Tao (2022) proved Col_min(N) < f(N) for almost all N in logarithmic density and asked (Remark 1.16) whether this holds in natural density. This preprint answers affirmatively, with an explicit rate and a time bound: the witnessing iterate occurs within (1+o(1)) ln(N)/ln(4/3) Syracuse steps, i.e. under 12 ln(N) Collatz steps.

Zenodo (PDF + LaTeX + numerics): https://doi.org/10.5281/zenodo.21340150

Error reports welcome.

Follow-up to Bases of starting 5-tuples are Proth primes and form cycles mod 100 : r/Collatz.

I finally dared to ask my new friend, Google AI to connect the two (Collatz was never mentioned before). Here is its answer:

"Proth numbers and the Collatz conjecture are deeply linked because the formula for a Proth number k*2^n+1 directly represents an idealized "reverse path" or "highway" in the Collatz process. [1]

When you run the Collatz conjecture (3n+1), odd numbers grow and even numbers shrink by dividing by 2. If you trace the sequence backward, Proth numbers represent a perfect, uninterrupted cascade of divisions. [1, 2]

1. Reverse Collatz Paths

In the standard Collatz function, if an odd integer \(x\) passes through a single (3n+1) phase to reach a power of 2, the algebra looks like this:
3x+1=2^n

Solving for x, you get:
x={2^n-1})3
This is the exact definition of a Mersenne-like Proth number where k=1/3. Any time a Collatz trajectory hits an integer of this form, it drops directly down to 1 via division by 2, without hitting any more odd numbers. [1, 2]

1. The General Proth "Highway"

If a Collatz sequence hits an odd number that requires m structural loops before collapsing into a clean stream of divisions, it aligns precisely with the broader Proth equation:
N=k*2^n+1

When a number fits this algebraic structure:

• It guarantees that the number can be reached by working backward from k through exactly n consecutive multiplications by 2 (the inverse of the even Collatz rule).
• Proth numbers function as major structural nodes or "highways" in the Collatz tree visualization. [1, 2]

1. Modulo 100 Constraints in Collatz

Because Proth numbers dictate these paths, the mod 100 cycle restrictions discussed earlier inject a structural limit into Collatz trajectories. Since a Proth number with n>=2 can never result in residues like 31, 51, 71, or 91, it mathematically proves that a Collatz sequence can never transition directly from an even division into a Proth "highway" landing on those specific terminal digits."

To the best of my understanding, this hold for the right side of any dome. Does this mean that a sequence alternates from bridges from one side of the dome to the other side of another dome ? I suspect it for a while, but I now need to check this.

Project "Tuples and segments" in 13 pages : r/Collatz

Follow-up to All but one bases of starting 5-tuples are Proth primes : u/No_Assist4814 and Bases of 5-tuples in domes follow the Swiss Watch (mod 100) : r/Collatz.

My new friend, Goggle AI, provides the following,

"Proth primes mod 100 do not form a single cyclic sequence because they are primes, but the underlying Proth numbers (k*2^n + 1\)) are governed by a strict 20-step cycle.

Because (2^n) (mod 100) has a repeating period of 20 starting at n=2, the last two digits of any Proth number with a fixed k will cycle through a predictable sequence every 20 powers.

The 20-Step Base Cycle

For any fixed odd multiplier k, the value of (2^n) (mod 100) loops through these 20 repeating remainders:
[4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52]

Proth Endings Constraints

Multiplying this cycle by an odd k and adding 1 restricts the possible last two digits of all Proth numbers (and thus Proth primes) where n ≥ 2. They can only end in these 25 specific modular values:

• Ending in 1: 01, 21, 41, 61, 81
• Ending in 3: 13, 33, 53, 73, 93
• Ending in 5: 05 (Only occurs when N=5; higher values are multiples of 5)
• Ending in 7: 17, 37, 57, 77, 97
• Ending in 9: 09, 29, 49, 69, 89

Because all Proth primes (except 3) satisfy N≡1 (mod 4), they can never end in 31, 51, 71, 91, 03, 23, 43, 63, 83, 25, 45, 65, 85, 07, 27, 47, 67, 87, 19, 39, 59, 79, or 99."

It has not found the Swiss watch yet (The Collatz procedure works like a Swiss watch (mod 100) : r/Collatz), but is very close. Note that the remaining numbers mentioned just above appear in blue-green bridge series, while those mentioned before do the same on the yellow bridge series.

Project "Tuples and segments" in 13 pages : r/Collatz

Note: This applies for natural numbers above two, i.e., {3, 4, 5, ...} . If anyone wants the calculations, do let me know.

Part 1: Setting the boundaries

It can be said that the given number decreases on x operations (of which (3x+1) is done 'H' times and (x/2) is done 't' times) if 3^H < 2^T .

It is also known that x = H + T

So, the boundaries can be set as follows:

(H/x) < (ln2/ln6)

(T/x) > (ln3/ln6)

Part 2: Finding a general trend for h and t as x changes

Since (3x+1) for an odd number always gives an even number, (x/2) always comes after (3x+1) . So, no two (3x+1) operations can happen consecutively. This can be rewritten as:

If a coin is flipped x times, what is the average number of heads, if no two heads appear consecutively?

An example to ensure proper understanding: Let's say the coin is flipped thrice. The possible outcomes are TTT, TTH, THT, HTT, HTH. (THH, HHT, HHH are excluded because they have consecutive heads.)

There are 5 total outcomes.

The average number of heads = (0+1+1+1+2)/5 = 1

1/5 is less than ln2/ln6 (H/x < 0.38)

Finding the formula for the total number for outcomes for x throws will lead to the formula for H/x .

After some solving, I arrived at the formula shown below (If someone knows how to format math on reddit, do let me know):

To show that the above equation does not cross (ln2/ln6), I used desmos to plot a graph (see below):

It is seen that (from 3 and onwards) the graph continues to decrease. The maximum is at (3, 1/3) - meaning that it will not cross (ln2/ln6 = 0.38)

Can someone tell me why this approach might not work?

Some users asked me for more mathematical content and fewer 3D renderings. I hope that this one-page summary better reflects the mathematical language behind the LNL/LZR project.

https://github.com/lortzingring-tech/LNL---LZR-Collatz

The repository contains the computational core of the Lange Number Line project, including deterministic search algorithms and validation tools.

Some terms used here — such as selection, forced, and Σ(-5) — are currently project-specific working concepts. Formal definitions and examples for individual Collatz peaks will be added in future updates.

Feedback, criticism, and independent verification are welcome.

For each f(N)→∞, almost every N falls below f(N) within 436 ln N steps. New: natural density and one explicit clock. Not the full conjecture. Lean-verified.

Lean formalization: 128k lines for Tao's theorem.
Extension: +60k lines.

Two upgrades to Terence Tao's result:
- Coverage: natural (ordinary counting) density rather than logarithmic density.
- Speed: one explicit clock, 436 ln N raw steps, for every diverging threshold.

Tao's log average treats multiplicative scales equally, though they contain more numbers. Ours counts every number equally: below X, success tends to 100%.

This is strictly stronger. Tao called the refinement plausible (Remarks 1.4, 1.16).

Hi All,

I may have some meaningful math regarding Tao's Syracuse random variables.

Long time listener; first time caller. Also, terrified to be here. First and foremost, I'd like to give a qualifier: no idea what I'm talking about. Complete imposter syndrome and I did most of this through metaphor with the assistance of LLM. Now for the headline and then the backstory:

My github repository is completely public and the more recent arc is furthering the qx+1 work and linking Dr. Maxwell Siegel's work with my own.

https://github.com/mrnathanhumphrey-droid/Collatz

Above is the repository. It is over 125k files in totality so I have left the necessary documents for replication. There are documents chronicling everything by timeline and by field as well.

https://doi.org/10.5281/zenodo.20673500

Here is a preprint I am faking my way through. I've taught myself enough Lean 4 to verify everything. That has its own DOI and repository that you can find through my profile.

Now for the Storytime because if I'm putting myself out there; I'm doing so authentically. This all started originally when I was 11 and I hand plotted the first 800 digits on a TI-83. I started mapping ln(x) to the pattern and saw outliers. I thought "if my computer was powerful enough I could calculate a ton and there would be a pattern in the noise".

27 years later my best friend and I were making NBA projections for fun when I re-discovered "The Dwight Effect" in block data and the Bayesian fit reminded me of Collatz when I was 11. So I did a Bayesian fit of the stopping times with the intention of finding a pattern. From there, I just kept probing the pattern and trying to retroactively teach myself anything I found. There are a million and a half dead ends. There are stupid repetitive probes that are a waste. But, I have tried to prove every single thread and finding wrong a million times over before presenting it.

If it's all fugazi; it's okay. If it's not; hopefully it will be of help to someone. I'm having fun regardless.

​

I have been investigating the Collatz conjecture from a local structural perspective. Rather than trying to prove the entire conjecture directly, I analyze the final approach to the largest value reached by a trajectory.

For a starting value n, define

x0 = n

and

x(k+1) = C(x(k))

where

C(x) = 3x + 1, if x is odd

C(x) = x / 2, if x is even.

Let W denote the maximum value reached by the trajectory.

I examine the nine values immediately preceding W:

D9(W) = (n-9, n-8, ..., n-1),

where

C(n-1) = W.

Computational experiments for all starting values up to

10^6

revealed a surprisingly rigid local structure.

The observed pattern can be written as

D9 = D4(selection) ⊗ Σ(-5) ⊗ D4(forced).

The three components are:

D4(selection) = (n-9, n-8, n-7, n-6),

Σ(-5) = n-5,

and

D4(forced) = (n-4, n-3, n-2, n-1).

The first four positions contain variation, the value at position -5 acts as a switch, and the last four positions exhibit a forced structure.

--------------------------------------------------

1. The forced four-step structure

For every tested trajectory, the final four predecessors satisfy

E → O → E → O → W,

where E denotes an even number and O denotes an odd number.

Thus,

n-4 is even,

n-3 is odd,

n-2 is even,

n-1 is odd,

and therefore

W = 3n-1 + 1

is even.

Once the trajectory enters this configuration, the last four operations are completely determined by parity.

--------------------------------------------------

1. The switch at position -5

The value five operations before the final maximum is highly constrained.

The parity conditions imply that the switch value must belong to one of the following eight residue classes modulo 64:

7, 12, 23, 28, 39, 44, 55, 60 (mod 64).

However, only five of these classes can actually occur before a final maximum:

Σ(-5)^(64) = {7, 12, 23, 44, 55} (mod 64).

The remaining three classes,

{28, 39, 60} (mod 64),

never occur.

These classes are not merely absent statistically; they can be excluded algebraically.

--------------------------------------------------

1. Why 28 (mod 64) is impossible

Let

n-5 = 64q + 28.

The next five Collatz operations are:

64q + 28

→ 32q + 14

→ 16q + 7

→ 48q + 22

→ 24q + 11

→ 72q + 34.

Thus the local peak candidate is

W = 72q + 34.

But the trajectory continues:

72q + 34

→ 36q + 17

→ 108q + 52.

Since

108q + 52 > 72q + 34

for all q ≥ 0, the first peak candidate is always exceeded.

Therefore,

28 (mod 64)

cannot occur before the final maximum.

--------------------------------------------------

1. Why 39 (mod 64) is impossible

Let

n-5 = 64q + 39.

Then

64q + 39

→ 192q + 118

→ 96q + 59

→ 288q + 178

→ 144q + 89

→ 432q + 268.

The local peak candidate is

W = 432q + 268.

A later value is

486q + 304.

Because

486q + 304 > 432q + 268,

the candidate peak is necessarily exceeded.

Therefore,

39 (mod 64)

cannot occur before the final maximum.

--------------------------------------------------

1. Why 60 (mod 64) is impossible

Let

n-5 = 64q + 60.

Then

64q + 60

→ 32q + 30

→ 16q + 15

→ 48q + 46

→ 24q + 23

→ 72q + 70.

The local peak candidate is

W = 72q + 70.

The trajectory then continues:

72q + 70

→ 36q + 35

→ 108q + 106.

Since

108q + 106 > 72q + 70,

the candidate peak is always surpassed.

Therefore,

60 (mod 64)

cannot occur before the final maximum.

--------------------------------------------------

1. Proven local exclusion result

The three exclusions follow from explicit symbolic calculations:

28, 39, 60 (mod 64)

⇒ the local peak is necessarily exceeded.

Consequently, every final maximum satisfying the observed nine-step structure must obey

n-5 ≡ 7, 12, 23, 44, or 55 (mod 64).

This is the local structural result established here.

This is not a proof of the Collatz conjecture itself. The result concerns the local structure immediately preceding final trajectory maxima.

--------------------------------------------------

1. Local peak creation versus peak survival

The five remaining classes are locally admissible, but they do not automatically guarantee that the resulting peak is the final maximum.

Define local admissibility by

L(r) = 1

if the residue class r (mod 64) is not eliminated by the immediate continuation.

Then

L(r) = 1 if and only if

r ∈ {7, 12, 23, 44, 55}.

Define global survival by

S(s) = 1

if the peak generated by the switch value s is never exceeded later in the trajectory.

Then

S(s) = 1 ⇒ L(s mod 64) = 1,

but

L(s mod 64) = 1 does not imply S(s) = 1.

Thus modulo 64 determines local admissibility but not complete survival.

--------------------------------------------------

1. A nested binary structure

The analysis was refined using increasingly large powers of two:

64, 128, 256, 512, 1024, 2048, ...

This yields a hierarchy

Σ(6) ⊃ Σ(7) ⊃ Σ(8) ⊃ Σ(9) ⊃ ...

where Σ(k) contains information modulo 2^k.

Each additional binary digit determines more of the future parity sequence.

This suggests the working model

Σ = Σ(6) ⊗ Σ(7) ⊗ Σ(8) ⊗ ...

The switch therefore appears to be a layered binary code rather than a single residue class.

Modulo 64 determines whether a candidate peak is locally admissible.

Higher powers of two reveal additional subclasses whose future behavior forces them to lose.

--------------------------------------------------

1. Interpretation

The current interpretation is that the low-order binary digits of the switch encode part of the future Collatz trajectory.

The local structure near a final maximum may therefore be written as

D9 = D4(selection) ⊗ Σ(-5)^(2^k) ⊗ D4(forced).

The first four positions select a local family.

The switch at position -5 contains binary information.

The final four positions execute the forced approach to the peak.

The remaining trajectory determines whether that peak survives.

--------------------------------------------------

1. Open questions

Can all losing subclasses be excluded by finite affine calculations?

Does every non-surviving switch eventually enter a provably forbidden dyadic subclass?

Is there a finite binary depth that determines peak survival?

I would be very interested to know whether similar residue-class restrictions around trajectory maxima have already been studied.

I have been working on an alternative structural view of the Collatz problem, which I call the LNL/LZR framework (Lange Number Line / LZR Engine).

The central idea is not to search for new arithmetic operations, but to analyze the geometric structure of Collatz trajectories in terms of mountains, valleys, peaks, trajectories and deterministic backward paths.

My current hypothesis is the following:

Every Collatz trajectory belongs to exactly one mountain.

Every peak is an even number.

The last 9 backward steps before a peak follow a deterministic pattern.

These 9 steps uniquely determine the predecessor structure of the mountain.

The mountain–valley decomposition may explain why trajectories cannot escape to infinity.

I have created several visualizations to illustrate the idea. At the moment, this is still an experimental framework rather than a formal proof, and I would be very interested in critical feedback from mathematicians.

For the mathematicians: please attack the logic, the assumptions and the definitions. 🧮

For the treasure hunters: please remember that GitHub and Zenodo timestamps exist. 😄

Constructive criticism, counterexamples and alternative interpretations are highly appreciated.

GitHub repository:

https://github.com/lortzingring-tech/LNL---LZR-Collatz

Hi. I'm just a first-year in uni, not looking to solve Collatz, but am looking to provide some hopefully interesting results. Here is the ruleset that u might not be used to but still connects back to good ol' Collatz.
Let O: n -> (3n+1)/2 for odd n and E: n -> n/2 for even n. U: n -> 4n+1 for odd n. E(-1) is the inverse of E, O(-1) is the inverse of O, and U(-1) is the inverse of U.

(Explanation for U: let n = 2k+1. Applying O to it would yield 3k+2. Let N = 4n+1 => O on N would yield 12k + 8. So applying E twice would yield 3k+2. Therefore, 4n+1 is a BACKWARD number that leads to the same number n goes to when moving FORWARD. Note: Any number that requires dividing thrice or more means that it originates from a number of the 8k+5 type, where k is a positive integer.)

Now, as far as Collatz's rules are concerned, there is no such even number that is simultaneously a multiple of 3 AND originating from an odd number. This makes sense. An even multiple of three would be 6k, while an even number in the form 3n+1 would be 6k+4. So this means that multiples of 3 can NEVER be divided by another number when doing forward operations.

This makes them the root nodes of the Collatz tree, i.e. they can only generate numbers but never be generated themselves. So I asked myself this question. Assuming that a set of odd multiples of 3 goes to 1, can we generate other odd integers using O, E, and U?

Maybe. Here's my insight. Note: This is about an implication, where IF "ALL odd multiples of 3 go to 1" -> "Every other odd number goes to 1". So, No Circular Argument. I am fully aware that if the first statement isn't proven, then the second isn't. I'm just potentially showing that an implication is there and that if we prove X, we can prove Y.

Let's first see how many numbers do odd multiples of 3 there are based on the natural set. Odd numbers comprise half of the natural set, and there can only be one multiple of 3 among every 3 consecutive odd numbers {2k-1, 2k+1, 2k+3} by the Pigeonhole Principle. So 1/2 x 1/3 = 1/6 of all natural numbers.

First, we must assume that the set of ALL odd multiples of 3 S(1/6) leads to one.

Now pick a number outside of the set. Say 7. 7 can be reversed by multiplying by 2 twice to get 28. (28-1)/3 = 9. So, a number can be rooted by a multiple of 3 through regular reverse operations of the Collatz Conjecture. But what if an odd integer doesn't need to have a root of an odd multiple of 3? Then it could never need to reach 1, assuming that S(1/6) lead to 1. So we have an infinite backwards AND forwards staircase of odd numbers. Well....
Multiply the first backwards odd integer by 4, then add 1. And u can generate a new number that tends to n. So if n is an integer that supposedly never reaches one, then u can do E(-1) and O(-1) to generate a new odd number, let's call it p. Then, you use U on p to make 4p+1, and you do it again. And again. And again and again and again and ..... The thing about 4p+1 is that it creates two more E events. But 4p+1 = 3p + (p+1). So p mod 3 can determine the modulus of 4p+1 with respect to 3. In this particular instance, it cycles through the remainders {1, 2, 0} as p shuffles in ascending order {0, 1, 2} i.e. if p is an odd integer with 0 mod 3, well, then it wouldn't even have an infinite backwards chain to begin with. if p is an odd integer with 1 mod 3, 4p+1 would be 2 mod 3. Applying U to an odd number with 2 mod 3 and you can get an odd multiple of 3 that leads to the same number that supposedly never diverges i.e. there will always be a multiple of 3 leading to any other number either through inverse Collatz operations or supercharged U operations (which are just cyclical O(-1)'s and E(-1)'s in disguise but calling them supercharged sounds cool). This contradicts the idea that a number could diverge or never end up at 1. So, in order to prove the Collatz conjecture for all numbers, you would have to prove it for odd multiples of 3. Then the argument has solid ground.

The reason why I'm bringing this up is because of Terence Tao's stabilisation theorem. He said that for "almost all numbers" his theorem worked, but didn't in the exceptional set where the logarithmic density is 0. But if there was some statistical or probabilistic way where without needing to bother with anything outside odd multiples of 3, then we may have something interesting all right.

Here's the thing. I have created a table where applying a set of O's and E's can help you map one number to another. Say that the number is n = 16a + b. Where a = 16c + d. Based on the remainders of a and c (256 possible combinations), this table allows you to reach a number k = M*a + N. M, N are natural numbers that don't necessarily have to be powers of 2.

Follow-up to Bases of 5-tuples in domes follow the Swiss Watch (mod 100) : r/Collatz.

The tables below are based on the first odd number of each right-side bridges series in every dome analyzed so far. This number is a potential base for a 5-tuple series. As we have seen, it is not always the case (without explanation so far).

They show that domes respect an double almost homothety:

• Within a dome: consecutive numbers are related by y=(x+3)/4.
• Between domes: numbers having the same position (p. q) in a dome are related by the ratio between their root m (m/m).

Our hypothesis is that the whole Collatz tree is almost homothetic by segments.

Project "Tuples and segments" in 13 pages : r/Collatz

It is well known that all Collatz paths satisfy this identity:

b/2^e - a.3^o = K

where K is sometimes referred to as the carry term.

This 3D visualiser plots (a,b, log_2(K)) for every pair of odd points (a,b) in the path from a to 1.

The points are coloured according to their mod 8 residue.

You will notice that with log_2(K) pointing at you (so the plane has y=b, x=a) is that the rightmost stacks will always be red (5 mod 8) or green (1 mod 8). Of course, this is a consequence of that fact the paths that start highest must necessarily start with a descent (any such path that doesn't start with a descent is necessarily not a local maxima)

The above is a screen capture - the full interactive version can be accessed at the link below.

(https://wildducktheories.github.io/collatz/apps/collatz-a-b-k/dist/?a=27).

​

This is a factual progress report from the Lange Number Line Project (LNL). It describes a computational sample, not a proof of the Collatz conjecture and not a world-record claim.

Search configuration:

Program: LNL/LZR 1024-bit C++ engine V2

Integer width: 1024 bits

Search region: 299-bit starting values near 10^90

Deterministic range: approximately 5.09259e89 to 1.01852e90

Workers: 18 independent C++ processes

Available hardware: 9 logical CPUs

Samples per worker: 250,000

Total starting values tested: 4,500,000

Total evaluated Collatz transitions: 9,438,626,254

Embedded endgame hits: 3,617,336

External elapsed time: approximately 4.179 seconds

The 18 processes shared 9 logical CPUs. This was deliberate oversubscription and should not be interpreted as performance from 18 physical processors.

The strongest sampled result was found in deterministic block 1000009015.

Starting value:

546155887574231641925572447161502272099920080353468344867984210086557977943510820502121831

Maximum trajectory value:

295057791277230372652050617924929578075815655945850998329262334547300781130347118135532972451999904

Main trajectory data:

Starting-value digits: 90

Peak digits: 99

Peak magnitude: approximately 2.9505779127723037e98

Peak-to-start ratio: approximately 540,244,640.752183

Step at which the peak occurs: 382

Total steps required to reach 1: 2,511

The trajectory reached 1 successfully

Verification:

The complete winning block of 250,000 starting values was recalculated using multiple independent C++ calculation paths.

Scalar calculation time: 3,275.218 ms

Batched calculation time: 1,649.836 ms

Engine-fast calculation time: 1,718.601 ms

Engine-fast speedup over scalar: approximately 1.906

Verification failures: 0

Reproducibility checksum: 1552236215461419092

The strongest candidate was also checked with a separate arbitrary-precision integer implementation. It reproduced:

The exact maximum value

Peak step 382

Total step count 2,511

Final convergence to 1

Residue information was recorded for classification only. It was not used to skip trajectories, declare convergence, or identify cache entries.

For example:

Start modulo 512: 359

Start is divisible by 3 exactly once

Peak modulo 512: 160

Peak is divisible by 2 exactly five times

Number-line scale comparison:

As a visualization, suppose consecutive natural numbers are positioned 1 millimeter apart on a straight number line.

The direct distance from coordinate 1 to the sampled peak would then be approximately:

2.9505779127723037e95 meters

At the physical speed of light, approximately 299,792.458 km/s, the direct journey would require approximately:

3.118763314654934e79 years

That is approximately 2.26e69 times the current estimated age of the universe.

This distance comparison is only a coordinate-scale illustration. It is not the length of the Collatz trajectory and has no effect on computation time.

Reproduction command format:

./search_lnl_1024_engine_native search 250000 299 BLOCK engine-fast 0

Blocks used:

1000009000 through 1000009017

Binary SHA-256:

c0ae494420fa475b2536ac5267148b40a3fb68ee365f6a257099639fb6b3ae84

Scientific limitation:

This result is the strongest candidate found within these 18 deterministic sampled blocks. It is not an exhaustive search of all natural numbers below 10^90, not a verified global Collatz record, and not a proof of the Collatz conjecture.

A Modulo-12 Markov Approximation for Collatz Dynamics

Imagine a Galton's board of 12 equidistributed silos of sand grains. Instead of random interactions or random obstacles yielding us normal distribution, the state change flows via the Collatz function yields us what and what insight can be extrapolated from the same is what this exploratory paper attempts to do. This is no way proof of Collatz conjecture and I doubt I am competent enough to prove or disprove it.

I would appreciate some feedback. "Roast" was me being hyperbolic.
(P.S. Shall I put it on Arxiv if so can someone endorse my account? I apparently have lost my old university account to the abyss)

Alt Link: https://www.researchgate.net/publication/410612929_A_Modulo-12_Markov_Approximation_for_Collatz_Dynamics

Follow-up to Do these non-prime factors have an impact ? V : r/Collatz.

I do not dare refer to primes anymore in the title, but this post used the same framework.

The figure below shows the cycles in bases of 5-tuples series mod 100 (see The Collatz procedure works like a Swiss watch (mod 100) : r/Collatz), using the same color code.

It started with the domes with a a root m divisible by 5, all ending by a 1. m=25 is the sole user, for the time being, of 01 mod 100 (red). The others use a cycle 21-41-81-61 mod 100 (blue).

I was more or less expecting something similar for the other domes, but it was slightly more complex. So, I started clustering by color, as usual and came with the figure below.

One can observe that:

• Domes form six groups, each starting with a different color, consistent with the Swiss watch.
• Each group then follows the same cycle of colors: yellow-orange-dark blue-dark red-green-yellow.
• The dome with root m=7 remains undecided for the time being.

LNL Number Signature

In addition to the exact 1024-bit integer, the LNL/LZR project records an experimental classification signature for selected starting values and trajectory peaks.

The current signature contains:

- Bit length and parity

- Residues modulo 12, 16, 32, 48, 64, 128, 256 and 512

- v2(n), the exponent of 2 dividing n

- v3(n), the exponent of 3 dividing n

- core6 = n / (2^v2(n) * 3^v3(n))

- Exact peak value

- Step at which the peak occurs

- Total steps to 1

The identity

n = core6 * 2^v2(n) * 3^v3(n)

allows the original number to be reconstructed exactly.

This signature is currently classification metadata. It is not used as proof of convergence, as a substitute for full-number equality, or as justification for skipping a trajectory. Whether particular signature classes have statistically meaningful relationships with stopping times or peak values remains an open experimental question.

I've been noodling around with Fable 5 before it's gone. Asked 5.6 Sol for verification.

https://claude.ai/code/artifact/05587812-3be7-4be9-a8b7-96326cb042f4

https://github.com/benpham3206/an-atlas-of-the-collatz-conjecture

​

PURPOSE OF THE PROJECT

The Lange Number Line Project, abbreviated LNL, is an independent computational project for calculating, encoding and examining Collatz trajectories.

LNL does not claim to prove the Collatz conjecture.

The current goals are:

Calculate individual Collatz trajectories exactly.

Create reproducible operation codes for every tested trajectory.

Detect trajectory peaks and possible path-record candidates.

Compare the software with established reference results.

Publish code and data so that other people can reproduce or challenge the calculations.

The LNL terminology is an additional descriptive layer. Every numerical result remains based on the standard Collatz operations.

STANDARD COLLATZ RULE

For every positive integer n:

If n is even:

T(n) = n / 2

If n is odd:

T(n) = 3*n + 1

The trajectory of n is:

n, T(n), T(T(n)), and so on.

The calculation continues until the trajectory reaches 1.

TRAJECTORY PEAK

The trajectory peak P(n) is the largest number reached by the complete trajectory, including the starting value.

In plain notation:

P(n) = maximum value in the trajectory starting at n

The peak position is the first step at which this maximum is reached.

PATH RECORD

A starting value n is a path record only if its peak is higher than the peak of every smaller positive starting value.

In plain notation:

P(n) > P(m) for every m < n

This distinction is important.

Finding a very large peak for one sampled starting value does not prove that the starting value is the next path record.

To confirm the next path record, every smaller starting value must first be checked.

STANDARD OPERATION CODE

LNL records every standard operation with one letter:

U = odd operation: n becomes 3*n + 1

D = even operation: n becomes n / 2

A complete trajectory can therefore be stored as a sequence of U and D operations.

ACCELERATED ODD-STEP CODE

For every odd number x, LNL calculates:

d = v2(3*x + 1)

Here, v2 means the number of times that 3*x + 1 can be divided by 2 before the result becomes odd.

The next odd number is:

next odd x = (3*x + 1) / 2^d

The resulting list of d-values is a compact encoding of the standard Collatz trajectory.

This is not a new Collatz rule. It is only a shorter representation of the same operations.

CURRENT SOFTWARE

The current LNL software includes:

A deterministic path-record scanner

Exact 128-bit calculations

Exact 256-bit calculations

Exact 512-bit calculations

U and D trajectory codes

Accelerated d-value codes

Reproducible independent seed blocks

Overflow protection

Python arbitrary-precision verification

JavaScript BigInt verification

Residue-class and transition tests

The 512-bit C++ scanner compiles without warnings using:

-Wall -Wextra -Wconversion -Werror

REFERENCE VALIDATION

Before performing experimental searches, we tested the core software against established Collatz path records.

An exhaustive calculation of every starting value from 1 through 100,000,000 reproduced 41 successive path records.

The final path record in this interval is:

Start:

80,049,391

Peak:

2,185,143,829,170,100

Peak position:

164

Total steps to 1:

572

The results agree with the published reference sequences OEIS A006884 and OEIS A006885.

Reference links:

https://oeis.org/A006884

https://oeis.org/A006885

https://pcbarina.fit.vutbr.cz/path-records.htm

ARBITRARY-PRECISION EXAMPLE

As a separate large-number test, LNL calculated the trajectory of:

n = 10^1001

This number is exactly a 1 followed by 1,001 zeros.

It has 1,002 decimal digits.

Because:

10^1001 = 2^1001 * 5^1001

the trajectory begins with exactly 1,001 divisions by 2.

After these divisions, the odd core is:

5^1001

Exact results:

Starting value:

10^1001

Decimal digits:

1,002

Binary length:

3,326 bits

Trajectory peak:

10^1001

Peak position:

0

Total steps to 1:

17,840

Odd 3*n + 1 steps:

5,615

Even n / 2 steps:

12,225

Control calculation:

5,615 + 12,225 = 17,840

The starting value itself is the peak. The trajectory never returns to or exceeds it.

Python arbitrary-precision integers and JavaScript BigInt independently produced the same result.

SAMPLED HIGH-NUMBER SEARCHES

LNL has also calculated individual trajectories for sampled starting values far beyond the exhaustively tested interval.

Some of these trajectories reach extremely large exact values.

These experiments demonstrate that:

The software can process very large integers.

The individual trajectories can be reproduced.

The 512-bit engine works beyond the normal 64-bit range.

These results do not establish new path records.

At very large starting values, the starting value itself may already be larger than an older published peak.

For this reason, these results are described only as:

Exact sampled trajectories

or:

Sampled height exceeders

They are not described as confirmed world records.

WHAT LNL DOES NOT CLAIM

LNL does not currently claim:

A proof of the Collatz conjecture

Proof that every positive integer reaches 1

Discovery of the next official path record

Exhaustive verification beyond the stated limits

That sampled searches can replace exhaustive calculations

That every descriptive LNL term is mathematically new

NEXT SCIENTIFIC GOAL

The next goal is not simply to generate a larger number.

The next goal is to improve reproducibility.

PHASE 1: PUBLIC SOFTWARE RELEASE

Publish:

C++ source code

Python verification code

Compiler commands

Exact definitions

Test results

CSV result tables

SHA-256 file hashes

PHASE 2: EXHAUSTIVE EXTENSION

Extend the deterministic path-record calculation from:

100,000,000

toward:

1,000,000,000

No starting values may be skipped.

PHASE 3: INDEPENDENT VERIFICATION

Every detected path record must be recalculated with a separate arbitrary-precision implementation.

A result will be called a path record only after every smaller starting value in the interval has been checked.

PHASE 4: STRUCTURAL ANALYSIS

After the numerical baseline is secure, we will examine whether U/D codes, d-value sequences and inverse structures contain useful predictive information.

INVITATION TO REPRODUCE OR CRITICIZE

Technical criticism is welcome.

A reproducible countercheck should include:

The exact starting value.

The Collatz convention used.

The source code or algorithm.

The first trajectory position where the results differ.

The integer type and overflow protection.

Whether the starting value is included in the peak definition.

The purpose of LNL is not to replace established mathematics with graphics or terminology.

The purpose is to build a transparent computational framework whose results can be independently reproduced, tested, criticized and improved.

Copyright (c) 2026 Mike Lange

Lange Number Line Project (LNL)

All rights reserved.

Follow-up to Do these non-prime factors have an impact ? IV : r/Collatz.z

The figure below is based on the same information as in the previous posts, with some changes:

• The base b underlying a 5-tuple is presented here as b mod 6 (see previous post) and colored according to the color of the corresponding 5-tuple.
• The domes with root m (rows) form classes m mod 16. Every third dome is embedded in another one when m+16k is divisible by 3.

One can observe that:

• 5-tuples series of a given row start (on the right) with a given color,
• Within a class of domes, the starting color alternate.
• Classes of domes have a propensity to form 5-tuples series of certain lengths.

Project "Tuples and segments" in 13 pages : r/Collatz

​

This is my final full peak report posted directly on Reddit before moving the Lange Number Line Project documentation, source code, datasets, and reproducible releases to GitHub and Zenodo.

The latest reproducible 200-bit sampled search produced a new strongest LNL trajectory.

Start value:

1084820695304608898376597485794546984528180820629394323505151

Exact peak:

12324763716514253071447088281192419703733391394845870197521733855368

Main results:

Peak reached at step 142

Total steps to 1: 1,593

Odd steps: 539

Even steps: 1,054

Peak length: 68 digits

Approximately 3,254,796.718309 times above the previous LNL candidate

The candidate was found using exact 512-bit C++ arithmetic. Its complete trajectory was independently recalculated to 1 using JavaScript BigInt.

Important scientific limitation:

This is the strongest trajectory found in the documented LNL sampled search detected on 20th of July 2026. It is not a proven global Collatz record and not a proof of the Collatz conjecture. Confirming a global record would require exhaustive verification of every smaller starting value.

Future code, datasets, manifests, checksums, and reproducible releases will be published through GitHub and Zenodo. I will continue using Reddit for occasional progress updates and discussion.

Thank you for the criticism, questions, and constructive advice.

Best regards,

Mike Lange

LZS SHORT KEY CODE FOR n = 1 TO 1000

WHAT THE CODE MEANS

For a current lock value z, the smallest odd reverse Collatz key k is selected by z mod 3:

4: If z mod 3 = 1, use k = (4z - 1) / 3. This is an S4 key.

2: If z mod 3 = 2, use k = (2z - 1) / 3. This is an S2 key.

D: If z mod 3 = 0, no odd key exists under this minimal-key rule. The reverse chain ends at a dead end.

F1: The chain reaches the fixed root 1.

Repeated symbols are compressed:

2x3 means 2.2.2.

The number after D is the terminal dead-end value.

The symbol > separates the key signature from its terminal state.

Example:

7829:2x3>D2319

This expands to:

7829 -> 5219 -> 3479 -> 2319

The three S2 keys are followed by the dead end 2319.

IMPORTANT SCOPE

This is a deterministic and reproducible reverse Key-Lock code. It is not, by itself, a proof of the Collatz conjecture. It records the minimal odd-key ancestry defined above. The forward LVR trajectory code remains a separate layer.

DATASET: n = 1 TO 1000

1:F1

2:2>F1

3:D3

4:4.2>D3

5:2>D3

6:D6

7:4>D9

8:2x2>D3

9:D9

10:4x2.2x2.4>D9

11:2.4>D9

12:D12

13:4.2x2.4>D9

14:2>D9

15:D15

16:4>D21

17:2x2.4>D9

18:D18

19:4x2>D33

20:2.4.2x2.4>D9

21:D21

22:4.2.4x2>D33

23:2>D15

24:D24

25:4>D33

26:2x3.4>D9

27:D27

28:4x3.2.4>D57

29:2.4x2>D33

30:D30

31:4.2>D27

32:2>D21

33:D33

34:4>D45

35:2x2>D15

36:D36

37:4x2.2.4>D57

38:2.4>D33

39:D39

40:4.2x3>D15

41:2>D27

42:D42

43:4>D57

44:2x2.4x2>D33

45:D45

46:4x2>D81

47:2.4.2>D27

48:D48

49:4.2.4>D57

50:2>D33

51:D51

52:4>D69

53:2x3>D15

54:D54

55:4x3>D129

56:2.4x2.2.4>D57

57:D57

58:4.2>D51

59:2>D39

60:D60

61:4>D81

62:2x2>D27

63:D63

64:4x2.2>D75

65:2.4>D57

66:D66

67:4.2x2>D39

68:2>D45

69:D69

70:4>D93

71:2x2.4.2>D27

72:D72

73:4x2>D129

74:2.4.2.4>D57

75:D75

76:4.2.4.2x2>D39

77:2>D51

78:D78

79:4>D105

80:2x4>D15

81:D81

82:4x4.2>D171

83:2.4x3>D129

84:D84

85:4.2>D75

86:2>D57

87:D87

88:4>D117

89:2x2>D39

90:D90

91:4x2.2x4.4.2>D27

92:2.4>D81

93:D93

94:4.2x2.4x3>D129

95:2>D63

96:D96

97:4>D129

98:2x2.4>D57

99:D99

100:4x2>D177

101:2.4.2x2>D39

102:D102

103:4.2.4x2.2x4.4.2>D27

104:2>D69

105:D105

106:4>D141

107:2x3.4.2>D27

108:D108

109:4x3.2>D171

110:2.4x2>D129

111:D111

112:4.2>D99

113:2>D75

114:D114

115:4>D153

116:2x2>D51

117:D117

118:4x2.2.4.2>D123

119:2.4>D105

120:D120

121:4.2x4.4.2>D27

122:2>D81

123:D123

124:4>D165

125:2x2.4x3>D129

126:D126

127:4x2>D225

128:2.4.2>D75

129:D129

130:4.2.4>D153

131:2>D87

132:D132

133:4>D177

134:2x3>D39

135:D135

136:4x3>D321

137:2.4x2.2x4.4.2>D27

138:D138

139:4.2>D123

140:2>D93

141:D141

142:4>D189

143:2x2>D63

144:D144

145:4x2.2>D171

146:2.4>D129

147:D147

148:4.2x2>D87

149:2>D99

150:D150

151:4>D201

152:2x2.4.2x2>D39

153:D153

154:4x2>D273

155:2.4.2.4x2.2x4.4.2>D27

156:D156

157:4.2.4.2>D123

158:2>D105

159:D159

160:4>D213

161:2x4.4.2>D27

162:D162

163:4x4>D513

164:2.4x3.2>D171

165:D165

166:4.2>D147

167:2>D111

168:D168

169:4>D225

170:2x2>D75

171:D171

172:4x2.2x2>D135

173:2.4>D153

174:D174

175:4.2x2.4.2.4x2.2x4.4.2>D27

176:2>D117

177:D177

178:4>D237

179:2x2.4>D105

180:D180

181:4x2>D321

182:2.4.2x4.4.2>D27

183:D183

184:4.2.4x4>D513

185:2>D123

186:D186

187:4>D249

188:2x3.4x3>D129

189:D189

190:4x3.2x2.4x2.2.4x2>D417

191:2.4x2>D225

192:D192

193:4.2>D171

194:2>D129

195:D195

196:4>D261

197:2x2>D87

198:D198

199:4x2.2.4x2>D417

200:2.4>D177

201:D201

202:4.2x3.4>D105

203:2>D135

204:D204

205:4>D273

206:2x2.4x2.2x4.4.2>D27

207:D207

208:4x2>D369

209:2.4.2>D123

210:D210

211:4.2.4>D249

212:2>D141

213:D213

214:4>D285

215:2x3>D63

216:D216

217:4x3>D513

218:2.4x2.2>D171

219:D219

220:4.2>D195

221:2>D147

222:D222

223:4>D297

224:2x2>D99

225:D225

226:4x2.2>D267

227:2.4>D201

228:D228

229:4.2x2>D135

230:2>D153

231:D231

232:4>D309

233:2x2.4.2.4x2.2x4.4.2>D27

234:D234

235:4x2>D417

236:2.4.2.4.2>D123

237:D237

238:4.2.4.2.4>D249

239:2>D159

240:D240

241:4>D321

242:2x5.4.2>D27

243:D243

244:4x5.2x3>D303

245:2.4x4>D513

246:D246

247:4.2>D219

248:2>D165

249:D249

250:4>D333

251:2x2>D111

252:D252

253:4x2.2x2.4x2.2.4x2>D417

254:2.4>D225

255:D255

256:4.2x2.4>D201

257:2>D171

258:D258

259:4>D345

260:2x2.4>D153

261:D261

262:4x2>D465

263:2.4.2x2.4.2.4x2.2x4.4.2>D27

264:D264

265:4.2.4x2>D417

266:2>D177

267:D267

268:4>D357

269:2x3.4>D105

270:D270

271:4x3.2.4.2.4x3>D897

272:2.4x2>D321

273:D273

274:4.2>D243

275:2>D183

276:D276

277:4>D369

278:2x2>D123

279:D279

280:4x2.2.4>D441

281:2.4>D249

282:D282

283:4.2x3>D111

284:2>D189

285:D285

286:4>D381

287:2x2.4x2>D225

288:D288

289:4x2>D513

290:2.4.2>D171

291:D291

292:4.2.4>D345

293:2>D195

294:D294

295:4>D393

296:2x3>D87

297:D297

298:4x3>D705

299:2.4x2.2.4x2>D417

300:D300

301:4.2>D267

302:2>D201

303:D303

304:4>D405

305:2x2>D135

306:D306

307:4x2.2>D363

308:2.4>D273

309:D309

310:4.2x2>D183

311:2>D207

312:D312

313:4>D417

314:2x2.4.2>D123

315:D315

316:4x2>D561

317:2.4.2.4>D249

318:D318

319:4.2.4.2x3>D111

320:2>D213

321:D321

322:4>D429

323:2x4>D63

324:D324

325:4x4.2x3>D303

326:2.4x3>D513

327:D327

328:4.2>D291

329:2>D219

330:D330

331:4>D441

332:2x2>D147

333:D333

334:4x2.2x3.4.2x2.4.2.4x2.2x4.4.2>D27

335:2.4>D297

336:D336

337:4.2x2.4x2.2.4x2>D417

338:2>D225

339:D339

340:4>D453

341:2x2.4>D201

342:D342

343:4x2>D609

344:2.4.2x2>D135

345:D345

346:4.2.4x2.2>D363

347:2>D231

348:D348

349:4>D465

350:2x3.4.2.4x2.2x4.4.2>D27

351:D351

352:4x3.2>D555

353:2.4x2>D417

354:D354

355:4.2>D315

356:2>D237

357:D357

358:4>D477

359:2x2>D159

360:D360

361:4x2.2.4.2.4x3>D897

362:2.4>D321

363:D363

364:4.2x5>D63

365:2>D243

366:D366

367:4>D489

368:2x2.4x4>D513

369:D369

370:4x2>D657

371:2.4.2>D219

372:D372

373:4.2.4>D441

374:2>D249

375:D375

376:4>D501

377:2x3>D111

378:D378

379:4x3>D897

380:2.4x2.2x2.4x2.2.4x2>D417

381:D381

382:4.2>D339

383:2>D255

384:D384

385:4>D513

386:2x2>D171

387:D387

388:4x2.2>D459

389:2.4>D345

390:D390

391:4.2x2>D231

392:2>D261

393:D393

394:4>D525

395:2x2.4.2x2.4.2.4x2.2x4.4.2>D27

396:D396

397:4x2>D705

398:2.4.2.4x2>D417

399:D399

400:4.2.4.2>D315

401:2>D267

402:D402

403:4>D537

404:2x4.4>D105

405:D405

406:4x4>D1281

407:2.4x3.2.4.2.4x3>D897

408:D408

409:4.2>D363

410:2>D273

411:D411

412:4>D549

413:2x2>D183

414:D414

415:4x2.2x2>D327

416:2.4>D369

417:D417

418:4.2x2.4.2>D219

419:2>D279

420:D420

421:4>D561

422:2x2.4>D249

423:D423

424:4x2>D753

425:2.4.2x3>D111

426:D426

427:4.2.4x3>D897

428:2>D285

429:D429

430:4>D573

431:2x3.4x2>D225

432:D432

433:4x3.2x3>D303

434:2.4x2>D513

435:D435

436:4.2>D387

437:2>D291

438:D438

439:4>D585

440:2x2>D195

441:D441

442:4x2.2.4x2.2.4>D825

443:2.4>D393

444:D444

445:4.2x3.4.2x2.4.2.4x2.2x4.4.2>D27

446:2>D297

447:D447

448:4>D597

449:2x2.4x2.2.4x2>D417

450:D450

451:4x2>D801

452:2.4.2>D267

453:D453

454:4.2.4>D537

455:2>D303

456:D456

457:4>D609

458:2x3>D135

459:D459

460:4x3>D1089

461:2.4x2.2>D363

462:D462

463:4.2>D411

464:2>D309

465:D465

466:4>D621

467:2x2>D207

468:D468

469:4x2.2>D555

470:2.4>D417

471:D471

472:4.2x2>D279

473:2>D315

474:D474

475:4>D633

476:2x2.4.2.4>D249

477:D477

478:4x2>D849

479:2.4.2.4.2x3>D111

480:D480

481:4.2.4.2.4x3>D897

482:2>D321

483:D483

484:4>D645

485:2x5>D63

486:D486

487:4x5>D2049

488:2.4x4.2x3>D303

489:D489

490:4.2>D435

491:2>D327

492:D492

493:4>D657

494:2x2>D219

495:D495

496:4x2.2x2.4.2x2>D231

497:2.4>D441

498:D498

499:4.2x2.4>D393

500:2>D333

501:D501

502:4>D669

503:2x2.4>D297

504:D504

505:4x2>D897

506:2.4.2x2.4x2.2.4x2>D417

507:D507

508:4.2.4x2>D801

509:2>D339

510:D510

511:4>D681

512:2x3.4>D201

513:D513

514:4x3.2.4x4.2>D1707

515:2.4x2>D609

516:D516

517:4.2>D459

518:2>D345

519:D519

520:4>D693

521:2x2>D231

522:D522

523:4x2.2.4>D825

524:2.4>D465

525:D525

526:4.2x3>D207

527:2>D351

528:D528

529:4>D705

530:2x2.4x2>D417

531:D531

532:4x2>D945

533:2.4.2>D315

534:D534

535:4.2.4>D633

536:2>D357

537:D537

538:4>D717

539:2x3>D159

540:D540

541:4x3>D1281

542:2.4x2.2.4.2.4x3>D897

543:D543

544:4.2>D483

545:2>D363

546:D546

547:4>D729

548:2x2>D243

549:D549

550:4x2.2>D651

551:2.4>D489

552:D552

553:4.2x2>D327

554:2>D369

555:D555

556:4>D741

557:2x2.4.2>D219

558:D558

559:4x2>D993

560:2.4.2.4>D441

561:D561

562:4.2.4.2x2.4>D393

563:2>D375

564:D564

565:4>D753

566:2x4>D111

567:D567

568:4x4.2.4>D1593

569:2.4x3>D897

570:D570

571:4.2>D507

572:2>D381

573:D573

574:4>D765

575:2x2>D255

576:D576

577:4x2.2x3>D303

578:2.4>D513

579:D579

580:4.2x2.4x2>D609

581:2>D387

582:D582

583:4>D777

584:2x2.4>D345

585:D585

586:4x2>D1041

587:2.4.2x2>D231

588:D588

589:4.2.4x2.2.4>D825

590:2>D393

591:D591

592:4>D789

593:2x3.4.2x2.4.2.4x2.2x4.4.2>D27

594:D594

595:4x3.2>D939

596:2.4x2>D705

597:D597

598:4.2>D531

599:2>D399

600:D600

601:4>D801

602:2x2>D267

603:D603

604:4x2.2.4.2x2>D423

605:2.4>D537

606:D606

607:4.2x4>D159

608:2>D405

609:D609

610:4>D813

611:2x2.4x3.2.4.2.4x3>D897

612:D612

613:4x2>D1089

614:2.4.2>D363

615:D615

616:4.2.4>D729

617:2>D411

618:D618

619:4>D825

620:2x3>D183

621:D621

622:4x3>D1473

623:2.4x2.2x2>D327

624:D624

625:4.2>D555

626:2>D417

627:D627

628:4>D837

629:2x2>D279

630:D630

631:4x2.2>D747

632:2.4>D561

633:D633

634:4.2x2>D375

635:2>D423

636:D636

637:4>D849

638:2x2.4.2x3>D111

639:D639

640:4x2>D1137

641:2.4.2.4x3>D897

642:D642

643:4.2.4.2>D507

644:2>D429

645:D645

646:4>D861

647:2x4.4x2>D225

648:D648

649:4x4>D2049

650:2.4x3.2x3>D303

651:D651

652:4.2>D579

653:2>D435

654:D654

655:4>D873

656:2x2>D291

657:D657

658:4x2.2x2>D519

659:2.4>D585

660:D660

661:4.2x2.4.2x2>D231

662:2>D441

663:D663

664:4>D885

665:2x2.4>D393

666:D666

667:4x2>D1185

668:2.4.2x3.4.2x2.4.2.4x2.2x4.4.2>D27

669:D669

670:4.2.4x3.2>D939

671:2>D447

672:D672

673:4>D897

674:2x3.4x2.2.4x2>D417

675:D675

676:4x3.2x2>D711

677:2.4x2>D801

678:D678

679:4.2>D603

680:2>D453

681:D681

682:4>D909

683:2x2>D303

684:D684

685:4x2.2.4x4.2>D1707

686:2.4>D609

687:D687

688:4.2x3.4x3.2.4.2.4x3>D897

689:2>D459

690:D690

691:4>D921

692:2x2.4x2.2>D363

693:D693

694:4x2>D1233

695:2.4.2>D411

696:D696

697:4.2.4>D825

698:2>D465

699:D699

700:4>D933

701:2x3>D207

702:D702

703:4x3>D1665

704:2.4x2.2>D555

705:D705

706:4.2>D627

707:2>D471

708:D708

709:4>D945

710:2x2>D315

711:D711

712:4x2.2>D843

713:2.4>D633

714:D714

715:4.2x2>D423

716:2>D477

717:D717

718:4>D957

719:2x2.4.2.4.2x3>D111

720:D720

721:4x2>D1281

722:2.4.2.4.2.4x3>D897

723:D723

724:4.2.4.2.4.2>D507

725:2>D483

726:D726

727:4>D969

728:2x6>D63

729:D729

730:4x6.2.4.2>D2427

731:2.4x5>D2049

732:D732

733:4.2>D651

734:2>D489

735:D735

736:4>D981

737:2x2>D327

738:D738

739:4x2.2x2.4>D777

740:2.4>D657

741:D741

742:4.2x2.4>D585

743:2>D495

744:D744

745:4>D993

746:2x2.4>D441

747:D747

748:4x2>D1329

749:2.4.2x2.4>D393

750:D750

751:4.2.4x2>D1185

752:2>D501

753:D753

754:4>D1005

755:2x3.4>D297

756:D756

757:4x3.2.4>D1593

758:2.4x2>D897

759:D759

760:4.2>D675

761:2>D507

762:D762

763:4>D1017

764:2x2>D339

765:D765

766:4x2.2.4>D1209

767:2.4>D681

768:D768

769:4.2x3>D303

770:2>D513

771:D771

772:4>D1029

773:2x2.4x2>D609

774:D774

775:4x2>D1377

776:2.4.2>D459

777:D777

778:4.2.4>D921

779:2>D519

780:D780

781:4>D1041

782:2x3>D231

783:D783

784:4x3>D1857

785:2.4x2.2.4>D825

786:D786

787:4.2>D699

788:2>D525

789:D789

790:4>D1053

791:2x2>D351

792:D792

793:4x2.2>D939

794:2.4>D705

795:D795

796:4.2x2>D471

797:2>D531

798:D798

799:4>D1065

800:2x2.4.2>D315

801:D801

802:4x2>D1425

803:2.4.2.4>D633

804:D804

805:4.2.4.2x2>D423

806:2>D537

807:D807

808:4>D1077

809:2x4>D159

810:D810

811:4x4.2>D1707

812:2.4x3>D1281

813:D813

814:4.2>D723

815:2>D543

816:D816

817:4>D1089

818:2x2>D363

819:D819

820:4x2.2x6.4x2>D225

821:2.4>D729

822:D822

823:4.2x2.4x5>D2049

824:2>D549

825:D825

826:4>D1101

827:2x2.4>D489

828:D828

829:4x2>D1473

830:2.4.2x2>D327

831:D831

832:4.2.4x2.2x2.4>D777

833:2>D555

834:D834

835:4>D1113

836:2x3.4.2>D219

837:D837

838:4x3.2>D1323

839:2.4x2>D993

840:D840

841:4.2>D747

842:2>D561

843:D843

844:4>D1125

845:2x2>D375

846:D846

847:4x2.2.4.2>D891

848:2.4>D753

849:D849

850:4.2x4.4>D297

851:2>D567

852:D852

853:4>D1137

854:2x2.4x3>D897

855:D855

856:4x2>D1521

857:2.4.2>D507

858:D858

859:4.2.4>D1017

860:2>D573

861:D861

862:4>D1149

863:2x3>D255

864:D864

865:4x3>D2049

866:2.4x2.2x3>D303

867:D867

868:4.2>D771

869:2>D579

870:D870

871:4>D1161

872:2x2>D387

873:D873

874:4x2.2>D1035

875:2.4>D777

876:D876

877:4.2x2>D519

878:2>D585

879:D879

880:4>D1173

881:2x2.4.2x2>D231

882:D882

883:4x2>D1569

884:2.4.2.4x2.2.4>D825

885:D885

886:4.2.4.2>D699

887:2>D591

888:D888

889:4>D1185

890:2x4.4.2x2.4.2.4x2.2x4.4.2>D27

891:D891

892:4x4>D2817

893:2.4x3.2>D939

894:D894

895:4.2>D795

896:2>D597

897:D897

898:4>D1197

899:2x2>D399

900:D900

901:4x2.2x2>D711

902:2.4>D801

903:D903

904:4.2x2.4.2.4>D633

905:2>D603

906:D906

907:4>D1209

908:2x2.4>D537

909:D909

910:4x2>D1617

911:2.4.2x4>D159

912:D912

913:4.2.4x4.2>D1707

914:2>D609

915:D915

916:4>D1221

917:2x3.4x3.2.4.2.4x3>D897

918:D918

919:4x3.2x2.4.2.4.2.4>D1017

920:2.4x2>D1089

921:D921

922:4.2>D819

923:2>D615

924:D924

925:4>D1233

926:2x2>D411

927:D927

928:4x2.2.4x2>D1953

929:2.4>D825

930:D930

931:4.2x3.4>D489

932:2>D621

933:D933

934:4>D1245

935:2x2.4x2.2x2>D327

936:D936

937:4x2>D1665

938:2.4.2>D555

939:D939

940:4.2.4>D1113

941:2>D627

942:D942

943:4>D1257

944:2x3>D279

945:D945

946:4x3>D2241

947:2.4x2.2>D747

948:D948

949:4.2>D843

950:2>D633

951:D951

952:4>D1269

953:2x2>D423

954:D954

955:4x2.2>D1131

956:2.4>D849

957:D957

958:4.2x2>D567

959:2>D639

960:D960

961:4>D1281

962:2x2.4.2.4x3>D897

963:D963

964:4x2>D1713

965:2.4.2.4.2>D507

966:D966

967:4.2.4.2.4>D1017

968:2>D645

969:D969

970:4>D1293

971:2x5.4x2>D225

972:D972

973:4x5.2.4.2>D2427

974:2.4x4>D2049

975:D975

976:4.2>D867

977:2>D651

978:D978

979:4>D1305

980:2x2>D435

981:D981

982:4x2.2x2.4x2>D1377

983:2.4>D873

984:D984

985:4.2x2.4>D777

986:2>D657

987:D987

988:4>D1317

989:2x2.4>D585

990:D990

991:4x2>D1761

992:2.4.2x2.4.2x2>D231

993:D993

994:4.2.4x2>D1569

995:2>D663

996:D996

997:4>D1329

998:2x3.4>D393

999:D999

1000:4x3.2.4.2x4.4x2.2x2>D327

END OF DATASET

@Copyrights by Mike Lange

I've finally built the first visualizer I've been wanting to do for ages now that it's so much easier to do.

This explainer demo is an attempt to show the relationship between base3 and base2 (as base4 because it was just way easier) for any number. I gave each node (or digit) a specific rule for how the next node should be moved or connected. I wanted to basically handle resolving a collatz chain using only the individual digits (as nodes) rather than using the number the digits represents. In the picture example, I started with column 10212, which is base 3 for 104, a multiple of 13 (104/2=52, 52/2=26, 26/2=13)

Yellow circles: The last column's green number is 10212 base3, and the row numbers in red are 122 which is base4 for 26, or 13. Notice how the row can expand further to the right? if we add a 0 to 122, it becomes 1220, which is 104, or, a multiple of 13. That's the starting number we'll use.

Purple circles: the column's green number is 2222 which is base3 for 80, (aka 5 times some multiple of 2). and the row is 1100 base4 (5 times some multiple of 2). The relationship always holds although the columns can skip numbers and land on a multiple of the odd number in the sequence.

How it works:

Quick recap similar to wikipedia's example on collatz conjecture in base 2, rather using base 4 gives better access to use the carry as useful info instead of it causing chaos. That info is recognised as a part of some number's base3. So for example:

Collatz sequence: 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274

          1011110  94 47
        10001110   142 71
       11010110    214 107
     101000010     322 161
    111100100      484 242 121
  101101100        364 182 91
100010010          274

which is laid out in such a way similar to as if I never divided by 2. But that'd be hard to read each line with so many trailing 0s so for simplicity, I kept them all aligned but removed the trailing 0s.
I then choose a column. Any column as long as I start from the beginning and finish to the end (if not, then you need more numbers):

        |  |1011110  94 47
        |10|001110   142 71
       1|10|10110    214 107
     101|00|0010     322 161
    1111|00|100      484 242 121
  101101|10|0        364 182 91
10001001|0 |         274

My column's digits are {00, 10, 10, 00, 00, 10, 00} or in base4, {0, 2, 2, 0, 0, 2, 0} Then using the table to get how much of a carry there is to the next set of 2:

	∅ (c=0)	+ (c=1)	++ (c=2)
d=0	0 / ∅	1 / ∅	2 / ∅
d=1	3 / ∅	0 / +	1 / +
d=2	2 / +	3 / +	0 / ++
d=3	1 / ++	2 / ++	3 / ++

We then get {0, ∅, +, ++, ∅, ∅, 2} which in base3 is 12002 or 137 (274/2). We just got the base3 from specific digits across multiple numbers in a sequence without even having to know what those numbers were. Of course, the reverse is also true where we can jump to a number way ahead of a base4 without needing to complete every step (once we get the column, we can skip ahead to fill out a row in base4 knowing it exists down the chain.) Which I think is a pretty neat trick, although I have no idea if this will speed up searching through numbers.
eg.

        |  |1011110  94 47
        |10|001110   142 71
       -|10|10110    ? ?
     ---|00|0010     ? ?
    ----|00|100      ? ?
  ------|10|0        ? ?
--------|0 |         ?
~10001001            137

A cool thing about rewriting that matrix into something more visible, we can then see limitations in the column's digits.

Which we can then see that any base4 digit in any column, is limited to 3 other digits and the opposite is always impossible to reach. Eg if the base4 is 3, the base4 digit below can never be a 1.

Which I wonder if there's something useful that can be gained out of this or if it's a cool party trick (dont use this trick at parties)

Extra note: The last node of a column has a special rule for adding another node below it. If it ends on any non zero, then it's not the end yet. So the rule is: 1->+0, 2->++0, 3->++2->++0.

Some stuff I still want to explore or ideas for anyone else to try:
- The last digits of any base4 number determine the exact steps for the following numbers. The longer the number itself, the more info we have about the exact path it'll take. I wonder if we reduce the entire column to a path on the circle diagram thingy, if we can then predict the result of the next digit in the same row to the left (Which we can start at 0)
- The column for every number in base4 has to have some meaning too, because it's always unique for every number, but adding 0 still somehow relates to the base3 number by also just adding a 0. eg. the column 022002 base4 (321 decimal) related in some way to 12002 base3 (137 deciml). We can multiply the base3's decimal number by any multiple of 3, and it'll still give us 022002(...0...) base4. Meaning every starting point that's a number that's a multiple of 3, has a very specific single base4 number related to it... Needs exploring...

PS: Sorry for not using proper math terminology. I'm not a mathematician but I welcome any help if you're willing. I have about 4 other ideas I want to do regarding collatz conjecture visualizers including one to show the relationship of collatz conjecture number paths on a Sierpinski triangle, so share any feedback.

Interactive demo: https://dvilicate.github.io/dvilicate_home/explainer