Chapter 2.3:
Alternate Collatz Worlds
Recall that to prove the Collatz Conjecture, one must show that the x3+1 Collatz Space both has no divergence and also is "1-Looped", meaning that all the input n values have orbits which converge to the same one convergent loop. When reading the mathematical literature on the Collatz Conjecture, one often encounters discussions on two different but related Collatz Worlds, the two "neighbors" of the classical x3+1 Collatz Space — the x3+5 space and the x5+1 space. These two neighboring worlds prove surprisingly relevant to the original problem.
The x3+5 World
Where we defined Cx3+1(n) as 3n+1 when n is odd
n/2 when n is even,
let us now define Cx3+5(n) as 3n+5 when n is odd
n/2 when n is even.
Let's give an example of this alternate x3+5 world in action. Let's try out the number n = 1.
1 >> 8 >> 4 >> 2 >> 1 >>
8 >> 4 >> 2 >> 1 >>
8 >> 4 >> 2 >> 1 >>
etc...
Let's try out the next number n = 3.
3 >> 14 >> 7 >> 26 >> 13 >> 44 >> 22 >> 11 >> 38 >> 19 >>
62 >> 31 >> 98 >> 49 >> 152 >> 76 >> 38 >> 19 >>
62 >> 31 >> 98 >> 49 >> 152 >> 76 >> 38 >> 19 >>
etc...
Note that this system, unlike the x3+1 world, has more than one convergent loop! It turns out that this x3+5 system has (at least) six different convergent loops, one at 1, one at 5, one at 19, one at 23, one at 187, and one at 347. (Note: To keep track of which loops are which, different authors use different techniques. I, myself, have chosen to identify convergent loops by their least element. I call the least element in a loop the "loop seed". It is the x3+5 world's loop seeds that I have listed above.) In the x3+5 world, all the orbits seems to always fall into one of these six loops.
The various loops are:
1 >> 8 >> 4 >> 2 >> 1
5 >> 20 >> 10 >> 5
19 >> 62 >> 31 >> 98 >> 49 >> 152 >> 76 >> 38 >> 19
23 >> 74 >> 37 >> 116 >> 58 >> 29 >> 92 >> 46 >> 23
187 >> 566 >> 283 >> 854 >> 427 >> 1286 >> 643 >> 1934 >> 967 >> 2906 >> 1453 >> 4364 >>
>> 2182 >> 1091 >> 3278 >> 1639 >> 4922 >> 2461 >> 7388 >> 3694 >> 1847 >>
>> 5546 >> 2773 >> 8324 >> 4162 >> 2081 >> 6248 >> 3124 >> 1562 >> 781 >>
>> 2348 >> 1174 >> 587 >> 1766 >> 883 >> 2654 >> 1327 >> 3986 >> 1993 >>
>> 5984 >> 2992 >> 1496 >> 748 >> 374 >> 187
347 >> 1046 >> 523 >> 1574 >> 787 >> 2366 >> 1183 >> 3554 >> 1777 >> 5336 >> 2668 >>
>> 1334 >> 667 >> 2006 >> 1003 >> 3014 >> 1507 >> 4526 >> 2263 >> 6794 >>
>> 3397 >> 10196 >> 5098 >> 2549 >> 7652 >> 3826 >> 1913 >> 5744 >> 2872 >>
>> 1436 >> 718 >> 359 >> 1082 >> 541 >> 1628 >> 814 >> 407 >> 1226 >> 613 >>
>> 1844 >> 922 >> 461 >> 1388 >> 694 >> 347
This is the first world we have looked at which has more than one loop. Examining these various different loops can shed a bit of light on what it might be like if the conjecture is false and the x3+1 world does if fact have some other loop than 1 >> 4 >> 2 >> 1, somewhere way out there.
The x5+1 World
Let us now define Cx5+1(n) as 5n+1 when n is odd
n/2 when n is even.
The x5+1 Collatz world has three different convergent loops, one at 1, one at 13, and one at 17.
1 >> 6 >> 3 >> 16 >> 8 >> 4 >> 2 >> 1
13 >> 66 >> 33 >> 166 >> 83 >> 416 >> 208 >> 104 >> 52 >> 26 >> 13
17 >> 86 >> 43 >> 216 >> 108 >> 54 >> 27 >> 136 >> 68 >> 34 >> 17
But fascinatingly enough, with the x5+1 world, not ever positive integer has an orbit which appears to converge to one of these three loops! Many of the inputs you start with just simply grow and grow, appearing to diverge to infinity. We have not seen this before with either the x3+1 world or the x3+5 one. Let me give you an example. Try n = 7:
7 >> 36 >> 18 >> 9 >> 46 >> 23 >> 116 >> 58 >> 29 >> 146 >> 73 >> 366 >> 183 >> 916 >>
>> 458 >> 229 >> 1146 >> 573 >> 2866 >> 1433 >> 7166 >> 3583 >> 17916 >>
>> 8958 >> 4479 >> 22396 >> 11198 >> 5599 >> 27996 >> 13998 >> 6999 >>
>> 34996 >> 17498 >> 8749 >> 43746 >> 21873 >> 109366 >> 54683 >>
>> 273416 >> 136708 >> 68354 >> 34177 >> 170886 >> 85443 >> 427216 >>
>>213608 >> 106804 >> 53402 >> 26701 >> 133506 >> 66753 >> 333766 >>
>> 166883 >> 834416 >> 417208 >> 208644 >> 104302 >> 52151 >> 260756 >>
>> 130378 >> 65189 >> 325946 >> 162973 >> 814866 >> 407433 >> 2037166 >>
etc ...
n=7 certainly appears to diverge. After 215 steps, n=7 has risen above 10^10. After 3254 steps, it has risen above 10^100. After 32282 steps, it has gone above 10^1000. As far as humans have ever been able to look it appears to just keep rising.
However, n=7 has never been proven to diverge. No matter how high up the hailstone rises it is always distantly possible that it might (coincidentally?) land on some massive power of 2 and then tumble all the way back down to the loop at 1. Within the tumultuous, chaotic, unpredictable Collatz system, how can one prove that this never happens?
The statement "in the x5+1 world, n=7 diverges" is another famous conjecture out in the Collatz Wilderness. It's an absolutely fascinating math problem all on its own. It may well be as beautiful, haunting, and impossible as the original Collatz Conjecture.
A Question...
While reading time and again in the mathematics literature discussions about the x3+5 and x5+1 worlds, I was left with a burning question in my mind. What do these two worlds look like? I've mentioned before that I was, at the time, utterly fascinated with the images of the x3+1 Stopping Times and Max Value Graphs. I so badly wanted to know... what do the x3+5 and x5+1 Stopping Times and Max Value Graphs look like? Do they still have the characteristic mysterious patterns in them? Can one see anything different about the x5+1 space since it has divergence?
I needed to know!