The continuation of the last passage.
(资料图片)
3. sin(2*n*pi/53)
Case 1: n^13=1 mod 53, n=1, 10, 47, 46, 36, 42, 49, 13, 24, 28, 15, 44, 16
r0=2*(sin(2*pi/53)+sin(20*pi/53)+sin(94*pi/53)+sin(92*pi/53)+sin(72*pi/53)+sin(84*pi/53)+sin(98*pi/53)+sin(26*pi/53)+sin(48*pi/53)+sin(56*pi/53)+sin(30*pi/53)+sin(88*pi/53)+sin(32*pi/53))=-sqrt(106-14*sqrt(53))/2;
r1=2*(sin(2*pi/53)+t1*sin(20*pi/53)+t2*sin(94*pi/53)+t3*sin(92*pi/53)+t4*sin(72*pi/53)+t5*sin(84*pi/53)+t6*sin(98*pi/53)+t7*sin(26*pi/53)+t8*sin(48*pi/53)+t9*sin(56*pi/53)+t10*sin(30*pi/53)+t11*sin(88*pi/53)+t12*sin(32*pi/53));
r2=2*(sin(2*pi/53)+t2*sin(20*pi/53)+t4*sin(94*pi/53)+t6*sin(92*pi/53)+t8*sin(72*pi/53)+t10*sin(84*pi/53)+t12*sin(98*pi/53)+t1*sin(26*pi/53)+t3*sin(48*pi/53)+t5*sin(56*pi/53)+t7*sin(30*pi/53)+t9*sin(88*pi/53)+t11*sin(32*pi/53));
r3=2*(sin(2*pi/53)+t3*sin(20*pi/53)+t6*sin(94*pi/53)+t9*sin(92*pi/53)+t12*sin(72*pi/53)+t2*sin(84*pi/53)+t5*sin(98*pi/53)+t8*sin(26*pi/53)+t11*sin(48*pi/53)+t1*sin(56*pi/53)+t4*sin(30*pi/53)+t7*sin(88*pi/53)+t10*sin(32*pi/53));
r4=2*(sin(2*pi/53)+t4*sin(20*pi/53)+t8*sin(94*pi/53)+t12*sin(92*pi/53)+t3*sin(72*pi/53)+t7*sin(84*pi/53)+t11*sin(98*pi/53)+t2*sin(26*pi/53)+t6*sin(48*pi/53)+t10*sin(56*pi/53)+t1*sin(30*pi/53)+t5*sin(88*pi/53)+t9*sin(32*pi/53));
r5=2*(sin(2*pi/53)+t5*sin(20*pi/53)+t10*sin(94*pi/53)+t2*sin(92*pi/53)+t7*sin(72*pi/53)+t12*sin(84*pi/53)+t4*sin(98*pi/53)+t9*sin(26*pi/53)+t1*sin(48*pi/53)+t6*sin(56*pi/53)+t11*sin(30*pi/53)+t3*sin(88*pi/53)+t8*sin(32*pi/53));
r6=2*(sin(2*pi/53)+t6*sin(20*pi/53)+t12*sin(94*pi/53)+t5*sin(92*pi/53)+t11*sin(72*pi/53)+t4*sin(84*pi/53)+t10*sin(98*pi/53)+t3*sin(26*pi/53)+t9*sin(48*pi/53)+t2*sin(56*pi/53)+t8*sin(30*pi/53)+t1*sin(88*pi/53)+t7*sin(32*pi/53));
r7=2*(sin(2*pi/53)+t7*sin(20*pi/53)+t1*sin(94*pi/53)+t8*sin(92*pi/53)+t2*sin(72*pi/53)+t9*sin(84*pi/53)+t3*sin(98*pi/53)+t10*sin(26*pi/53)+t4*sin(48*pi/53)+t11*sin(56*pi/53)+t5*sin(30*pi/53)+t12*sin(88*pi/53)+t6*sin(32*pi/53));
r8=2*(sin(2*pi/53)+t8*sin(20*pi/53)+t3*sin(94*pi/53)+t11*sin(92*pi/53)+t6*sin(72*pi/53)+t1*sin(84*pi/53)+t9*sin(98*pi/53)+t4*sin(26*pi/53)+t12*sin(48*pi/53)+t7*sin(56*pi/53)+t2*sin(30*pi/53)+t10*sin(88*pi/53)+t5*sin(32*pi/53));
r9=2*(sin(2*pi/53)+t9*sin(20*pi/53)+t5*sin(94*pi/53)+t1*sin(92*pi/53)+t10*sin(72*pi/53)+t6*sin(84*pi/53)+t2*sin(98*pi/53)+t11*sin(26*pi/53)+t7*sin(48*pi/53)+t3*sin(56*pi/53)+t12*sin(30*pi/53)+t8*sin(88*pi/53)+t4*sin(32*pi/53));
r10=2*(sin(2*pi/53)+t10*sin(20*pi/53)+t7*sin(94*pi/53)+t4*sin(92*pi/53)+t1*sin(72*pi/53)+t11*sin(84*pi/53)+t8*sin(98*pi/53)+t5*sin(26*pi/53)+t2*sin(48*pi/53)+t12*sin(56*pi/53)+t9*sin(30*pi/53)+t6*sin(88*pi/53)+t3*sin(32*pi/53));
r11=2*(sin(2*pi/53)+t11*sin(20*pi/53)+t9*sin(94*pi/53)+t7*sin(92*pi/53)+t5*sin(72*pi/53)+t3*sin(84*pi/53)+t1*sin(98*pi/53)+t12*sin(26*pi/53)+t10*sin(48*pi/53)+t8*sin(56*pi/53)+t6*sin(30*pi/53)+t4*sin(88*pi/53)+t2*sin(32*pi/53));
r12=2*(sin(2*pi/53)+t12*sin(20*pi/53)+t11*sin(94*pi/53)+t10*sin(92*pi/53)+t9*sin(72*pi/53)+t8*sin(84*pi/53)+t7*sin(98*pi/53)+t6*sin(26*pi/53)+t5*sin(48*pi/53)+t4*sin(56*pi/53)+t3*sin(30*pi/53)+t2*sin(88*pi/53)+t1*sin(32*pi/53)).
Case 2: n^13=23 mod 53, n=23, 18, 21, 51, 33, 12, 14, 34, 22, 8, 27, 5, 50
s0=2*(sin(46*pi/53)+sin(36*pi/53)+sin(42*pi/53)+sin(102*pi/53)+sin(66*pi/53)+sin(24*pi/53)+sin(28*pi/53)+sin(68*pi/53)+sin(44*pi/53)+sin(16*pi/53)+sin(54*pi/53)+sin(10*pi/53)+sin(100*pi/53))=sqrt(106+14*sqrt(53))/2;
s1=2*(sin(46*pi/53)+t1*sin(36*pi/53)+t2*sin(42*pi/53)+t3*sin(102*pi/53)+t4*sin(66*pi/53)+t5*sin(24*pi/53)+t6*sin(28*pi/53)+t7*sin(68*pi/53)+t8*sin(44*pi/53)+t9*sin(16*pi/53)+t10*sin(54*pi/53)+t11*sin(10*pi/53)+t12*sin(100*pi/53));
s2=2*(sin(46*pi/53)+t2*sin(36*pi/53)+t4*sin(42*pi/53)+t6*sin(102*pi/53)+t8*sin(66*pi/53)+t10*sin(24*pi/53)+t12*sin(28*pi/53)+t1*sin(68*pi/53)+t3*sin(44*pi/53)+t5*sin(16*pi/53)+t7*sin(54*pi/53)+t9*sin(10*pi/53)+t11*sin(100*pi/53));
s3=2*(sin(46*pi/53)+t3*sin(36*pi/53)+t6*sin(42*pi/53)+t9*sin(102*pi/53)+t12*sin(66*pi/53)+t2*sin(24*pi/53)+t5*sin(28*pi/53)+t8*sin(68*pi/53)+t11*sin(44*pi/53)+t1*sin(16*pi/53)+t4*sin(54*pi/53)+t7*sin(10*pi/53)+t10*sin(100*pi/53));
s4=2*(sin(46*pi/53)+t4*sin(36*pi/53)+t8*sin(42*pi/53)+t12*sin(102*pi/53)+t3*sin(66*pi/53)+t7*sin(24*pi/53)+t11*sin(28*pi/53)+t2*sin(68*pi/53)+t6*sin(44*pi/53)+t10*sin(16*pi/53)+t1*sin(54*pi/53)+t5*sin(10*pi/53)+t9*sin(100*pi/53));
s5=2*(sin(46*pi/53)+t5*sin(36*pi/53)+t10*sin(42*pi/53)+t2*sin(102*pi/53)+t7*sin(66*pi/53)+t12*sin(24*pi/53)+t4*sin(28*pi/53)+t9*sin(68*pi/53)+t1*sin(44*pi/53)+t6*sin(16*pi/53)+t11*sin(54*pi/53)+t3*sin(10*pi/53)+t8*sin(100*pi/53));
s6=2*(sin(46*pi/53)+t6*sin(36*pi/53)+t12*sin(42*pi/53)+t5*sin(102*pi/53)+t11*sin(66*pi/53)+t4*sin(24*pi/53)+t10*sin(28*pi/53)+t3*sin(68*pi/53)+t9*sin(44*pi/53)+t2*sin(16*pi/53)+t8*sin(54*pi/53)+t1*sin(10*pi/53)+t7*sin(100*pi/53));
s7=2*(sin(46*pi/53)+t7*sin(36*pi/53)+t1*sin(42*pi/53)+t8*sin(102*pi/53)+t2*sin(66*pi/53)+t9*sin(24*pi/53)+t3*sin(28*pi/53)+t10*sin(68*pi/53)+t4*sin(44*pi/53)+t11*sin(16*pi/53)+t5*sin(54*pi/53)+t12*sin(10*pi/53)+t6*sin(100*pi/53));
s8=2*(sin(46*pi/53)+t8*sin(36*pi/53)+t3*sin(42*pi/53)+t11*sin(102*pi/53)+t6*sin(66*pi/53)+t1*sin(24*pi/53)+t9*sin(28*pi/53)+t4*sin(68*pi/53)+t12*sin(44*pi/53)+t7*sin(16*pi/53)+t2*sin(54*pi/53)+t10*sin(10*pi/53)+t5*sin(100*pi/53));
s9=2*(sin(46*pi/53)+t9*sin(36*pi/53)+t5*sin(42*pi/53)+t1*sin(102*pi/53)+t10*sin(66*pi/53)+t6*sin(24*pi/53)+t2*sin(28*pi/53)+t11*sin(68*pi/53)+t7*sin(44*pi/53)+t3*sin(16*pi/53)+t12*sin(54*pi/53)+t8*sin(10*pi/53)+t4*sin(100*pi/53));
s10=2*(sin(46*pi/53)+t10*sin(36*pi/53)+t7*sin(42*pi/53)+t4*sin(102*pi/53)+t1*sin(66*pi/53)+t11*sin(24*pi/53)+t8*sin(28*pi/53)+t5*sin(68*pi/53)+t2*sin(44*pi/53)+t12*sin(16*pi/53)+t9*sin(54*pi/53)+t6*sin(10*pi/53)+t3*sin(100*pi/53));
s11=2*(sin(46*pi/53)+t11*sin(36*pi/53)+t9*sin(42*pi/53)+t7*sin(102*pi/53)+t5*sin(66*pi/53)+t3*sin(24*pi/53)+t1*sin(28*pi/53)+t12*sin(68*pi/53)+t10*sin(44*pi/53)+t8*sin(16*pi/53)+t6*sin(54*pi/53)+t4*sin(10*pi/53)+t2*sin(100*pi/53));
s12=2*(sin(46*pi/53)+t12*sin(36*pi/53)+t11*sin(42*pi/53)+t10*sin(102*pi/53)+t9*sin(66*pi/53)+t8*sin(24*pi/53)+t7*sin(28*pi/53)+t6*sin(68*pi/53)+t5*sin(44*pi/53)+t4*sin(16*pi/53)+t3*sin(54*pi/53)+t2*sin(10*pi/53)+t1*sin(100*pi/53));
To simplify our calculation, let a1=r1^13; a2=r2^13; a3=r3^13; a4=r4^13; a5=r5^13; a6=r6^13; a7=r7^13; a8=r8^13; a9=r9^13; a10=r10^13; a11=r11^13; a12=r12^13; a0=r0^13; as well as b1=s1^13; b2=s2^13; b3=s3^13; b4=s4^13; b5=s5^13; b6=s6^13; b7=s7^13; b8=s8^13; b9=s9^13; b10=s10^13; b11=s11^13; b12=s12^13; b0=s0^13.
j0=(a1+a2+a3+a4+a5+a6+a7+a8+a9+a10+a11+a12+a0)/sqrt(223925-30758*sqrt(53))=(-273391472465-37713493611*sqrt(53))/2;
k0=(b1+b2+b3+b4+b5+b6+b7+b8+b9+b10+b11+b12+b0)/sqrt(223925+30758*sqrt(53))=(-273391472465+37713493611*sqrt(53))/2;
j1=(a1/t1+a2/t2+a3/t3+a4/t4+a5/t5+a6/t6+a7/t7+a8/t8+a9/t9+a10/t10+a11/t11+a12/t12+a0)/sqrt(223925-30758*sqrt(53))=(-163806041776-22659469612*sqrt(53))/2;
k1=(b1/t1+b2/t2+b3/t3+b4/t4+b5/t5+b6/t6+b7/t7+b8/t8+b9/t9+b10/t10+b11/t11+b12/t12+b0)/sqrt(223925+30758*sqrt(53))=(-163806041776+22659469612*sqrt(53))/2;
j2=(a1/t2+a2/t4+a3/t6+a4/t8+a5/t10+a6/t12+a7/t1+a8/t3+a9/t5+a10/t7+a11/t9+a12/t11+a0)/sqrt(223925-30758*sqrt(53))=(15500116320+2147928666*sqrt(53))/2;
k2=(b1/t2+b2/t4+b3/t6+b4/t8+b5/t10+b6/t12+b7/t1+b8/t3+b9/t5+b10/t7+b11/t9+b12/t11+b0)/sqrt(223925+30758*sqrt(53))=(15500116320-2147928666*sqrt(53))/2;
j3=(a1/t3+a2/t6+a3/t9+a4/t12+a5/t2+a6/t5+a7/t8+a8/t11+a9/t1+a10/t4+a11/t7+a12/t10+a0)/sqrt(223925-30758*sqrt(53))=(-6183969454-796495154*sqrt(53))/2;
k3=(b1/t3+b2/t6+b3/t9+b4/t12+b5/t2+b6/t5+b7/t8+b8/t11+b9/t1+b10/t4+b11/t7+b12/t10+b0)/sqrt(223925+30758*sqrt(53))=(-6183969454+796495154*sqrt(53))/2;
j4=(a1/t4+a2/t8+a3/t12+a4/t3+a5/t7+a6/t11+a7/t2+a8/t6+a9/t10+a10/t1+a11/t5+a12/t9+a0)/sqrt(223925-30758*sqrt(53))=(173578671552+23742889014*sqrt(53))/2;
k4=(b1/t4+b2/t8+b3/t12+b4/t3+b5/t7+b6/t11+b7/t2+b8/t6+b9/t10+b10/t1+b11/t5+b12/t9+b0)/sqrt(223925+30758*sqrt(53))=(173578671552-23742889014*sqrt(53))/2;
j5=(a1/t5+a2/t10+a3/t2+a4/t7+a5/t12+a6/t4+a7/t9+a8/t1+a9/t6+a10/t11+a11/t3+a12/t8+a0)/sqrt(223925-30758*sqrt(53))=(-50721445801-7146503819*sqrt(53))/2;
k5=(b1/t5+b2/t10+b3/t2+b4/t7+b5/t12+b6/t4+b7/t9+b8/t1+b9/t6+b10/t11+b11/t3+b12/t8+b0)/sqrt(223925+30758*sqrt(53)))=(-50721445801+7146503819*sqrt(53))/2;
j6=(a1/t6+a2/t12+a3/t5+a4/t11+a5/t4+a6/t10+a7/t3+a8/t9+a9/t2+a10/t8+a11/t1+a12/t7+a0)/sqrt(223925-30758*sqrt(53)))=(194027256475+26595930101*sqrt(53))/2;
k6=(b1/t6+b2/t12+b3/t5+b4/t11+b5/t4+b6/t10+b7/t3+b8/t9+b9/t2+b10/t8+b11/t1+b12/t7+b0)/sqrt(223925+30758*sqrt(53)))=(194027256475-26595930101*sqrt(53))/2;
j7=(a1/t7+a2/t1+a3/t8+a4/t2+a5/t9+a6/t3+a7/t10+a8/t4+a9/t11+a10/t5+a11/t12+a12/t6+a0)/sqrt(223925-30758*sqrt(53)))=(-13692971136-1808558440*sqrt(53))/2;
k7=(b1/t7+b2/t1+b3/t8+b4/t2+b5/t9+b6/t3+b7/t10+b8/t4+b9/t11+b10/t5+b11/t12+b12/t6+b0)/sqrt(223925+30758*sqrt(53)))=(-13692971136+1808558440*sqrt(53))/2;
j8=(a1/t8+a2/t3+a3/t11+a4/t6+a5/t1+a6/t9+a7/t4+a8/t12+a9/t7+a10/t2+a11/t10+a12/t5+a0)/sqrt(223925-30758*sqrt(53)))=(304879953794+41849434354*sqrt(53))/2;
k8=(b1/t8+b2/t3+b3/t11+b4/t6+b5/t1+b6/t9+b7/t4+b8/t12+b9/t7+b10/t2+b11/t10+b12/t5+b0)/sqrt(223925+30758*sqrt(53)))=(304879953794-41849434354*sqrt(53))/2;
j9=(a1/t9+a2/t5+a3/t1+a4/t10+a5/t6+a6/t2+a7/t11+a8/t7+a9/t3+a10/t12+a11/t8+a12/t4+a0)/sqrt(223925-30758*sqrt(53)))=(38155556998+5051576400*sqrt(53))/2;
k9=(b1/t9+b2/t5+b3/t1+b4/t10+b5/t6+b6/t2+b7/t11+b8/t7+b9/t3+b10/t12+b11/t8+b12/t4+b0)/sqrt(223925+30758*sqrt(53)))=(38155556998-5051576400*sqrt(53))/2;
j10=(a1/t10+a2/t7+a3/t4+a4/t1+a5/t11+a6/t8+a7/t5+a8/t2+a9/t12+a10/t9+a11/t6+a12/t3+a0)/sqrt(223925-30758*sqrt(53)))=(158735903742+21689661214*sqrt(53))/2;
k10=(b1/t10+b2/t7+b3/t4+b4/t1+b5/t11+b6/t8+b7/t5+b8/t2+b9/t12+b10/t9+b11/t6+b12/t3+b0)/sqrt(223925+30758*sqrt(53)))=(158735903742-21689661214*sqrt(53))/2;
j11=(a1/t11+a2/t9+a3/t7+a4/t5+a5/t3+a6/t1+a7/t12+a8/t10+a9/t8+a10/t6+a11/t4+a12/t2+a0)/sqrt(223925-30758*sqrt(53)))=(-194945288928-26716597804*sqrt(53))/2;
k11=(b1/t11+b2/t9+b3/t7+b4/t5+b5/t3+b6/t1+b7/t12+b8/t10+b9/t8+b10/t6+b11/t4+b12/t2+b0)/sqrt(223925+30758*sqrt(53)))=(-194945288928+26716597804*sqrt(53))/2;
j12=(a1/t12+a2/t11+a3/t10+a4/t9+a5/t8+a6/t7+a7/t6+a8/t5+a9/t4+a10/t3+a11/t2+a12/t1+a0)/sqrt(223925-30758*sqrt(53)))=(-179373558832-24615788782*sqrt(53))/2;
k12=(b1/t12+b2/t11+b3/t10+b4/t9+b5/t8+b6/t7+b7/t6+b8/t5+b9/t4+b10/t3+b11/t2+b12/t1+b0)/sqrt(223925+30758*sqrt(53)))=(-179373558832+24615788782*sqrt(53))/2.
Then according to the formulae 1+t1+……+t12=0, we find that
a1=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t1+(194873675152+26763717448*sqrt(53))*t2+(173189589378+23819293628*sqrt(53))*t3+(352952230384+48358677796*sqrt(53))*t4+(128652113031+17469284963*sqrt(53))*t5+(373400815307+51211718883*sqrt(53))*t6+(165680587696+22807230342*sqrt(53))*t7+(484253512626+66465223136*sqrt(53))*t8+(217529115830+29667365182*sqrt(53))*t9+(338109462574+46305449996*sqrt(53))*t10+(-15571730096-2100809022*sqrt(53))*t11);
a2=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t2+(194873675152+26763717448*sqrt(53))*t4+(173189589378+23819293628*sqrt(53))*t6+(352952230384+48358677796*sqrt(53))*t8+(128652113031+17469284963*sqrt(53))*t10+(373400815307+51211718883*sqrt(53))*t12+(165680587696+22807230342*sqrt(53))*t1+(484253512626+66465223136*sqrt(53))*t3+(217529115830+29667365182*sqrt(53))*t5+(338109462574+46305449996*sqrt(53))*t7+(-15571730096-2100809022*sqrt(53))*t9);
a3=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t3+(194873675152+26763717448*sqrt(53))*t6+(173189589378+23819293628*sqrt(53))*t9+(352952230384+48358677796*sqrt(53))*t12+(128652113031+17469284963*sqrt(53))*t2+(373400815307+51211718883*sqrt(53))*t5+(165680587696+22807230342*sqrt(53))*t8+(484253512626+66465223136*sqrt(53))*t11+(217529115830+29667365182*sqrt(53))*t1+(338109462574+46305449996*sqrt(53))*t4+(-15571730096-2100809022*sqrt(53))*t7); (very slight)
a4=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t4+(194873675152+26763717448*sqrt(53))*t8+(173189589378+23819293628*sqrt(53))*t12+(352952230384+48358677796*sqrt(53))*t3+(128652113031+17469284963*sqrt(53))*t7+(373400815307+51211718883*sqrt(53))*t11+(165680587696+22807230342*sqrt(53))*t2+(484253512626+66465223136*sqrt(53))*t6+(217529115830+29667365182*sqrt(53))*t10+(338109462574+46305449996*sqrt(53))*t1+(-15571730096-2100809022*sqrt(53))*t5);
a5=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t5+(194873675152+26763717448*sqrt(53))*t10+(173189589378+23819293628*sqrt(53))*t2+(352952230384+48358677796*sqrt(53))*t7+(128652113031+17469284963*sqrt(53))*t12+(373400815307+51211718883*sqrt(53))*t4+(165680587696+22807230342*sqrt(53))*t9+(484253512626+66465223136*sqrt(53))*t1+(217529115830+29667365182*sqrt(53))*t6+(338109462574+46305449996*sqrt(53))*t11+(-15571730096-2100809022*sqrt(53))*t3);
a6=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t6+(194873675152+26763717448*sqrt(53))*t12+(173189589378+23819293628*sqrt(53))*t5+(352952230384+48358677796*sqrt(53))*t11+(128652113031+17469284963*sqrt(53))*t4+(373400815307+51211718883*sqrt(53))*t10+(165680587696+22807230342*sqrt(53))*t3+(484253512626+66465223136*sqrt(53))*t9+(217529115830+29667365182*sqrt(53))*t2+(338109462574+46305449996*sqrt(53))*t8+(-15571730096-2100809022*sqrt(53))*t1);
a7=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t7+(194873675152+26763717448*sqrt(53))*t1+(173189589378+23819293628*sqrt(53))*t8+(352952230384+48358677796*sqrt(53))*t2+(128652113031+17469284963*sqrt(53))*t9+(373400815307+51211718883*sqrt(53))*t3+(165680587696+22807230342*sqrt(53))*t10+(484253512626+66465223136*sqrt(53))*t4+(217529115830+29667365182*sqrt(53))*t11+(338109462574+46305449996*sqrt(53))*t5+(-15571730096-2100809022*sqrt(53))*t12);
a8=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t8+(194873675152+26763717448*sqrt(53))*t3+(173189589378+23819293628*sqrt(53))*t11+(352952230384+48358677796*sqrt(53))*t6+(128652113031+17469284963*sqrt(53))*t1+(373400815307+51211718883*sqrt(53))*t9+(165680587696+22807230342*sqrt(53))*t4+(484253512626+66465223136*sqrt(53))*t12+(217529115830+29667365182*sqrt(53))*t7+(338109462574+46305449996*sqrt(53))*t2+(-15571730096-2100809022*sqrt(53))*t10);
a9=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t9+(194873675152+26763717448*sqrt(53))*t5+(173189589378+23819293628*sqrt(53))*t1+(352952230384+48358677796*sqrt(53))*t10+(128652113031+17469284963*sqrt(53))*t6+(373400815307+51211718883*sqrt(53))*t2+(165680587696+22807230342*sqrt(53))*t11+(484253512626+66465223136*sqrt(53))*t7+(217529115830+29667365182*sqrt(53))*t3+(338109462574+46305449996*sqrt(53))*t12+(-15571730096-2100809022*sqrt(53))*t8);
a10=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t10+(194873675152+26763717448*sqrt(53))*t7+(173189589378+23819293628*sqrt(53))*t4+(352952230384+48358677796*sqrt(53))*t1+(128652113031+17469284963*sqrt(53))*t11+(373400815307+51211718883*sqrt(53))*t8+(165680587696+22807230342*sqrt(53))*t5+(484253512626+66465223136*sqrt(53))*t2+(217529115830+29667365182*sqrt(53))*t12+(338109462574+46305449996*sqrt(53))*t9+(-15571730096-2100809022*sqrt(53))*t6); (very slight)
a11=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t11+(194873675152+26763717448*sqrt(53))*t9+(173189589378+23819293628*sqrt(53))*t7+(352952230384+48358677796*sqrt(53))*t5+(128652113031+17469284963*sqrt(53))*t3+(373400815307+51211718883*sqrt(53))*t1+(165680587696+22807230342*sqrt(53))*t12+(484253512626+66465223136*sqrt(53))*t10+(217529115830+29667365182*sqrt(53))*t8+(338109462574+46305449996*sqrt(53))*t6+(-15571730096-2100809022*sqrt(53))*t4);
a12=0.5*sqrt(1325-182*sqrt(53))*(-94017913633-13097704829*sqrt(53)+(15567517056+1956319170*sqrt(53))*t12+(194873675152+26763717448*sqrt(53))*t11+(173189589378+23819293628*sqrt(53))*t10+(352952230384+48358677796*sqrt(53))*t9+(128652113031+17469284963*sqrt(53))*t8+(373400815307+51211718883*sqrt(53))*t7+(165680587696+22807230342*sqrt(53))*t6+(484253512626+66465223136*sqrt(53))*t5+(217529115830+29667365182*sqrt(53))*t4+(338109462574+46305449996*sqrt(53))*t3+(-15571730096-2100809022*sqrt(53))*t2);
and
b1=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t1+(194873675152-26763717448*sqrt(53))*t2+(173189589378-23819293628*sqrt(53))*t3+(352952230384-48358677796*sqrt(53))*t4+(128652113031-17469284963*sqrt(53))*t5+(373400815307-51211718883*sqrt(53))*t6+(165680587696-22807230342*sqrt(53))*t7+(484253512626-66465223136*sqrt(53))*t8+(217529115830-29667365182*sqrt(53))*t9+(338109462574-46305449996*sqrt(53))*t10+(-15571730096+2100809022*sqrt(53))*t11);
b2=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t2+(194873675152-26763717448*sqrt(53))*t4+(173189589378-23819293628*sqrt(53))*t6+(352952230384-48358677796*sqrt(53))*t8+(128652113031-17469284963*sqrt(53))*t10+(373400815307-51211718883*sqrt(53))*t12+(165680587696-22807230342*sqrt(53))*t1+(484253512626-66465223136*sqrt(53))*t3+(217529115830-29667365182*sqrt(53))*t5+(338109462574-46305449996*sqrt(53))*t7+(-15571730096+2100809022*sqrt(53))*t9);
b3=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t3+(194873675152-26763717448*sqrt(53))*t6+(173189589378-23819293628*sqrt(53))*t9+(352952230384-48358677796*sqrt(53))*t12+(128652113031-17469284963*sqrt(53))*t2+(373400815307-51211718883*sqrt(53))*t5+(165680587696-22807230342*sqrt(53))*t8+(484253512626-66465223136*sqrt(53))*t11+(217529115830-29667365182*sqrt(53))*t1+(338109462574-46305449996*sqrt(53))*t4+(-15571730096+2100809022*sqrt(53))*t7);
b4=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t4+(194873675152-26763717448*sqrt(53))*t8+(173189589378-23819293628*sqrt(53))*t12+(352952230384-48358677796*sqrt(53))*t3+(128652113031-17469284963*sqrt(53))*t7+(373400815307-51211718883*sqrt(53))*t11+(165680587696-22807230342*sqrt(53))*t2+(484253512626-66465223136*sqrt(53))*t6+(217529115830-29667365182*sqrt(53))*t10+(338109462574-46305449996*sqrt(53))*t1+(-15571730096+2100809022*sqrt(53))*t5);
b5=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t5+(194873675152-26763717448*sqrt(53))*t10+(173189589378-23819293628*sqrt(53))*t2+(352952230384-48358677796*sqrt(53))*t7+(128652113031-17469284963*sqrt(53))*t12+(373400815307-51211718883*sqrt(53))*t4+(165680587696-22807230342*sqrt(53))*t9+(484253512626-66465223136*sqrt(53))*t1+(217529115830-29667365182*sqrt(53))*t6+(338109462574-46305449996*sqrt(53))*t11+(-15571730096+2100809022*sqrt(53))*t3);
b6=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t6+(194873675152-26763717448*sqrt(53))*t12+(173189589378-23819293628*sqrt(53))*t5+(352952230384-48358677796*sqrt(53))*t11+(128652113031-17469284963*sqrt(53))*t4+(373400815307-51211718883*sqrt(53))*t10+(165680587696-22807230342*sqrt(53))*t3+(484253512626-66465223136*sqrt(53))*t9+(217529115830-29667365182*sqrt(53))*t2+(338109462574-46305449996*sqrt(53))*t8+(-15571730096+2100809022*sqrt(53))*t1);
b7=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t7+(194873675152-26763717448*sqrt(53))*t1+(173189589378-23819293628*sqrt(53))*t8+(352952230384-48358677796*sqrt(53))*t2+(128652113031-17469284963*sqrt(53))*t9+(373400815307-51211718883*sqrt(53))*t3+(165680587696-22807230342*sqrt(53))*t10+(484253512626-66465223136*sqrt(53))*t4+(217529115830-29667365182*sqrt(53))*t11+(338109462574-46305449996*sqrt(53))*t5+(-15571730096+2100809022*sqrt(53))*t12);
b8=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t8+(194873675152-26763717448*sqrt(53))*t3+(173189589378-23819293628*sqrt(53))*t11+(352952230384-48358677796*sqrt(53))*t6+(128652113031-17469284963*sqrt(53))*t1+(373400815307-51211718883*sqrt(53))*t9+(165680587696-22807230342*sqrt(53))*t4+(484253512626-66465223136*sqrt(53))*t12+(217529115830-29667365182*sqrt(53))*t7+(338109462574-46305449996*sqrt(53))*t2+(-15571730096+2100809022*sqrt(53))*t10);
b9=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t9+(194873675152-26763717448*sqrt(53))*t5+(173189589378-23819293628*sqrt(53))*t1+(352952230384-48358677796*sqrt(53))*t10+(128652113031-17469284963*sqrt(53))*t6+(373400815307-51211718883*sqrt(53))*t2+(165680587696-22807230342*sqrt(53))*t11+(484253512626-66465223136*sqrt(53))*t7+(217529115830-29667365182*sqrt(53))*t3+(338109462574-46305449996*sqrt(53))*t12+(-15571730096+2100809022*sqrt(53))*t8);
b10=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t10+(194873675152-26763717448*sqrt(53))*t7+(173189589378-23819293628*sqrt(53))*t4+(352952230384-48358677796*sqrt(53))*t1+(128652113031-17469284963*sqrt(53))*t11+(373400815307-51211718883*sqrt(53))*t8+(165680587696-22807230342*sqrt(53))*t5+(484253512626-66465223136*sqrt(53))*t2+(217529115830-29667365182*sqrt(53))*t12+(338109462574-46305449996*sqrt(53))*t9+(-15571730096+2100809022*sqrt(53))*t6);
b11=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t11+(194873675152-26763717448*sqrt(53))*t9+(173189589378-23819293628*sqrt(53))*t7+(352952230384-48358677796*sqrt(53))*t5+(128652113031-17469284963*sqrt(53))*t3+(373400815307-51211718883*sqrt(53))*t1+(165680587696-22807230342*sqrt(53))*t12+(484253512626-66465223136*sqrt(53))*t10+(217529115830-29667365182*sqrt(53))*t8+(338109462574-46305449996*sqrt(53))*t6+(-15571730096+2100809022*sqrt(53))*t4);
b12=0.5*sqrt(1325+182*sqrt(53))*(-94017913633+13097704829*sqrt(53)+(15567517056-1956319170*sqrt(53))*t12+(194873675152-26763717448*sqrt(53))*t11+(173189589378-23819293628*sqrt(53))*t10+(352952230384-48358677796*sqrt(53))*t9+(128652113031-17469284963*sqrt(53))*t8+(373400815307-51211718883*sqrt(53))*t7+(165680587696-22807230342*sqrt(53))*t6+(484253512626-66465223136*sqrt(53))*t5+(217529115830-29667365182*sqrt(53))*t4+(338109462574-46305449996*sqrt(53))*t3+(-15571730096+2100809022*sqrt(53))*t2).
Meanwhile
r1/a1^(1/13)=t11; r2/a2^(1/13)=t1; r3/a3^(1/13)=t10; r4/a4^(1/13)=t1; r5/a5^(1/13)=t9; r6/a6^(1/13)=t7;
s1/b1^(1/13)=t11; s2/b2^(1/13)=t1; s3/b3^(1/13)=t3; s4/b4^(1/13)=t8; s5/b5^(1/13)=t2; s6/b6^(1/13)=1.
We conclude that
sin(2*pi/53)=(-sqrt(106-14*sqrt(53))+2*(t11*a1^(1/13)+t2*a12^(1/13)+t1*a2^(1/13)+t12*a11^(1/13)+t10*a3^(1/13)+t3*a10^(1/13)+t1*a4^(1/13)+t12*a9^(1/13)+t9*a5^(1/13)+t4*a8^(1/13)+t7*a6^(1/13)+t6*a7^(1/13)))/52;
sin(20*pi/53)=(-sqrt(106-14*sqrt(53))+2*(t10*a1^(1/13)+t3*a12^(1/13)+t12*a2^(1/13)+t1*a11^(1/13)+t7*a3^(1/13)+t6*a10^(1/13)+t10*a4^(1/13)+t3*a9^(1/13)+t4*a5^(1/13)+t9*a8^(1/13)+t1*a6^(1/13)+t12*a7^(1/13)))/52;
……
sin(46*pi/53)=(sqrt(106+14*sqrt(53))+2*(t11*b1^(1/13)+t2*b12^(1/13)+t1*b2^(1/13)+t12*b11^(1/13)+t3*b3^(1/13)+t10*b10^(1/13)+t8*b4^(1/13)+t5*b9^(1/13)+t2*b5^(1/13)+t11*b8^(1/13)+b6^(1/13)+b7^(1/13)))/52;
sin(36*pi/53)=(sqrt(106+14*sqrt(53))+2*(t10*b1^(1/13)+t3*b12^(1/13)+t12*b2^(1/13)+t1*b11^(1/13)+b3^(1/13)+b10^(1/13)+t4*b4^(1/13)+t9*b9^(1/13)+t10*b5^(1/13)+t3*b8^(1/13)+t7*b6^(1/13)+t6*b7^(1/13)))/52;
……
4. The complex solutions of x^53=1
Some examples like:
exp(2*pi*j/53)=(-1+sqrt(53)+2*(t2*A1^(1/13)+t11*A12^(1/13)+t12*A2^(1/13)+t1*A11^(1/13)+t6*A3^(1/13)+t7*A10^(1/13)+t12*A4^(1/13)+t1*A9^(1/13)+t1*A5^(1/13)+t12*A8^(1/13)+t1*A6^(1/13)+t12*A7^(1/13))-j*(sqrt(106-14*sqrt(53))-2*(t11*a1^(1/13)+t2*a12^(1/13)+t1*a2^(1/13)+t12*a11^(1/13)+t10*a3^(1/13)+t3*a10^(1/13)+t1*a4^(1/13)+t12*a9^(1/13)+t9*a5^(1/13)+t4*a8^(1/13)+t7*a6^(1/13)+t6*a7^(1/13))))/52;
exp(104*pi*j/53)=(-1+sqrt(53)+2*(t2*A1^(1/13)+t11*A12^(1/13)+t12*A2^(1/13)+t1*A11^(1/13)+t6*A3^(1/13)+t7*A10^(1/13)+t12*A4^(1/13)+t1*A9^(1/13)+t1*A5^(1/13)+t12*A8^(1/13)+t1*A6^(1/13)+t12*A7^(1/13))+j*(sqrt(106-14*sqrt(53))-2*(t11*a1^(1/13)+t2*a12^(1/13)+t1*a2^(1/13)+t12*a11^(1/13)+t10*a3^(1/13)+t3*a10^(1/13)+t1*a4^(1/13)+t12*a9^(1/13)+t9*a5^(1/13)+t4*a8^(1/13)+t7*a6^(1/13)+t6*a7^(1/13))))/52;
exp(20*pi*j/53)=(-1+sqrt(53)+2*(t1*A1^(1/13)+t12*A12^(1/13)+t10*A2^(1/13)+t3*A11^(1/13)+t3*A3^(1/13)+t10*A10^(1/13)+t8*A4^(1/13)+t5*A9^(1/13)+t9*A5^(1/13)+t4*A8^(1/13)+t8*A6^(1/13)+t5*A7^(1/13))-j*(sqrt(106-14*sqrt(53))-2*(t10*a1^(1/13)+t3*a12^(1/13)+t12*a2^(1/13)+t1*a11^(1/13)+t7*a3^(1/13)+t6*a10^(1/13)+t10*a4^(1/13)+t3*a9^(1/13)+t4*a5^(1/13)+t9*a8^(1/13)+t1*a6^(1/13)+t12*a7^(1/13))))/52;
exp(86*pi*j/53)=(-1+sqrt(53)+2*(t1*A1^(1/13)+t12*A12^(1/13)+t10*A2^(1/13)+t3*A11^(1/13)+t3*A3^(1/13)+t10*A10^(1/13)+t8*A4^(1/13)+t5*A9^(1/13)+t9*A5^(1/13)+t4*A8^(1/13)+t8*A6^(1/13)+t5*A7^(1/13))+j*(sqrt(106-14*sqrt(53))-2*(t10*a1^(1/13)+t3*a12^(1/13)+t12*a2^(1/13)+t1*a11^(1/13)+t7*a3^(1/13)+t6*a10^(1/13)+t10*a4^(1/13)+t3*a9^(1/13)+t4*a5^(1/13)+t9*a8^(1/13)+t1*a6^(1/13)+t12*a7^(1/13))))/52;
……
exp(46*pi*j/53)=(-1-sqrt(53)+2*(t12*B1^(1/13)+t1*B12^(1/13)+t9*B2^(1/13)+t4*B11^(1/13)+t9*B3^(1/13)+t4*B10^(1/13)+t9*B4^(1/13)+t4*B9^(1/13)+t4*B5^(1/13)+t9*B8^(1/13)+t4*B6^(1/13)+t9*B7^(1/13))+j*(sqrt(106+14*sqrt(53))+2*(t11*b1^(1/13)+t2*b12^(1/13)+t1*b2^(1/13)+t12*b11^(1/13)+t3*b3^(1/13)+t10*b10^(1/13)+t8*b4^(1/13)+t5*b9^(1/13)+t2*b5^(1/13)+t11*b8^(1/13)+b6^(1/13)+b7^(1/13))))/52;
exp(60*pi*j/53)=(-1-sqrt(53)+2*(t12*B1^(1/13)+t1*B12^(1/13)+t9*B2^(1/13)+t4*B11^(1/13)+t9*B3^(1/13)+t4*B10^(1/13)+t9*B4^(1/13)+t4*B9^(1/13)+t4*B5^(1/13)+t9*B8^(1/13)+t4*B6^(1/13)+t9*B7^(1/13))-j*(sqrt(106+14*sqrt(53))+2*(t11*b1^(1/13)+t2*b12^(1/13)+t1*b2^(1/13)+t12*b11^(1/13)+t3*b3^(1/13)+t10*b10^(1/13)+t8*b4^(1/13)+t5*b9^(1/13)+t2*b5^(1/13)+t11*b8^(1/13)+b6^(1/13)+b7^(1/13))))/52;
exp(36*pi*j/53)=(-1-sqrt(53)+2*(t11*B1^(1/13)+t2*B12^(1/13)+t7*B2^(1/13)+t6*B11^(1/13)+t6*B3^(1/13)+t7*B10^(1/13)+t5*B4^(1/13)+t8*B9^(1/13)+t12*B5^(1/13)+t1*B8^(1/13)+t11*B6^(1/13)+t2*B7^(1/13))+j*(sqrt(106+14*sqrt(53))+2*(t10*b1^(1/13)+t3*b12^(1/13)+t12*b2^(1/13)+t1*b11^(1/13)+b3^(1/13)+b10^(1/13)+t4*b4^(1/13)+t9*b9^(1/13)+t10*b5^(1/13)+t3*b8^(1/13)+t7*b6^(1/13)+t6*b7^(1/13))))/52;
exp(70*pi*j/53)=(-1-sqrt(53)+2*(t11*B1^(1/13)+t2*B12^(1/13)+t7*B2^(1/13)+t6*B11^(1/13)+t6*B3^(1/13)+t7*B10^(1/13)+t5*B4^(1/13)+t8*B9^(1/13)+t12*B5^(1/13)+t1*B8^(1/13)+t11*B6^(1/13)+t2*B7^(1/13))-j*(sqrt(106+14*sqrt(53))+2*(t10*b1^(1/13)+t3*b12^(1/13)+t12*b2^(1/13)+t1*b11^(1/13)+b3^(1/13)+b10^(1/13)+t4*b4^(1/13)+t9*b9^(1/13)+t10*b5^(1/13)+t3*b8^(1/13)+t7*b6^(1/13)+t6*b7^(1/13))))/52;
……
This is the LAST passage by Hatsune Miku 589825of the series x^N=1. Thanks for your reading!
凡注有"实况网-重新发现生活"或电头为"实况网-重新发现生活"的稿件，均为实况网-重新发现生活独家版权所有，未经许可不得转载或镜像；授权转载必须注明来源为"实况网-重新发现生活"，并保留"实况网-重新发现生活"的电头。
热点
- 作为新型零售业态之一 即时零售近年来正成为撬动消费增长不可忽视的新动能
- 更具有沉浸性和体验性 农业旅游已经不再是“传统”意义上的农家乐
- 脱发年轻化带来了市场扩增 各大厂商通过不同形式入局期望分得一杯羹
- 上半年投诉数量呈上升趋势 预制菜是花高价吃料理包吗？
- 在国民支出中占比将达46% 服务型消费快速地迭代发展持续焕发生命力
- 成了资本眼中的“好生意” 不少知名企业都在积极跨界布局咖啡赛道
- 新增超千家成为投资者关注的焦点 国庆期间预制菜整体销量较去年大为增加
- 值得借鉴和共享 上海、重庆出境游业务有助于国内旅游业进一步走向世界
- 假日经济按捺不住的“火” 今年“十一”消费有了新特色和新亮点
- 前八月亏数千万九月扭亏为盈 车市的“金九银十”要来了吗？
头条
- 进博会各展台上各种直播设备林立 直播成为越来越多外商的选择
- 在整体上保持了政策的延续性和稳定性 全国碳排放权交易继续实行配额免费分配
- 改变人类生产生活方式 虚拟现实产业正迈入以产品升级和融合应用为主线的战略窗口期
- 很多具体制度安排含金量高短期可见成效 有利于解决个体工商户“急难愁盼”问题
- 具有权威性和规范性 社会公众可以通过互联网访问市场监管法律法规规章数据库
- 巩固成果守底线为保民生防通胀 前三季度农业收成为稳定经济大盘提供了坚实支撑
- 流动性优化定价效率进一步得到提升 科创板做市望迎来增量资金持续入场
- 一般贸易进出口比重提升7.5个百分点 前三季度陕西外贸结构明显优化
- 有力有序有效推进各项重点工作 中国碳达峰碳中和取得显著成效
- 增速较前8个月略降 我国进出口仍有信心克服国外市场压力安全承受能力不会脱离
