2022-09-08, 19:37 | #1 |
Mar 2016
2^{5}·13 Posts |
rational points on the unit circle / rotation matrix
A peaceful and pleasant night for you, wherever you stay
if p is a prime > 3 and all calculation is performed by modulo p then all rational points u/v with the same norm n=u²+v² and as non quadratic residii (jacobi (n,p)==-1) which are connected with 3 different rotation matrixs, are mapped after exponention with the group order (2^p) on the same rational point on the unit circle. The practical calculation for my homework failed: a) u1=2, v1=3; u2=4, v2=11; 2²+3²=4²+11²=13 mod 31 with jacobi (13, 31)=-1, b) the belonging rotation matrixs are (10,26)*(2)=(5) (5,10) (3)=(9) (27,4)*(2)=(4) (27,27) (3)=(11) (2,11)*(2)=(6) (20,2) (3)=(15) c) Order of the group for p=31 is 32. As both bases are non quadratic residues, you can divide the group order by 2 and the according exponent is 16 Calculation in radian: α = arctan (u1/v1)=arctan (3/2) = 0,982793723247 β = arctan (u2/v2)=arctan (11/4) = 1,22202532321 A:=16α=15,724699572 =0,702849249836 mod (pi/4) B:=16β=19,5524051714!=0,702849249836 mod (pi/4) Perhaps someone could help me. |
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