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 Complex Ratio and Number Resolution (Posted on 2016-04-13)
Consider the function:
G(Z) = (Z+i)/(Z-i) for all complex numbers Z ≠ i, and:
The sequence {Zn} is defined as:
G(Zn-1) = Zn, whenever n is a positive integer, and:
Z0 = 1/137 + i

Find each of A, B, C and D, given that:
Z2014 = A+B*i, and:
Z2015 = C+D*i

 No Solution Yet Submitted by K Sengupta No Rating

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 computer assisted solution | Comment 1 of 2
10   Real=1//137:Imag=1
20   for I=1 to 2015
30     a=real:c=real
40     b=imag+1:d=imag-1
50     real=(a*c+b*d)//(c*c+d*d)
60     imag=(b*c-a*d)//(c*c+d*d)
70     if i=2014 or i=2015 then
80       :print i,real;imag
90    next

Uses symbols a,b,c and d as in Wolfram Mathworld's article on complex division.

It finds

2014:  1 + 274*i
2015:  37538/37265 + 1/37265 * i

In fact the 2014th value is preceded by

2013:  1/137 + i

so actually it has been a 3-cycle since:

0:   1/137 + i
1:   1 + 274*i
2:   37538/37265 + 1/37265 * i
3:   1/137 + i

 Posted by Charlie on 2016-04-13 14:37:15

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