479245-49748*sqrt(73))+6*sqrt(3)*j*(146-13*sqrt(73)+sqrt(14965-1748*sqrt(73))))^(1/3)+(-876+158*sqrt(73)-2*sqrt(479245-49748*sqrt(73))-6*sqrt(3)*j*(146-13*sqrt(73)+sqrt(14965-1748*sqrt(73))))^(1/3))/6;
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B121=(-sqrt(73)+sqrt(73-8*sqrt(73))+w2*(-876+158*sqrt(73)+2*sqrt(479245-49748*sqrt(73))+6*sqrt(3)*j*(146-13*sqrt(73)-sqrt(14965-1748*sqrt(73))))^(1/3)+w1*(-876+158*sqrt(73)+2*sqrt(479245-49748*sqrt(73))-6*sqrt(3)*j*(146-13*sqrt(73)-sqrt(14965-1748*sqrt(73))))^(1/3))/6;
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B122=(-sqrt(73)+sqrt(73-8*sqrt(73))+w1*(-876+158*sqrt(73)+2*sqrt(479245-49748*sqrt(73))+6*sqrt(3)*j*(146-13*sqrt(73)-sqrt(14965-1748*sqrt(73))))^(1/3)+w2*(-876+158*sqrt(73)+2*sqrt(479245-49748*sqrt(73))-6*sqrt(3)*j*(146-13*sqrt(73)-sqrt(14965-1748*sqrt(73))))^(1/3))/6;
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B123=(-sqrt(73)+sqrt(73-8*sqrt(73))+(-876+158*sqrt(73)+2*sqrt(479245-49748*sqrt(73))+6*sqrt(3)*j*(146-13*sqrt(73)-sqrt(14965-1748*sqrt(73))))^(1/3)+(-876+158*sqrt(73)+2*sqrt(479245-49748*sqrt(73))-6*sqrt(3)*j*(146-13*sqrt(73)-sqrt(14965-1748*sqrt(73))))^(1/3))/6;
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B211=(sqrt(73)-sqrt(73+8*sqrt(73))+w1*(-876-158*sqrt(73)+2*sqrt(479245+49748*sqrt(73))+6*sqrt(3)*j*(146+13*sqrt(73)-sqrt(14965+1748*sqrt(73))))^(1/3)+w2*(-876-158*sqrt(73)+2*sqrt(479245+49748*sqrt(73))-6*sqrt(3)*j*(146+13*sqrt(73)-sqrt(14965+1748*sqrt(73))))^(1/3))/6;
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B212=(sqrt(73)-sqrt(73+8*sqrt(73))+(-876-158*sqrt(73)+2*sqrt(479245+49748*sqrt(73))+6*sqrt(3)*j*(146+13*sqrt(73)-sqrt(14965+1748*sqrt(73))))^(1/3)+(-876-158*sqrt(73)+2*sqrt(479245+49748*sqrt(73))-6*sqrt(3)*j*(146+13*sqrt(73)-sqrt(14965+1748*sqrt(73))))^(1/3))/6;
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B213=(sqrt(73)-sqrt(73+8*sqrt(73))+w2*(-876-158*sqrt(73)+2*sqrt(479245+49748*sqrt(73))+6*sqrt(3)*j*(146+13*sqrt(73)-sqrt(14965+1748*sqrt(73))))^(1/3)+w1*(-876-158*sqrt(73)+2*sqrt(479245+49748*sqrt(73))-6*sqrt(3)*j*(146+13*sqrt(73)-sqrt(14965+1748*sqrt(73))))^(1/3))/6;
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B221=(sqrt(73)+sqrt(73+8*sqrt(73))+(-876-158*sqrt(73)-2*sqrt(479245+49748*sqrt(73))+6*sqrt(3)*j*(146+13*sqrt(73)+sqrt(14965+1748*sqrt(73))))^(1/3)+(-876-158*sqrt(73)-2*sqrt(479245+49748*sqrt(73))-6*sqrt(3)*j*(146+13*sqrt(73)+sqrt(14965+1748*sqrt(73))))^(1/3))/6;
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B222=(sqrt(73)+sqrt(73+8*sqrt(73))+w2*(-876-158*sqrt(73)-2*sqrt(479245+49748*sqrt(73))+6*sqrt(3)*j*(146+13*sqrt(73)+sqrt(14965+1748*sqrt(73))))^(1/3)+w1*(-876-158*sqrt(73)-2*sqrt(479245+49748*sqrt(73))-6*sqrt(3)*j*(146+13*sqrt(73)+sqrt(14965+1748*sqrt(73))))^(1/3))/6;
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B223=(sqrt(73)+sqrt(73+8*sqrt(73))+w1*(-876-158*sqrt(73)-2*sqrt(479245+49748*sqrt(73))+6*sqrt(3)*j*(146+13*sqrt(73)+sqrt(14965+1748*sqrt(73))))^(1/3)+w2*(-876-158*sqrt(73)-2*sqrt(479245+49748*sqrt(73))-6*sqrt(3)*j*(146+13*sqrt(73)+sqrt(14965+1748*sqrt(73))))^(1/3))/6;
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III.三次根號內的數(shù)據(jù)
2*實部：M=2*A^3-9*A*B+27*C
2*虛部/sqrt(-27)：N=±sqrt(A^2*B^2-27*C^2+18*A*B*C-4*A^3*C-4*B^3);
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M111=2*A111^3-9*A111*B111+27*C111;
M112=2*A112^3-9*A112*B112+27*C112;
M113=2*A113^3-9*A113*B113+27*C113;
M121=2*A121^3-9*A121*B121+27*C121;
M122=2*A122^3-9*A122*B122+27*C122;
M123=2*A123^3-9*A123*B123+27*C123;
M211=2*A211^3-9*A211*B211+27*C211;
M212=2*A212^3-9*A212*B212+27*C212;
M213=2*A213^3-9*A213*B213+27*C213;
M221=2*A221^3-9*A221*B221+27*C221;
M222=2*A222^3-9*A222*B222+27*C222;
M223=2*A223^3-9*A223*B223+27*C223;
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(M111+M112+M113)^2 + (M121+M122+M123)^2 + (M211+M212+M213)^2 + (M221+M222+M223)^2 =24382;
(M111+M112+M113)^2 + (M121+M122+M123)^2 - (M211+M212+M213)^2 - (M221+M222+M223)^2
=-1000*sqrt(73);
((M111+M112+M113)^2-(M121+M122+M123)^2)^2+((M211+M212+M213)^2-(M221+M222+M223)^2)^2=57959810;
((M111+M112+M113)^2-(M121+M122+M123)^2)^2-((M211+M212+M213)^2-(M221+M222+M223)^2)^2=590032*sqrt(73);
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a11=(M111+M112+M113)=-sqrt(24382-1000*sqrt(73)-2*sqrt(28979905+295016*sqrt(73)))/2;
a12=(M121+M122+M123)=-sqrt(24382-1000*sqrt(73)+2*sqrt(28979905+295016*sqrt(73)))/2;
a21=(M211+M212+M213)=-sqrt(24382+1000*sqrt(73)+2*sqrt(28979905-295016*sqrt(73)))/2;
a22=(M221+M222+M223)=-sqrt(24382+1000*sqrt(73)-2*sqrt(28979905-295016*sqrt(73)))/2;
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M111*M112+M112*M113+M113*M111+M121*M122+M122*M123+M123*M121+M211*M212+M212*M213+M213*M211+M221*M222+M222*M223+M223*M221=4818;
M111*M112+M112*M113+M113*M111+M121*M122+M122*M123+M123*M121-(M211*M212+M212*M213+M213*M211+M221*M222+M222*M223+M223*M221)=-64*sqrt(73);
(M111*M112+M112*M113+M113*M111-(M121*M122+M122*M123+M123*M121))^2+(M211*M212+M212*M213+M213*M211-(M221*M222+M222*M223+M223*M221))^2=8688898;
(M111*M112+M112*M113+M113*M111-(M121*M122+M122*M123+M123*M121))^2-(M211*M212+M212*M213+M213*M211-(M221*M222+M222*M223+M223*M221))^2=-351552*sqrt(73);
b11=M111*M112+M112*M113+M113*M111=(2409-32*sqrt(73)-sqrt(4344449-175776*sqrt(73)))/2;
b12=M121*M122+M122*M123+M123*M121=(2409-32*sqrt(73)+sqrt(4344449-175776*sqrt(73)))/2;
b21=M211*M212+M212*M213+M213*M211=(2409+32*sqrt(73)+sqrt(4344449+175776*sqrt(73)))/2;
b22=M211*M212+M212*M213+M213*M211=(2409+32*sqrt(73)-sqrt(4344449+175776*sqrt(73)))/2;
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(M111*M112*M113)^2 + (M121*M122*M123)^2 + (M211*M212*M213)^2 + (M221*M222*M223)^2 =375370964;
(M111*M112*M113)^2 + (M121*M122*M123)^2 - (M211*M212*M213)^2 - (M221*M222*M223)^2 =-451368*sqrt(73);
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((M111*M112*M113)^2-(M121*M122*M123)^2)^2+((M211*M212*M213)^2-(M221*M222*M223)^2)^2 =70429682701883200;
((M111*M112*M113)^2-(M121*M122*M123)^2)^2-((M211*M212*M213)^2-(M221*M222*M223)^2)^2=-172444776163776*sqrt(73);
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c11=M111*M112*M113=-sqrt(93842741-112842*sqrt(73)-2*sqrt(2200927584433850-5388899255118*sqrt(73)));
c12=M121*M122*M123=-sqrt(93842741-112842*sqrt(73)+2*sqrt(2200927584433850-5388899255118*sqrt(73)));
c21=M211*M212*M213=-sqrt(93842741+112842*sqrt(73)+2*sqrt(2200927584433850+5388899255118*sqrt(73)));
c22=M221*M222*M223=-sqrt(93842741+112842*sqrt(73)-2*sqrt(2200927584433850+5388899255118*sqrt(73)));
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實部內的三次根號內：都小于0
實部f11=2*a11^3-9*a11*b11+27*c11;
f12=2*a12^3-9*a12*b12+27*c12;
f21=2*a21^3-9*a21*b21+27*c21;
f22=2*a22^3-9*a22*b22+27*c22;
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f11^2+f12^2+f21^2+f22^2 = 641143109068;
f11^2+f12^2-f21^2-f22^2 = -73955454496*sqrt(73);
(f11^2-f12^2)^2+(f21^2-f22^2)^2 = 279677368778228715099776;
(f11^2-f12^2)^2-(f21^2-f22^2)^2 = -32729642415614441739776*sqrt(73);
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f11=-sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)));
f12=-sqrt(160285777267-18488863624*sqrt(73)+4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)));
f21=-sqrt(160285777267+18488863624*sqrt(73)+4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)));
f22=-sqrt(160285777267+18488863624*sqrt(73)-4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)));
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虛部：g11=a11^2*b11^2-27*c11^2+18*a11*b11*c11-4*a11^3*c11-4*b11^3;
g12=a12^2*b12^2-27*c12^2+18*a12*b12*c12-4*a12^3*c12-4*b12^3;
g21=a21^2*b21^2-27*c21^2+18*a21*b21*c21-4*a21^3*c21-4*b21^3;
g22=a22^2*b22^2-27*c22^2+18*a22*b22*c22-4*a22^3*c22-4*b22^3;
g11+g12 + g21+g22 =2672199164;
g11+g12 - g21-g22 =-310854976*sqrt(73);
(g11-g12)^2 + (g21-g22)^2=6942502868789189120;
((g11-g12)^2 - (g21-g22)^2)/sqrt(73)=-812543041578735616*sqrt(73);
g11=668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73));
g12=668049791-77713744*sqrt(73)-8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73));
g21=668049791+77713744*sqrt(73)-8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73));
g22=668049791+77713744*sqrt(73)+8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73));
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M111/2=-(sqrt(24382-1000*sqrt(73)-2*sqrt(28979905+295016*sqrt(73)))+w2*(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w1*(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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M112/2=-(sqrt(24382-1000*sqrt(73)-2*sqrt(28979905+295016*sqrt(73)))+(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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M113/2=-(sqrt(24382-1000*sqrt(73)-2*sqrt(28979905+295016*sqrt(73)))+w1*(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w2*(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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M121/2=-(sqrt(24382-1000*sqrt(73)+2*sqrt(28979905+295016*sqrt(73)))+w2*(4*sqrt(160285777267-18488863624*sqrt(73)+4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)-8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w1*(4*sqrt(160285777267-18488863624*sqrt(73)+4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)-8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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M122/2=-(sqrt(24382-1000*sqrt(73)+2*sqrt(28979905+295016*sqrt(73)))+(4*sqrt(160285777267-18488863624*sqrt(73)+4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)-8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+(4*sqrt(160285777267-18488863624*sqrt(73)+4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)-8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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M123/2=-(sqrt(24382-1000*sqrt(73)+2*sqrt(28979905+295016*sqrt(73)))+w1*(4*sqrt(160285777267-18488863624*sqrt(73)+4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)-8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w2*(4*sqrt(160285777267-18488863624*sqrt(73)+4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)-8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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M211/2=-(sqrt(24382+1000*sqrt(73)+2*sqrt(28979905-295016*sqrt(73)))+(4*sqrt(160285777267+18488863624*sqrt(73)+4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)-8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+(4*sqrt(160285777267+18488863624*sqrt(73)+4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)-8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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M212/2=-(sqrt(24382+1000*sqrt(73)+2*sqrt(28979905-295016*sqrt(73)))+w2*(4*sqrt(160285777267+18488863624*sqrt(73)+4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)-8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w1*(4*sqrt(160285777267+18488863624*sqrt(73)+4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)-8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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M213/2=-(sqrt(24382+1000*sqrt(73)+2*sqrt(28979905-295016*sqrt(73)))+w1*(4*sqrt(160285777267+18488863624*sqrt(73)+4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)-8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w2*(4*sqrt(160285777267+18488863624*sqrt(73)+4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)-8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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M221/2=-(sqrt(24382+1000*sqrt(73)-2*sqrt(28979905-295016*sqrt(73)))+w2*(4*sqrt(160285777267+18488863624*sqrt(73)-4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)+8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w1*(4*sqrt(160285777267+18488863624*sqrt(73)-4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)+8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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M222/2=-(sqrt(24382+1000*sqrt(73)-2*sqrt(28979905-295016*sqrt(73)))+w1*(4*sqrt(160285777267+18488863624*sqrt(73)-4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)+8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w2*(4*sqrt(160285777267+18488863624*sqrt(73)-4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)+8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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M223/2=-(sqrt(24382+1000*sqrt(73)-2*sqrt(28979905-295016*sqrt(73)))+(4*sqrt(160285777267+18488863624*sqrt(73)-4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)+8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+(4*sqrt(160285777267+18488863624*sqrt(73)-4*(3900024464-513733999*sqrt(73))*sqrt(9125+1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791+77713744*sqrt(73)+8*(3405824-541295*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12
N111=sqrt(A111^2*B111^2-27*C111^2+18*A111*B111*C111-4*A111^3*C111-4*B111^3);
N112=sqrt(A112^2*B112^2-27*C112^2+18*A112*B112*C112-4*A112^3*C112-4*B112^3);
N113=sqrt(A113^2*B113^2-27*C113^2+18*A113*B113*C113-4*A113^3*C113-4*B113^3);
N121=-sqrt(A121^2*B121^2-27*C121^2+18*A121*B121*C121-4*A121^3*C121-4*B121^3);
N122=sqrt(A122^2*B122^2-27*C122^2+18*A122*B122*C122-4*A122^3*C122-4*B122^3);
N123=-sqrt(A123^2*B123^2-27*C123^2+18*A123*B123*C123-4*A123^3*C123-4*B123^3);
N211=sqrt(A211^2*B211^2-27*C211^2+18*A211*B211*C211-4*A211^3*C211-4*B211^3);
N212=sqrt(A212^2*B212^2-27*C212^2+18*A212*B212*C212-4*A212^3*C212-4*B212^3);
N213=sqrt(A213^2*B213^2-27*C213^2+18*A213*B213*C213-4*A213^3*C213-4*B213^3);
N221=-sqrt(A221^2*B221^2-27*C221^2+18*A221*B221*C221-4*A221^3*C221-4*B221^3);
N222=-sqrt(A222^2*B222^2-27*C222^2+18*A222*B222*C222-4*A222^3*C222-4*B222^3);
N223=sqrt(A223^2*B223^2-27*C223^2+18*A223*B223*C223-4*A223^3*C223-4*B223^3);
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(N111+N112+N113)^2 + (N121+N122+N123)^2 =219+20*sqrt(73);
(N211+N212+N213)^2 + (N221+N222+N223)^2 =219-20*sqrt(73);
((N111+N112+N113)^2-(N121+N122+N123)^2)^2+((N211+N212+N213)^2-(N221+N222+N223)^2)^2
=149650;
((N111+N112+N113)^2-(N121+N122+N123)^2)^2-((N211+N212+N213)^2-(N221+N222+N223)^2)^2
=17328*sqrt(73);
k11=N111+N112+N113=sqrt(438+40*sqrt(73)+2*sqrt(74825+8664*sqrt(73)))/2;
k12=N121+N122+N123=sqrt(438+40*sqrt(73)-2*sqrt(74825+8664*sqrt(73)))/2;
k21=N211+N212+N213=sqrt(438-40*sqrt(73)+2*sqrt(74825-8664*sqrt(73)))/2;
k22=N221+N222+N223=sqrt(438-40*sqrt(73)-2*sqrt(74825-8664*sqrt(73)))/2;
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N111*N112+N112*N113+N113*N111+N121*N122+N122*N123+N123*N121+N211*N212+N212*N213+N213*N211+N221*N222+N222*N223+N223*N221 = 0;
N111*N112+N112*N113+N113*N111+N121*N122+N122*N123+N123*N121= -sqrt(73);
(N111*N112+N112*N113+N113*N111-(N121*N122+N122*N123+N123*N121))^2+(N211*N212+N212*N213+N213*N211-(N221*N222+N222*N223+N223*N221))^2=22338;
(N111*N112+N112*N113+N113*N111-(N121*N122+N122*N123+N123*N121))^2-(N211*N212+N212*N213+N213*N211-(N221*N222+N222*N223+N223*N221))^2=2592*sqrt(73);
l11=N111*N112+N112*N113+N113*N111=(-sqrt(73)+3*sqrt(1241+144*sqrt(73)))/2;
l12=N121*N122+N122*N123+N123*N121=(-sqrt(73)-3*sqrt(1241+144*sqrt(73)))/2;
l21=N211*N212+N212*N213+N213*N211=(sqrt(73)+3*sqrt(1241-144*sqrt(73)))/2;
l22=N221*N222+N222*N223+N223*N221=(sqrt(73)-3*sqrt(1241-144*sqrt(73)))/2;
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(N111*N112*N113)^2+(N121*N122*N123)^2+(N211*N212*N213)^2+(N221*N222*N223)^2=1460;
(N111*N112*N113)^2+(N121*N122*N123)^2-(N211*N212*N213)^2-(N221*N222*N223)^2=168*sqrt(73);
((N111*N112*N113)^2-(N121*N122*N123)^2)^2+((N211*N212*N213)^2-(N221*N222*N223)^2)^2=2022976;
(((N111*N112*N113)^2-(N121*N122*N123)^2)^2-((N211*N212*N213)^2-(N221*N222*N223)^2)^2)/sqrt(73)=236736;
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m11=N111*N112*N113=sqrt(365+42*sqrt(73)-2*sqrt(63218+7398*sqrt(73)));
m12=N121*N122*N123=sqrt(365+42*sqrt(73)+2*sqrt(63218+7398*sqrt(73)));
m21=N211*N212*N213=sqrt(365-42*sqrt(73)+2*sqrt(63218-7398*sqrt(73)));
m22=N221*N222*N223=sqrt(365-42*sqrt(73)-2*sqrt(63218-7398*sqrt(73)));
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u11=2*k11^3-9*k11*l11+27*m11;
u12=2*k12^3-9*k12*l12+27*m12;
u21=2*k21^3-9*k21*l21+27*m21;
u22=2*k22^3-9*k22*l22+27*m22;
u11^2+u12^2+u21^2+u22^2=12659076;
u11^2+u12^2-u21^2-u22^2=1479352*sqrt(73);
(u11^2-u12^2)^2+(u21^2-u22^2)^2=18667365014080;
(u11^2-u12^2)^2-(u21^2-u22^2)^2=2184847282368*sqrt(73);
u11=sqrt(3164769+369838*sqrt(73)-(5010-2458*sqrt(73))*sqrt(9125+1068*sqrt(73)));
u12=sqrt(3164769+369838*sqrt(73)+(5010-2458*sqrt(73))*sqrt(9125+1068*sqrt(73)));
u21=sqrt(3164769-369838*sqrt(73)-(5010+2458*sqrt(73))*sqrt(9125-1068*sqrt(73)));
u22=sqrt(3164769-369838*sqrt(73)+(5010+2458*sqrt(73))*sqrt(9125-1068*sqrt(73)));
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v11=k11^2*l11^2-27*m11^2+18*k11*l11*m11-4*k11^3*m11-4*l11^3;
v12=k12^2*l12^2-27*m12^2+18*k12*l12*m12-4*k12^3*m12-4*l12^3;
v21=k21^2*l21^2-27*m21^2+18*k21*l21*m21-4*k21^3*m21-4*l21^3;
v22=k22^2*l22^2-27*m22^2+18*k22*l22*m22-4*k22^3*m22-4*l22^3;
v11+v12+v21+v22=2363740;
v11+v12-v21-v22=276640*sqrt(73);
(v11-v12)^2+(v21-v22)^2=1823599147520;
(v11-v12)^2-(v21-v22)^2=213436135424*sqrt(73);
v11=(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73)));
v12=(590935+69160*sqrt(73)+(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73)));
v21=(590935-69160*sqrt(73)-(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73)));
v22=(590935-69160*sqrt(73)+(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73)));
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N111/2=(sqrt(438+40*sqrt(73)+2*sqrt(74825+8664*sqrt(73)))+w2*(4*sqrt(3164769+369838*sqrt(73)-(5010-2458*sqrt(73))*sqrt(9125+1068*sqrt(73)))+12*sqrt(3)*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w1*(4*sqrt(3164769+369838*sqrt(73)-(5010-2458*sqrt(73))*sqrt(9125+1068*sqrt(73)))-12*sqrt(3)*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N111/2=3*(sqrt(1314+120*sqrt(73)+6*sqrt(74825+8664*sqrt(73)))+w2*(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))+108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w1*(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))-108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N112/2=3*(sqrt(1314+120*sqrt(73)+6*sqrt(74825+8664*sqrt(73)))+(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))+108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))-108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N113/2=3*(sqrt(1314+120*sqrt(73)+6*sqrt(74825+8664*sqrt(73)))+w1*(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))+108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w2*(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))-108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N121/2=3*(sqrt(1314+120*sqrt(73)-6*sqrt(74825+8664*sqrt(73)))+w1*(12*sqrt(9494307+1109514*sqrt(73)+(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))+108*j*sqrt(590935+69160*sqrt(73)+(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w2*(12*sqrt(9494307+1109514*sqrt(73)+(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))-108*j*sqrt(590935+69160*sqrt(73)+(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N122/2=3*(sqrt(1314+120*sqrt(73)-6*sqrt(74825+8664*sqrt(73)))+(12*sqrt(9494307+1109514*sqrt(73)+(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))+108*j*sqrt(590935+69160*sqrt(73)+(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+(12*sqrt(9494307+1109514*sqrt(73)+(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))-108*j*sqrt(590935+69160*sqrt(73)+(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N123/2=3*(sqrt(1314+120*sqrt(73)-6*sqrt(74825+8664*sqrt(73)))+w2*(12*sqrt(9494307+1109514*sqrt(73)+(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))+108*j*sqrt(590935+69160*sqrt(73)+(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w1*(12*sqrt(9494307+1109514*sqrt(73)+(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))-108*j*sqrt(590935+69160*sqrt(73)+(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N211/2=3*(sqrt(1314-120*sqrt(73)+6*sqrt(74825-8664*sqrt(73)))+(12*sqrt(9494307-1109514*sqrt(73)-(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))+108*j*sqrt(590935-69160*sqrt(73)-(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+(12*sqrt(9494307-1109514*sqrt(73)-(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))-108*j*sqrt(590935-69160*sqrt(73)-(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N212/2=3*(sqrt(1314-120*sqrt(73)+6*sqrt(74825-8664*sqrt(73)))+w1*(12*sqrt(9494307-1109514*sqrt(73)-(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))+108*j*sqrt(590935-69160*sqrt(73)-(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w2*(12*sqrt(9494307-1109514*sqrt(73)-(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))-108*j*sqrt(590935-69160*sqrt(73)-(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N213/2=3*(sqrt(1314-120*sqrt(73)+6*sqrt(74825-8664*sqrt(73)))+w2*(12*sqrt(9494307-1109514*sqrt(73)-(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))+108*j*sqrt(590935-69160*sqrt(73)-(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w1*(12*sqrt(9494307-1109514*sqrt(73)-(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))-108*j*sqrt(590935-69160*sqrt(73)-(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N221/2=3*(sqrt(1314-120*sqrt(73)-6*sqrt(74825-8664*sqrt(73)))+w2*(12*sqrt(9494307-1109514*sqrt(73)+(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))+108*j*sqrt(590935-69160*sqrt(73)+(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w1*(12*sqrt(9494307-1109514*sqrt(73)+(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))-108*j*sqrt(590935-69160*sqrt(73)+(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N222/2=3*(sqrt(1314-120*sqrt(73)-6*sqrt(74825-8664*sqrt(73)))+w1*(12*sqrt(9494307-1109514*sqrt(73)+(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))+108*j*sqrt(590935-69160*sqrt(73)+(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w2*(12*sqrt(9494307-1109514*sqrt(73)+(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))-108*j*sqrt(590935-69160*sqrt(73)+(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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3*sqrt(3)*N223/2=3*(sqrt(1314-120*sqrt(73)-6*sqrt(74825-8664*sqrt(73)))+(12*sqrt(9494307-1109514*sqrt(73)+(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))+108*j*sqrt(590935-69160*sqrt(73)+(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+(12*sqrt(9494307-1109514*sqrt(73)+(15030+7374*sqrt(73))*sqrt(9125-1068*sqrt(73)))-108*j*sqrt(590935-69160*sqrt(73)+(2264-320*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))/12;
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IV. 最后一步
sin(2*k*pi/73)的表達式形式是(A+z1*(M/2+N*sqrt(-27)/2)^(1/3)+z2*(M/2+N*sqrt(-27)/2)^(1/3))/6，且A、M、N的下標相同，同一個下標對應三個不同的k，任一個k唯一對應三次單位根z1、z2且z1*z2=1?？梢酝ㄟ^化簡最終組合出以下表達式，見下例，其他類推。
sin(2*pi/73)=(sqrt(73-sqrt(73)-sqrt(2482+38*sqrt(73)))+w1*(2*sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))+6*sqrt(-69423+8121*sqrt(73)-(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w2*(2*sqrt(261997-30571*sqrt(73)-(4123+935*sqrt(73))*sqrt(9125-1068*sqrt(73)))-6*sqrt(-69423+8121*sqrt(73)-(231-93*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w2*(-18*(sqrt(24382-1000*sqrt(73)-2*sqrt(28979905+295016*sqrt(73)))+w2*(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w1*(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))+54*j*(sqrt(1314+120*sqrt(73)+6*sqrt(74825+8664*sqrt(73)))+w2*(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))+108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w1*(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))-108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)))^(1/3)+w1*(-18*(sqrt(24382-1000*sqrt(73)-2*sqrt(28979905+295016*sqrt(73)))+w2*(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))+12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3)+w1*(4*sqrt(160285777267-18488863624*sqrt(73)-4*(3900024464+513733999*sqrt(73))*sqrt(9125-1068*sqrt(73)))-12*sqrt(-3)*sqrt(668049791-77713744*sqrt(73)+8*(3405824+541295*sqrt(73))*sqrt(9125-1068*sqrt(73))))^(1/3))-54*j*(sqrt(1314+120*sqrt(73)+6*sqrt(74825+8664*sqrt(73)))+w2*(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))+108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)+w1*(12*sqrt(9494307+1109514*sqrt(73)-(15030-7374*sqrt(73))*sqrt(9125+1068*sqrt(73)))-108*j*sqrt(590935+69160*sqrt(73)-(2264+320*sqrt(73))*sqrt(9125+1068*sqrt(73))))^(1/3)))^(1/3))/36
這個式子的形式同sin(k*pi/19)、sin(k*pi/37)的格式一樣，不過因為有了更多的二次根式，式子出奇地長而且和sin(k*pi/97)的表達式不相上下。
下集預告 求cos(2*k*pi/73) !!!!!
