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Old 2019-03-20, 08:40   #5
Cybertronic
 
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Jan 2007
Germany

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One loop I forgot

3rd critical loop: This can go to infinite. (0.5 * 3 * 0.5 * 0.5 * 3 ...)
10->01->00->10->11->10..

start with ..10: N=4k+2
I:halve it, to get a number ..01 : N=2k+1
II:3N+1 it , to get a number ..00 : N=6k+4
III:halve it, to get a number ..10 : N=3k+2
IV:halve it, to get a number ..11 : N=1.5k+1
V:3N+1 it, to get a number ..10 : N=4.5k+5

k=0,4,8,... "V" fails, because 4.5k+5 is not member of 4k+2 for ending ..10
k=1,5,9,... "I" fails, because 2k+1 is not member of 4k+1 for ending ..01
k=2,6,10,.. "III" fails, because 3k+2 is not member of 4k+2 for ending ..10
k=3,7,11,.. "II" fails, because 6k+4 is not member of 4k for ending ..00

For all k's , this loop can not run to infinite.

Summary:
possible dual combination -> after Collatz regulations

..00 -> ..00 or ..10
..01 -> ..00
..10 -> ..01 or ..11
..11 -> ..10


There are 2 critical loops where more increased threefold than halved. (condition for undless run)
These are:

loop 1: (0.5 * 3 * 0.5 * 0.5 * 3 ...) 10->01->00->10->11->10.. (9/8>1)

loop 2: (0.5 * 3 * 0.5 * 3 *...) 10->11->10->11-> 10.. (3/2)>1

Both I showed that for all numbers this is not possible.

All other loops have more halved than increased threefold. ( numbers get smaller and smaller und end to 4,2,1. )

Last fiddled with by Cybertronic on 2019-03-20 at 09:06
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