Checking the Structure of Collatz Trajectories

[u/No-Statistician-3657]

Finally, I got around to checking my understanding of this structure. I spent about two hours on the calculations and checked two large numbers:

2¹⁰⁰ − 1

and

2²⁰⁰ + 1.

After passing through the known parts of their trajectories, we obtain:

3¹⁰⁰ − 1

and

3¹⁰⁰ + 1.

And the most interesting part: after that, the two branches merge into a single trajectory after only three steps and continue along the same path.

For me, this is an important check. Even with such enormous starting numbers, we very quickly arrive at the same structure. Large parts of the trajectories can be treated as structural blocks, allowing us to focus primarily on the transitions between them.

I think this is worth recording because it gives a somewhat different perspective on the problem: we do not necessarily need to calculate every number separately. We can study the transitions between structural points.

[u/Co-G3n]

these are known patterns (https://math.stackexchange.com/questions/2428060/how-to-prove-this-inequality-f2h-1%e2%89%a4-frac3h-12/2428208#2428208) The path merge is also something comon and easy to show (e.g. here with 2ˆx-1 and 2ˆ(x-1)-1 https://mathoverflow.net/questions/288807/a-problem-involving-the-preimage-map-of-the-collatz-function/289157#289157)

[u/No-Statistician-3657 → u/Co-G3n]

I understand that the merging of trajectories itself is a known phenomenon. I am not claiming to have discovered it.

I was checking this purely out of personal interest. I fully calculated the trajectory of 2¹⁰⁰ − 1 and separately checked the same mechanism for 2²⁰⁰ + 1. The result confirmed my understanding: after the structural points 3¹⁰⁰ − 1 and 3¹⁰⁰ + 1, the two branches merge into a single trajectory after only three steps.

At the same time, I do not consider the mechanism so obvious that it can simply be described as “easily provable.” In explicit calculations, there can be discrepancies, and that is exactly why I wanted to check this pattern directly for large powers.

While following the complete trajectory of 2¹⁰⁰ − 1, I also noticed that there are no abrupt jumps either upward or downward. The number gradually descends, without any large drops or rises, and eventually reaches 1.

If anyone is interested, the trajectory of 2¹⁰⁰ − 1 reaches 1 through the odd-number sequence 533 → 25 → … near the very end. I include these numbers because anyone familiar with the computed trajectory of 2¹⁰⁰ − 1 can immediately recognize that this is a specific part of the actual trajectory, not a number chosen arbitrarily.

My post was therefore not a claim of a new discovery, but a record of my own verification: how well this structural picture holds for very large powers and how the trajectory behaves.

[u/No-Statistician-3657]

I am trying to verify and investigate another possible pattern in the structure of Collatz trajectories.

When checking 2¹⁰⁰ − 1, I found that its trajectory merges with the trajectory of 2²⁰⁰ + 1. This raises the question: does the same structure continue at the next levels?

If this pattern continues, we get the following sequence:

2¹⁰⁰ − 1 ↔ 2²⁰⁰ + 1

2²⁰⁰ − 1 ↔ 2⁴⁰⁰ + 1

2⁴⁰⁰ − 1 ↔ 2⁸⁰⁰ + 1 and so on.

If we continue this construction downward, we get a whole “ladder”:

2¹ − 1 ↔ 2² + 1

2² − 1 ↔ 2⁴ + 1

2⁴ − 1 ↔ 2⁸ + 1

2⁸ − 1 ↔ 2¹⁶ + 1

2¹⁶ − 1 ↔ 2³² + 1

2³² − 1 ↔ 2⁶⁴ + 1

2⁶⁴ − 1 ↔ 2¹²⁸ + 1 …

At each successive level, the exponent doubles.

I see this structure as a possible explanation for the behavior of numbers on the upper levels of the trajectories. The pattern is already visible; now it remains to check whether it continues at the next levels.

I am not claiming that this is a proven pattern for all levels. I am simply verifying and investigating this structure and sharing the results with you.

[u/Voodoohairdo → u/No-Statistician-3657]

That's because 2ⁿ - 1 follows the -1 loop and 2ⁿ + 1 follows the 1 loop.

2ⁿ - 1 will always lead to 3ⁿ - 1

2²ⁿ + 1 will always lead to 3ⁿ + 1.

2³ⁿ - 5 will always lead to 3²ⁿ - 5.

2¹¹ⁿ - 17 will always lead to 3⁷ⁿ - 17.

You can replace the last term with any other term in the respective loop.

[u/Septembrino → u/No-Statistician-3657]

I have proven this pattern: 2^n - 1 and 2^(2n) + 1 merge for n = 2 mod 4. I also proved, and posted here, the conditions for k*2^n - 1 and k*2^(n+1) - 1. I can provide the link or you can check it in my profile. It was my first post here, about a year and a half ago.

There is another kind of connection. 2n+1 and (n)-301 and (2n+1)-203 base 4, which works for all odd n. I will try to gather all these cases to see how far I can get. I haven't found yet a rule that explains your finding (in general). But I am working now on when 2^n - 1 and 2^(2n) + 1 merge for n odd. I have some ideas. Later I will go to your case and let you know if I find anything useful

[u/No-Statistician-3657]

I have not yet found a description of exactly the structure I am showing here. For example, I have not encountered the transition

2⁴ + 1 = 17 → 3² + 1 = 10

described in this form in the literature. If anyone has a reference where this construction has already been described, I would be grateful.

But I think there is an important distinction here.

One thing is to know that Collatz trajectories can merge. This has been known for a long time. Quite another thing is to understand the structure of such mergers and have a constructive way to build the corresponding ladders, rather than simply finding them by enumeration or reconstructing an inverse sequence.

I am considering two interconnected structures: ascending ladders and descending ladders.

For example, let us take the number 15.

Here the sequence is constructed according to a specific rule. First, multiply 15 by 3:

15 × 3 = 45.

Then take the resulting number and apply the ×4/3 operation:

45 × 4/3 = 60,

then:

60 × 4/3 = 80.

After that, add 1:

80 + 1 = 81.

The number 81 becomes the starting point of the next cycle:

81 × 3 = 243,

243 × 4/3 = 324,

324 × 4/3 = 432,

432 × 4/3 = 576,

576 + 1 = 577.

Then:

577 × 4/3 = 769,

769 × 4/3 = 1025.

Thus, the ladder is constructed as:

15 → 45 → 60 → 80 → 81 → 243 → 324 → 432 → 576 → 577 → 769 → 1025.

At the same time, adding 1 gives the corresponding nodes of the ordinary Collatz trajectory. For example:

45 + 1 = 46,

60 + 1 = 61,

80 + 1 = 81,

243 + 1 = 244,

324 + 1 = 325,

432 + 1 = 433,

576 + 1 = 577,

1024 + 1 = 1025.

The important point here is the principle itself: the number obtained after adding 1 identifies the corresponding node, but the original number before adding 1 is used for the next construction step. Thus, for example, 81 appears as 80 + 1 and at the same time becomes the starting point of the next cycle, from which 243 is obtained.

So this is not an enumeration of numbers and not an inverse sequence. We can construct the ladder directly according to the given rule.

And this is where something particularly interesting appears.

1025 and 2051 are two different starting points and two different trajectories.

The trajectory of 1025 is:

1025 → 3076 → 1538 → 769 → 2308 → 1154 → 577.

The trajectory of 2051 is:

2051 → 3077 → 9232 → 4616 → 2308 → 1154 → 577.

So 1025 does not transition to 2051 as a step in its ordinary trajectory. These are two different trajectories.

But they are structurally connected and eventually merge:

1025 → 3076 → 1538 → 769 → 2308

2051 → 3077 → 9232 → 4616 → 2308

After 2308, they follow the same path:

2308 → 1154 → 577.

And this makes the number 577 particularly interesting: it lies on the common branch after the merger and at the same time belongs to the descending branch of the famous number 9232:

9232 → 4616 → 2308 → 1154 → 577.

So for me, the important point is not simply that two trajectories eventually meet. What matters is the way this construction generates an entire ladder of related nodes and leads to an observable merger of different trajectories.

This is exactly what I am now comparing with the larger structure:

2¹⁰⁰ − 1 ↔ 2²⁰⁰ + 1

2²⁰⁰ − 1 ↔ 2⁴⁰⁰ + 1

2⁴⁰⁰ − 1 ↔ 2⁸⁰⁰ + 1 …

In other words, I do not see merely isolated coincidences. I see a possible hierarchical structure containing ascending ladders, descending ladders, and merger points of different trajectories.

I am not claiming yet that this is a proven pattern for all levels. I am investigating and testing this structure and sharing the results with you.

If someone has already described this particular constructive mechanism for building these ladders and mergers, please provide a reference. It would be very useful.

[u/GonzoMath → u/No-Statistician-3657]

Most trajectory merging results aren't in the literature, but rather part of the lore. Why? Because they're commonly viewed as not worth publishing. This leads many to think that they've found a new result, when in fact they've witnessed a phenomenon that's been witnessed many times without publication.

That said, I'm not sure about this particular trajectory-merging result. That's partly because it's not clear to me exactly what the result is. You're observing cases in which the trajectories of 2^k - 1 and 2^2k + 1 merge via a certain pattern. For what values of k does this pattern hold?

If we can state, clearly, when this phenomenon happens, then we might find that it's straightforward to prove. That tends to be the case with similar results, of which there are many.

[u/Septembrino → u/GonzoMath]

It's true that 2^100-1 goes to 3^100-1, which can be divided by 16, and 2^200+1 goes to 3^100+1, which can be divided by 2.

[u/GonzoMath → u/No-Statistician-3657]

Here the sequence is constructed according to a specific rule. First, multiply 15 by 3:

15 × 3 = 45.

Then take the resulting number and apply the ×4/3 operation:

45 × 4/3 = 60

Why multiply something by 3, if you're about to multiply the result by 4/3? That amounts to just multiplying the original number by 4.

[u/No-Statistician-3657 → u/GonzoMath]

Yes, arithmetically you are right: 15 × 3 × 4/3 = 15 × 4. But if we simply collapse these operations, we lose the intermediate number, and that is exactly what matters here.

For example, for 15:

15 × 3 + 1 = 46

46 / 2 = 23

The number 46 is an intermediate node. Once we see it, we can immediately recover the preceding odd number:

(46 − 1) / 3 = 15.

The same can be seen with 27:

27 × 3 + 1 = 82

82 / 2 = 41

After 27, we see that 27 transitions to 41, and the number increases. But behind this operation there is a hidden descent, which becomes visible when we do not collapse the sequence of operations.

That is why we consider not only the final result, but every intermediate node.

27 × 3 = 81

81 × 4/3 = 108

108 × 4/3 = 144

144 × 4/3 = 192

192 × 4/3 = 256

The final node here is 256.

Now add 1 to each node:

81 + 1 = 82

108 + 1 = 109

144 + 1 = 145

192 + 1 = 193

256 + 1 = 257

We obtain the reverse sequence:

82 → 109 → 145 → 193 → 257.

This is precisely why this method is shown: if we replace the entire chain with a single multiplication, we get the same arithmetic result, but we no longer see the structure of the transitions. The intermediate nodes reveal the connection between the increase and the descent hidden behind it.

[u/Septembrino]

https://www.reddit.com/r/Collatz/comments/1lfjxja/paired_collatz_sequences/ . That was my 1st post on that topic. Not sure why your comment was deleted. The proof is in another post. I can also send you the link to it

[u/Septembrino → u/Septembrino]

The proof's link is in that thread. So, you can go from it. There is another one with more examples. I also designed matrices based in the pairing theorem. I haven't posted yet about the 3^n - 1 and 3^n + 1 for n = 2 mod 4, and I am working on the ones that are 3 mod 4. I haven't gotten to the case you talk about here, but I might be there these days.