Thread: Collatz 3x+1 problem View Single Post
 2019-03-20, 08:40 #5 Cybertronic     Jan 2007 Germany 6418 Posts 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