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Old 2022-09-08, 19:37   #1
bhelmes
 
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Mar 2016

25×13 Posts
Default 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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