1. cos(2*n*pi/13)+j*sin(2*n*pi/13)

Let w1=(-1+sqrt(3)*j)/2; w2=(-1-sqrt(3)*j)/2, and the solutions of t^13=1 except from t=1 are tn=cos(2*n*pi/13)+j*sin(2*n*pi/13), written as:

t1=(-1+sqrt(13)+(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+(104-20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26-6*sqrt(13))-w2*(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-w1*(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

(资料图片仅供参考)

t2=(-1-sqrt(13)+(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+(104+20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26+6*sqrt(13))-w1*(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-w2*(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

t3=(-1+sqrt(13)+w2*(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+w1*(104-20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26-6*sqrt(13))-w1*(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-w2*(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

t4=(-1+sqrt(13)+w1*(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+w2*(104-20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26-6*sqrt(13))-(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

t5=(-1-sqrt(13)+w2*(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+w1*(104+20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26+6*sqrt(13))-w2*(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-w1*(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

t6=(-1-sqrt(13)+w1*(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+w2*(104+20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26+6*sqrt(13))-(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

t7=(-1-sqrt(13)+w1*(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+w2*(104+20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26+6*sqrt(13))-(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

t8=(-1-sqrt(13)+w2*(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+w1*(104+20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26+6*sqrt(13))-w2*(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-w1*(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

t9=(-1+sqrt(13)+w1*(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+w2*(104-20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26-6*sqrt(13))-(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

t10=(-1+sqrt(13)+w2*(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+w1*(104-20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26-6*sqrt(13))-w1*(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-w2*(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

t11=(-1-sqrt(13)+(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+(104+20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26+6*sqrt(13))-w1*(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-w2*(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

t12=(-1+sqrt(13)+(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+(104-20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26-6*sqrt(13))-w2*(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-w1*(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12.

2. cos(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*(cos(2*pi/53)+cos(20*pi/53)+cos(94*pi/53)+cos(92*pi/53)+cos(72*pi/53)+cos(84*pi/53)+cos(98*pi/53)+cos(26*pi/53)+cos(48*pi/53)+cos(56*pi/53)+cos(30*pi/53)+cos(88*pi/53)+cos(32*pi/53))=(-1+sqrt(53))/2;

R1=2*(cos(2*pi/53)+t1*cos(20*pi/53)+t2*cos(94*pi/53)+t3*cos(92*pi/53)+t4*cos(72*pi/53)+t5*cos(84*pi/53)+t6*cos(98*pi/53)+t7*cos(26*pi/53)+t8*cos(48*pi/53)+t9*cos(56*pi/53)+t10*cos(30*pi/53)+t11*cos(88*pi/53)+t12*cos(32*pi/53));

R2=2*(cos(2*pi/53)+t2*cos(20*pi/53)+t4*cos(94*pi/53)+t6*cos(92*pi/53)+t8*cos(72*pi/53)+t10*cos(84*pi/53)+t12*cos(98*pi/53)+t1*cos(26*pi/53)+t3*cos(48*pi/53)+t5*cos(56*pi/53)+t7*cos(30*pi/53)+t9*cos(88*pi/53)+t11*cos(32*pi/53));

R3=2*(cos(2*pi/53)+t3*cos(20*pi/53)+t6*cos(94*pi/53)+t9*cos(92*pi/53)+t12*cos(72*pi/53)+t2*cos(84*pi/53)+t5*cos(98*pi/53)+t8*cos(26*pi/53)+t11*cos(48*pi/53)+t1*cos(56*pi/53)+t4*cos(30*pi/53)+t7*cos(88*pi/53)+t10*cos(32*pi/53));

R4=2*(cos(2*pi/53)+t4*cos(20*pi/53)+t8*cos(94*pi/53)+t12*cos(92*pi/53)+t3*cos(72*pi/53)+t7*cos(84*pi/53)+t11*cos(98*pi/53)+t2*cos(26*pi/53)+t6*cos(48*pi/53)+t10*cos(56*pi/53)+t1*cos(30*pi/53)+t5*cos(88*pi/53)+t9*cos(32*pi/53));

R5=2*(cos(2*pi/53)+t5*cos(20*pi/53)+t10*cos(94*pi/53)+t2*cos(92*pi/53)+t7*cos(72*pi/53)+t12*cos(84*pi/53)+t4*cos(98*pi/53)+t9*cos(26*pi/53)+t1*cos(48*pi/53)+t6*cos(56*pi/53)+t11*cos(30*pi/53)+t3*cos(88*pi/53)+t8*cos(32*pi/53));

R6=2*(cos(2*pi/53)+t6*cos(20*pi/53)+t12*cos(94*pi/53)+t5*cos(92*pi/53)+t11*cos(72*pi/53)+t4*cos(84*pi/53)+t10*cos(98*pi/53)+t3*cos(26*pi/53)+t9*cos(48*pi/53)+t2*cos(56*pi/53)+t8*cos(30*pi/53)+t1*cos(88*pi/53)+t7*cos(32*pi/53));

R7=2*(cos(2*pi/53)+t7*cos(20*pi/53)+t1*cos(94*pi/53)+t8*cos(92*pi/53)+t2*cos(72*pi/53)+t9*cos(84*pi/53)+t3*cos(98*pi/53)+t10*cos(26*pi/53)+t4*cos(48*pi/53)+t11*cos(56*pi/53)+t5*cos(30*pi/53)+t12*cos(88*pi/53)+t6*cos(32*pi/53));

R8=2*(cos(2*pi/53)+t8*cos(20*pi/53)+t3*cos(94*pi/53)+t11*cos(92*pi/53)+t6*cos(72*pi/53)+t1*cos(84*pi/53)+t9*cos(98*pi/53)+t4*cos(26*pi/53)+t12*cos(48*pi/53)+t7*cos(56*pi/53)+t2*cos(30*pi/53)+t10*cos(88*pi/53)+t5*cos(32*pi/53));

R9=2*(cos(2*pi/53)+t9*cos(20*pi/53)+t5*cos(94*pi/53)+t1*cos(92*pi/53)+t10*cos(72*pi/53)+t6*cos(84*pi/53)+t2*cos(98*pi/53)+t11*cos(26*pi/53)+t7*cos(48*pi/53)+t3*cos(56*pi/53)+t12*cos(30*pi/53)+t8*cos(88*pi/53)+t4*cos(32*pi/53));

R10=2*(cos(2*pi/53)+t10*cos(20*pi/53)+t7*cos(94*pi/53)+t4*cos(92*pi/53)+t1*cos(72*pi/53)+t11*cos(84*pi/53)+t8*cos(98*pi/53)+t5*cos(26*pi/53)+t2*cos(48*pi/53)+t12*cos(56*pi/53)+t9*cos(30*pi/53)+t6*cos(88*pi/53)+t3*cos(32*pi/53));

R11=2*(cos(2*pi/53)+t11*cos(20*pi/53)+t9*cos(94*pi/53)+t7*cos(92*pi/53)+t5*cos(72*pi/53)+t3*cos(84*pi/53)+t1*cos(98*pi/53)+t12*cos(26*pi/53)+t10*cos(48*pi/53)+t8*cos(56*pi/53)+t6*cos(30*pi/53)+t4*cos(88*pi/53)+t2*cos(32*pi/53));

R12=2*(cos(2*pi/53)+t12*cos(20*pi/53)+t11*cos(94*pi/53)+t10*cos(92*pi/53)+t9*cos(72*pi/53)+t8*cos(84*pi/53)+t7*cos(98*pi/53)+t6*cos(26*pi/53)+t5*cos(48*pi/53)+t4*cos(56*pi/53)+t3*cos(30*pi/53)+t2*cos(88*pi/53)+t1*cos(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*(cos(46*pi/53)+cos(36*pi/53)+cos(42*pi/53)+cos(102*pi/53)+cos(66*pi/53)+cos(24*pi/53)+cos(28*pi/53)+cos(68*pi/53)+cos(44*pi/53)+cos(16*pi/53)+cos(54*pi/53)+cos(10*pi/53)+cos(100*pi/53))=(-1-sqrt(53))/2;

S1=2*(cos(46*pi/53)+t1*cos(36*pi/53)+t2*cos(42*pi/53)+t3*cos(102*pi/53)+t4*cos(66*pi/53)+t5*cos(24*pi/53)+t6*cos(28*pi/53)+t7*cos(68*pi/53)+t8*cos(44*pi/53)+t9*cos(16*pi/53)+t10*cos(54*pi/53)+t11*cos(10*pi/53)+t12*cos(100*pi/53));

S2=2*(cos(46*pi/53)+t2*cos(36*pi/53)+t4*cos(42*pi/53)+t6*cos(102*pi/53)+t8*cos(66*pi/53)+t10*cos(24*pi/53)+t12*cos(28*pi/53)+t1*cos(68*pi/53)+t3*cos(44*pi/53)+t5*cos(16*pi/53)+t7*cos(54*pi/53)+t9*cos(10*pi/53)+t11*cos(100*pi/53));

S3=2*(cos(46*pi/53)+t3*cos(36*pi/53)+t6*cos(42*pi/53)+t9*cos(102*pi/53)+t12*cos(66*pi/53)+t2*cos(24*pi/53)+t5*cos(28*pi/53)+t8*cos(68*pi/53)+t11*cos(44*pi/53)+t1*cos(16*pi/53)+t4*cos(54*pi/53)+t7*cos(10*pi/53)+t10*cos(100*pi/53));

S4=2*(cos(46*pi/53)+t4*cos(36*pi/53)+t8*cos(42*pi/53)+t12*cos(102*pi/53)+t3*cos(66*pi/53)+t7*cos(24*pi/53)+t11*cos(28*pi/53)+t2*cos(68*pi/53)+t6*cos(44*pi/53)+t10*cos(16*pi/53)+t1*cos(54*pi/53)+t5*cos(10*pi/53)+t9*cos(100*pi/53));

S5=2*(cos(46*pi/53)+t5*cos(36*pi/53)+t10*cos(42*pi/53)+t2*cos(102*pi/53)+t7*cos(66*pi/53)+t12*cos(24*pi/53)+t4*cos(28*pi/53)+t9*cos(68*pi/53)+t1*cos(44*pi/53)+t6*cos(16*pi/53)+t11*cos(54*pi/53)+t3*cos(10*pi/53)+t8*cos(100*pi/53));

S6=2*(cos(46*pi/53)+t6*cos(36*pi/53)+t12*cos(42*pi/53)+t5*cos(102*pi/53)+t11*cos(66*pi/53)+t4*cos(24*pi/53)+t10*cos(28*pi/53)+t3*cos(68*pi/53)+t9*cos(44*pi/53)+t2*cos(16*pi/53)+t8*cos(54*pi/53)+t1*cos(10*pi/53)+t7*cos(100*pi/53));

S7=2*(cos(46*pi/53)+t7*cos(36*pi/53)+t1*cos(42*pi/53)+t8*cos(102*pi/53)+t2*cos(66*pi/53)+t9*cos(24*pi/53)+t3*cos(28*pi/53)+t10*cos(68*pi/53)+t4*cos(44*pi/53)+t11*cos(16*pi/53)+t5*cos(54*pi/53)+t12*cos(10*pi/53)+t6*cos(100*pi/53));

S8=2*(cos(46*pi/53)+t8*cos(36*pi/53)+t3*cos(42*pi/53)+t11*cos(102*pi/53)+t6*cos(66*pi/53)+t1*cos(24*pi/53)+t9*cos(28*pi/53)+t4*cos(68*pi/53)+t12*cos(44*pi/53)+t7*cos(16*pi/53)+t2*cos(54*pi/53)+t10*cos(10*pi/53)+t5*cos(100*pi/53));

S9=2*(cos(46*pi/53)+t9*cos(36*pi/53)+t5*cos(42*pi/53)+t1*cos(102*pi/53)+t10*cos(66*pi/53)+t6*cos(24*pi/53)+t2*cos(28*pi/53)+t11*cos(68*pi/53)+t7*cos(44*pi/53)+t3*cos(16*pi/53)+t12*cos(54*pi/53)+t8*cos(10*pi/53)+t4*cos(100*pi/53));

S10=2*(cos(46*pi/53)+t10*cos(36*pi/53)+t7*cos(42*pi/53)+t4*cos(102*pi/53)+t1*cos(66*pi/53)+t11*cos(24*pi/53)+t8*cos(28*pi/53)+t5*cos(68*pi/53)+t2*cos(44*pi/53)+t12*cos(16*pi/53)+t9*cos(54*pi/53)+t6*cos(10*pi/53)+t3*cos(100*pi/53));

S11=2*(cos(46*pi/53)+t11*cos(36*pi/53)+t9*cos(42*pi/53)+t7*cos(102*pi/53)+t5*cos(66*pi/53)+t3*cos(24*pi/53)+t1*cos(28*pi/53)+t12*cos(68*pi/53)+t10*cos(44*pi/53)+t8*cos(16*pi/53)+t6*cos(54*pi/53)+t4*cos(10*pi/53)+t2*cos(100*pi/53));

S12=2*(cos(46*pi/53)+t12*cos(36*pi/53)+t11*cos(42*pi/53)+t10*cos(102*pi/53)+t9*cos(66*pi/53)+t8*cos(24*pi/53)+t7*cos(28*pi/53)+t6*cos(68*pi/53)+t5*cos(44*pi/53)+t4*cos(16*pi/53)+t3*cos(54*pi/53)+t2*cos(10*pi/53)+t1*cos(100*pi/53));

In order to facilitate 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 long 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.

Then by means of the Magma, we substantiate that

J0=(A1+A2+A3+A4+A5+A6+A7+A8+A9+A10+A11+A12+A0)/13=(25425188645+465582599*sqrt(53))/2;

K0=(B1+B2+B3+B4+B5+B6+B7+B8+B9+B10+B11+B12+B0)/13=(25425188645-465582599*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)/13=(12146715932+2532962328*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)/13=(12146715932-2532962328*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)/13=(14971909446+1970308912*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)/13=(14971909446+1970308912*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)/13=(-1212575156+2618368168*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)/13=(-1212575156-2618368168*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)/13=(13314570932-2180447126*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)/13=(13314570932+2180447126*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)/13=(-17763292885-293061171*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)/13=(-17763292885+293061171*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)/13=(-12729403765-2012034791*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)/13=(-12729403765+2012034792*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)/13=(-10299774108-2660825908*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)/13=(-10299774108+2660825908*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)/13=(-18453125886-1687438350*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)/13=(-18453125886+1687438350*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)/13=(-15046690594-1391437372*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)/13=(-15046690594+1391437372*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)/13=(-8256778868+7074652*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)/13=(-8256778868-7074652*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)/13=(20510346376-1129781068*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)/13=(20510346376+1129781068*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)/13=(-2709178238+3775544656*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)/13=(-2709178238-3775544656*sqrt(53))/2.

Consequently:

A1=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t1+(17681087684-1805235744*sqrt(53))*t2+(1496603082-1157176488*sqrt(53))*t3+(16023749170-5955991782*sqrt(53))*t4+(-15054114647-4068605827*sqrt(53))*t5+(-10020225527-5787579447*sqrt(53))*t6+(-7590595870-6436370564*sqrt(53))*t7+(-15743947648-5462983006*sqrt(53))*t8+(-12337512356-5166982028*sqrt(53))*t9+(-5547600630-3768470004*sqrt(53))*t10+(23219524614-4905325724*sqrt(53))*t11);

A2=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t2+(17681087684-1805235744*sqrt(53))*t4+(1496603082-1157176488*sqrt(53))*t6+(16023749170-5955991782*sqrt(53))*t8+(-15054114647-4068605827*sqrt(53))*t10+(-10020225527-5787579447*sqrt(53))*t12+(-7590595870-6436370564*sqrt(53))*t1+(-15743947648-5462983006*sqrt(53))*t3+(-12337512356-5166982028*sqrt(53))*t5+(-5547600630-3768470004*sqrt(53))*t7+(23219524614-4905325724*sqrt(53))*t9);

A3=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t3+(17681087684-1805235744*sqrt(53))*t6+(1496603082-1157176488*sqrt(53))*t9+(16023749170-5955991782*sqrt(53))*t12+(-15054114647-4068605827*sqrt(53))*t2+(-10020225527-5787579447*sqrt(53))*t5+(-7590595870-6436370564*sqrt(53))*t8+(-15743947648-5462983006*sqrt(53))*t11+(-12337512356-5166982028*sqrt(53))*t1+(-5547600630-3768470004*sqrt(53))*t4+(23219524614-4905325724*sqrt(53))*t7);

A4=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t4+(17681087684-1805235744*sqrt(53))*t8+(1496603082-1157176488*sqrt(53))*t12+(16023749170-5955991782*sqrt(53))*t3+(-15054114647-4068605827*sqrt(53))*t7+(-10020225527-5787579447*sqrt(53))*t11+(-7590595870-6436370564*sqrt(53))*t2+(-15743947648-5462983006*sqrt(53))*t6+(-12337512356-5166982028*sqrt(53))*t10+(-5547600630-3768470004*sqrt(53))*t1+(23219524614-4905325724*sqrt(53))*t5);

A5=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t5+(17681087684-1805235744*sqrt(53))*t10+(1496603082-1157176488*sqrt(53))*t2+(16023749170-5955991782*sqrt(53))*t7+(-15054114647-4068605827*sqrt(53))*t12+(-10020225527-5787579447*sqrt(53))*t4+(-7590595870-6436370564*sqrt(53))*t9+(-15743947648-5462983006*sqrt(53))*t1+(-12337512356-5166982028*sqrt(53))*t6+(-5547600630-3768470004*sqrt(53))*t11+(23219524614-4905325724*sqrt(53))*t3);

A6=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t6+(17681087684-1805235744*sqrt(53))*t12+(1496603082-1157176488*sqrt(53))*t5+(16023749170-5955991782*sqrt(53))*t11+(-15054114647-4068605827*sqrt(53))*t4+(-10020225527-5787579447*sqrt(53))*t10+(-7590595870-6436370564*sqrt(53))*t3+(-15743947648-5462983006*sqrt(53))*t9+(-12337512356-5166982028*sqrt(53))*t2+(-5547600630-3768470004*sqrt(53))*t8+(23219524614-4905325724*sqrt(53))*t1);

A7=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t7+(17681087684-1805235744*sqrt(53))*t1+(1496603082-1157176488*sqrt(53))*t8+(16023749170-5955991782*sqrt(53))*t2+(-15054114647-4068605827*sqrt(53))*t9+(-10020225527-5787579447*sqrt(53))*t3+(-7590595870-6436370564*sqrt(53))*t10+(-15743947648-5462983006*sqrt(53))*t4+(-12337512356-5166982028*sqrt(53))*t11+(-5547600630-3768470004*sqrt(53))*t5+(23219524614-4905325724*sqrt(53))*t12);

A8=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t8+(17681087684-1805235744*sqrt(53))*t3+(1496603082-1157176488*sqrt(53))*t11+(16023749170-5955991782*sqrt(53))*t6+(-15054114647-4068605827*sqrt(53))*t1+(-10020225527-5787579447*sqrt(53))*t9+(-7590595870-6436370564*sqrt(53))*t4+(-15743947648-5462983006*sqrt(53))*t12+(-12337512356-5166982028*sqrt(53))*t7+(-5547600630-3768470004*sqrt(53))*t2+(23219524614-4905325724*sqrt(53))*t10);

A9=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t9+(17681087684-1805235744*sqrt(53))*t5+(1496603082-1157176488*sqrt(53))*t1+(16023749170-5955991782*sqrt(53))*t10+(-15054114647-4068605827*sqrt(53))*t6+(-10020225527-5787579447*sqrt(53))*t2+(-7590595870-6436370564*sqrt(53))*t11+(-15743947648-5462983006*sqrt(53))*t7+(-12337512356-5166982028*sqrt(53))*t3+(-5547600630-3768470004*sqrt(53))*t12+(23219524614-4905325724*sqrt(53))*t8);

A10=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t10+(17681087684-1805235744*sqrt(53))*t7+(1496603082-1157176488*sqrt(53))*t4+(16023749170-5955991782*sqrt(53))*t1+(-15054114647-4068605827*sqrt(53))*t11+(-10020225527-5787579447*sqrt(53))*t8+(-7590595870-6436370564*sqrt(53))*t5+(-15743947648-5462983006*sqrt(53))*t2+(-12337512356-5166982028*sqrt(53))*t12+(-5547600630-3768470004*sqrt(53))*t9+(23219524614-4905325724*sqrt(53))*t6);

A11=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t11+(17681087684-1805235744*sqrt(53))*t9+(1496603082-1157176488*sqrt(53))*t7+(16023749170-5955991782*sqrt(53))*t5+(-15054114647-4068605827*sqrt(53))*t3+(-10020225527-5787579447*sqrt(53))*t1+(-7590595870-6436370564*sqrt(53))*t12+(-15743947648-5462983006*sqrt(53))*t10+(-12337512356-5166982028*sqrt(53))*t8+(-5547600630-3768470004*sqrt(53))*t6+(23219524614-4905325724*sqrt(53))*t4);

A12=0.5*((28134366883-3309962057*sqrt(53))+(14855894170-1242582328*sqrt(53))*t12+(17681087684-1805235744*sqrt(53))*t11+(1496603082-1157176488*sqrt(53))*t10+(16023749170-5955991782*sqrt(53))*t9+(-15054114647-4068605827*sqrt(53))*t8+(-10020225527-5787579447*sqrt(53))*t7+(-7590595870-6436370564*sqrt(53))*t6+(-15743947648-5462983006*sqrt(53))*t5+(-12337512356-5166982028*sqrt(53))*t4+(-5547600630-3768470004*sqrt(53))*t3+(23219524614-4905325724*sqrt(53))*t2);

and

B1=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t1+(17681087684+1805235744*sqrt(53))*t2+(1496603082+1157176488*sqrt(53))*t3+(16023749170+5955991782*sqrt(53))*t4+(-15054114647+4068605827*sqrt(53))*t5+(-10020225527+5787579447*sqrt(53))*t6+(-7590595870+6436370564*sqrt(53))*t7+(-15743947648+5462983006*sqrt(53))*t8+(-12337512356+5166982028*sqrt(53))*t9+(-5547600630+3768470004*sqrt(53))*t10+(23219524614+4905325724*sqrt(53))*t11);

B2=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t2+(17681087684+1805235744*sqrt(53))*t4+(1496603082+1157176488*sqrt(53))*t6+(16023749170+5955991782*sqrt(53))*t8+(-15054114647+4068605827*sqrt(53))*t10+(-10020225527+5787579447*sqrt(53))*t12+(-7590595870+6436370564*sqrt(53))*t1+(-15743947648+5462983006*sqrt(53))*t3+(-12337512356+5166982028*sqrt(53))*t5+(-5547600630+3768470004*sqrt(53))*t7+(23219524614+4905325724*sqrt(53))*t9);

B3=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t3+(17681087684+1805235744*sqrt(53))*t6+(1496603082+1157176488*sqrt(53))*t9+(16023749170+5955991782*sqrt(53))*t12+(-15054114647+4068605827*sqrt(53))*t2+(-10020225527+5787579447*sqrt(53))*t5+(-7590595870+6436370564*sqrt(53))*t8+(-15743947648+5462983006*sqrt(53))*t11+(-12337512356+5166982028*sqrt(53))*t1+(-5547600630+3768470004*sqrt(53))*t4+(23219524614+4905325724*sqrt(53))*t7);

B4=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t4+(17681087684+1805235744*sqrt(53))*t8+(1496603082+1157176488*sqrt(53))*t12+(16023749170+5955991782*sqrt(53))*t3+(-15054114647+4068605827*sqrt(53))*t7+(-10020225527+5787579447*sqrt(53))*t11+(-7590595870+6436370564*sqrt(53))*t2+(-15743947648+5462983006*sqrt(53))*t6+(-12337512356+5166982028*sqrt(53))*t10+(-5547600630+3768470004*sqrt(53))*t1+(23219524614+4905325724*sqrt(53))*t5);

B5=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t5+(17681087684+1805235744*sqrt(53))*t10+(1496603082+1157176488*sqrt(53))*t2+(16023749170+5955991782*sqrt(53))*t7+(-15054114647+4068605827*sqrt(53))*t12+(-10020225527+5787579447*sqrt(53))*t4+(-7590595870+6436370564*sqrt(53))*t9+(-15743947648+5462983006*sqrt(53))*t1+(-12337512356+5166982028*sqrt(53))*t6+(-5547600630+3768470004*sqrt(53))*t11+(23219524614+4905325724*sqrt(53))*t3);

B6=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t6+(17681087684+1805235744*sqrt(53))*t12+(1496603082+1157176488*sqrt(53))*t5+(16023749170+5955991782*sqrt(53))*t11+(-15054114647+4068605827*sqrt(53))*t4+(-10020225527+5787579447*sqrt(53))*t10+(-7590595870+6436370564*sqrt(53))*t3+(-15743947648+5462983006*sqrt(53))*t9+(-12337512356+5166982028*sqrt(53))*t2+(-5547600630+3768470004*sqrt(53))*t8+(23219524614+4905325724*sqrt(53))*t1);

B7=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t7+(17681087684+1805235744*sqrt(53))*t1+(1496603082+1157176488*sqrt(53))*t8+(16023749170+5955991782*sqrt(53))*t2+(-15054114647+4068605827*sqrt(53))*t9+(-10020225527+5787579447*sqrt(53))*t3+(-7590595870+6436370564*sqrt(53))*t10+(-15743947648+5462983006*sqrt(53))*t4+(-12337512356+5166982028*sqrt(53))*t11+(-5547600630+3768470004*sqrt(53))*t5+(23219524614+4905325724*sqrt(53))*t12);

B8=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t8+(17681087684+1805235744*sqrt(53))*t3+(1496603082+1157176488*sqrt(53))*t11+(16023749170+5955991782*sqrt(53))*t6+(-15054114647+4068605827*sqrt(53))*t1+(-10020225527+5787579447*sqrt(53))*t9+(-7590595870+6436370564*sqrt(53))*t4+(-15743947648+5462983006*sqrt(53))*t12+(-12337512356+5166982028*sqrt(53))*t7+(-5547600630+3768470004*sqrt(53))*t2+(23219524614+4905325724*sqrt(53))*t10);

B9=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t9+(17681087684+1805235744*sqrt(53))*t5+(1496603082+1157176488*sqrt(53))*t1+(16023749170+5955991782*sqrt(53))*t10+(-15054114647+4068605827*sqrt(53))*t6+(-10020225527+5787579447*sqrt(53))*t2+(-7590595870+6436370564*sqrt(53))*t11+(-15743947648+5462983006*sqrt(53))*t7+(-12337512356+5166982028*sqrt(53))*t3+(-5547600630+3768470004*sqrt(53))*t12+(23219524614+4905325724*sqrt(53))*t8);

B10=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t10+(17681087684+1805235744*sqrt(53))*t7+(1496603082+1157176488*sqrt(53))*t4+(16023749170+5955991782*sqrt(53))*t1+(-15054114647+4068605827*sqrt(53))*t11+(-10020225527+5787579447*sqrt(53))*t8+(-7590595870+6436370564*sqrt(53))*t5+(-15743947648+5462983006*sqrt(53))*t2+(-12337512356+5166982028*sqrt(53))*t12+(-5547600630+3768470004*sqrt(53))*t9+(23219524614+4905325724*sqrt(53))*t6);

B11=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t11+(17681087684+1805235744*sqrt(53))*t9+(1496603082+1157176488*sqrt(53))*t7+(16023749170+5955991782*sqrt(53))*t5+(-15054114647+4068605827*sqrt(53))*t3+(-10020225527+5787579447*sqrt(53))*t1+(-7590595870+6436370564*sqrt(53))*t12+(-15743947648+5462983006*sqrt(53))*t10+(-12337512356+5166982028*sqrt(53))*t8+(-5547600630+3768470004*sqrt(53))*t6+(23219524614+4905325724*sqrt(53))*t4);

B12=0.5*((28134366883+3309962057*sqrt(53))+(14855894170+1242582328*sqrt(53))*t12+(17681087684+1805235744*sqrt(53))*t11+(1496603082+1157176488*sqrt(53))*t10+(16023749170+5955991782*sqrt(53))*t9+(-15054114647+4068605827*sqrt(53))*t8+(-10020225527+5787579447*sqrt(53))*t7+(-7590595870+6436370564*sqrt(53))*t6+(-15743947648+5462983006*sqrt(53))*t5+(-12337512356+5166982028*sqrt(53))*t4+(-5547600630+3768470004*sqrt(53))*t3+(23219524614+4905325724*sqrt(53))*t2).

Besides

R1/A1^(1/13)=t2; R2/A2^(1/13)=t12; R3/A3^(1/13)=t6; R4/A4^(1/13)=t12; R5/A5^(1/13)=t1; R6/A6^(1/13)=t1;

S1/B1^(1/13)=t12; S2/B2^(1/13)=t9; S3/B3^(1/13)=t9; S4/B4^(1/13)=t9; S5/B5^(1/13)=t4; S6/B6^(1/13)=t4.

In summary

cos(2*pi/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)))/52;

cos(20*pi/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)))/52;

……

cos(46*pi/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)))/52;

cos(36*pi/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)))/52;

……

关键词： T-10 T-11