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 20190320 at 09:06
