Thread: rotation matrix View Single Post
 2022-10-02, 21:22 #14 bhelmes     Mar 2016 25×13 Posts A prediction of a rotation matrix for Mp A peaceful and pleasant night for you, If Mp is a Mersenne number with exponent p I can predict the following rotation matrix M=(Mp-2^[(p-1)/2];-2^[(p-1)/2) (2^[(p-1)/2]; Mp-2^[(p-1)/2]) with det (M)=1 Example: Mp=31 p=5 M=(27, -4) (4, 27) with det (M)=16+16=1 mod 31. This rotation matrix is not a result of two vectors (x1,y1), (x2, y2) with the same norm x1²+y²=x2²+y2²=n mod Mp, which needed to be found in advance, but it is completely predictive. I did not understand, why this matrix occurs one time for non quadratic residues and one time for quadratic residues: (27,4)* (2)=(4) (-4,27) (3)=(11) with 22+32 = 13 and jac (13, 31)=-1 (27,4)* (5)=(8) (-4,27) (7)=(14) 52+72 = 12 and jac (12, 31)=-1 (27,27)* (8)=(1) (-27,27) (15)=(3) 82+152 = 10 and jac (10, 31)=1 Is there a mathematical explication for this phenomenon ? More examples and as usual a web interface under http://devalco.de/unit_circle/system_tangens.php Enjoy the predictable rotation matrix,