Thread: rotation matrix
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Old 2022-10-02, 21:22   #14
bhelmes
 
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Mar 2016

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Default 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,

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