# An Algorithm for Linearizing the Collatz Convergence

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## Abstract

**:**

## 1. Introduction

## 2. Related Research

#### 2.1. Goodstein Sequences and Hydra Games

**Definition**

**1.**

#### 2.2. L-Systems and Analogies with Statistical Physics

## 3. Contributions to the State-of-the-Art

## 4. Binary and Ternary Trees as a Novel Coordinate System for the Collatz Basins of Attraction

**Note 1.**

**Syr(x)**or the

**“Syracuse action”**as

**“the next odd number in the forward Collatz orbit of**

**x**

**”**. Whenever two numbers a and b have a common number in their orbit, we also note a≡b, a relation that is self-evidently transitive:

**Definition 2.**

**Actions G, V**

**and**

**S**

**:**For any natural number a,

- 1.
- $G\left(a\right):=2a-1$
- 2.
- $S\left(a\right):=2a+1$.
- 3.
- $V\left(a\right):=4a+1=G\circ S\left(a\right)$

**Definition 3. Rank:**

**Definition 4. Types**

**A**

**,**

**B**

**, and**

**C**

**:**

- 1.
- A number a is of type A if its base 3 representation ends with the digit 2.
- 2.
- A number b is of type B if its base 3 representation ends with the digit 0.
- 3.
- A number c is of type C if its base 3 representation ends with the digit 1.

## 5. The Five Fundamental Rules of the Collatz Dynamical System

**Theorem**

**1.**

**The reader should note that, although we call them “rules” in anticipation of their use in programming our linearizing algorithm, they are in fact theorems, which we prove in the next subsections**, where operator ⋀ is defined as $\underset{i=1}{\overset{n}{\bigwedge}}}\left({x}_{i}\right)=\underset{n}{\underbrace{{x}_{1}\wedge ...\wedge {x}_{n}}$, with ∧ representing the “AND” boolean operator).

**Rule One:**$\forall x$ odd, $V\left(x\right)\equiv \left(x\right)$**Rule Two:**$\forall x\in \mathbb{N}$ if x is odd, then, ${S}^{k}V\left(x\right)\equiv {S}^{k+1}V\left(x\right)$ with k odd. If x is even ${S}^{k}V\left(x\right)\equiv {S}^{k+1}V\left(x\right)$ with k even.**Rule Three:**$\forall \{n;y\}\in {\mathbb{N}}^{2}$, $\forall x$ odd non B, ${3}^{n}x\equiv y\Rightarrow {\displaystyle \underset{i=1}{\overset{n}{\bigwedge}}}\left(V\left({4}^{i}{3}^{n-i}x\right)\right)\wedge S\left(V\left({4}^{i}{3}^{n-i}x\right)\right)\equiv y$**Rule Four:**$\forall \{n;y\}\in {\mathbb{N}}^{2}$, $\forall x$ odd non B, $S\left({3}^{n}x\right)\equiv y\Rightarrow {\displaystyle \underset{i=1}{\overset{n}{\bigwedge}}}(S\left({4}^{i}{3}^{n-i}x\right)\wedge {S}^{2}\left({4}^{i}{3}^{n-i}x\right))\equiv y$**Rule Five:**$\forall n\in \mathbb{N}$, $\forall y\in \mathbb{N}$, $\forall x$ odd non B where ${3}^{n}x$ is of rank 1, $a\equiv y$, $a=G\left({3}^{n}x\right)\Rightarrow {\displaystyle \underset{i=0}{\overset{n}{\bigwedge}}}({S}^{i}\left(G\left({3}^{n-i}x\right)\right)\wedge {S}^{i+1}\left(G\left({3}^{n-i}x\right)\right))\equiv y$

**Definition**

**5.**

**Figure 2**, we call

**"vertical odd"**a number that can be written $V\left(o\right)$, where o is odd, and

**"vertical even"**if it can be written $V\left(e\right)$, where e is even. For example, 5 is the first vertical odd in $\mathbb{N}$ because $5=4\times 1+1$ and 9 is the first vertical even number in $\mathbb{N}$ because $9=4\times 2+1$.

#### 5.1. Proving Rule One

#### 5.2. Proving Rule Two

**Lemma**

**2.**

**Proof.**

**Note**

**2.**

**exactly four consecutive times**. Thus, any strictly ascending Collatz orbit concerns only numbers a of rank $n>1$ and is defined by

**Lemma**

**4.**

**Proof.**

**Theorem 5.**

**Rule Two**) Let a be a number that is vertical even; then,$\left(a\right)\equiv S\left(a\right)$ and ${S}^{k}\left(a\right)\equiv {S}^{k+1}\left(a\right)$ for any even k. Let a be a number that is vertical odd; then, $S\left(a\right)\equiv {S}^{2}\left(a\right)$ and ${S}^{k}\left(a\right)\equiv {S}^{k+1}\left(a\right)$ for any odd k.

**Proof.**

**Note**

**3.**

**Rule Two**, we also demonstrated that any number of rank 2 or more is finitely turned into a rank 1 number of type A by the Collatz dynamic and that any number x of rank 2 or more so that $x\equiv S\left(x\right)$ under

**Rule Two**is finitely mapped to a type A number that is vertical even; therefore,

**proving the convergence of such numbers is enough to prove the Collatz Conjecture**. In the upcoming sections, they are called the “${A}_{g}$” numbers (which one may admit is more practical than calling them “the intersection of residue classes ${\left[1\right]}_{2}$, ${\left[2\right]}_{3}$, and ${\left[3\right]}_{4}$”), and identified with set $24{\mathbb{N}}^{*}+17$.

#### 5.3. Proving Rules Three and Four

**Theorem 7.**

**Rules Three and Four**) Let a be a vertical even number with $a={G}^{n+2}\left(S\left(b\right)\right)$, where n and b are odd; then, $a\equiv {3}^{\frac{n+1}{2}}\left(b\right)$. Let a be a vertical even number with $a={G}^{m+2}\left(S\left(b\right)\right)$, where m is even (zero included) and b is odd; then, $a\equiv S\left({3}^{\frac{m}{2}}\left(b\right)\right)$

**Proof.**

**Definition**

**6.**

- All
**“Variety S”**numbers above x are written $V(x\xb7{2}^{2k-1})\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}or\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}S(x\xb7{2}^{2k})={2}^{2k+1}\xb7x+1$ and - all
**“Variety V”**numbers above x are written $V(x\xb7{4}^{k})$ or equivalently $S(x\xb7{2}^{2k+1})={4}^{k+1}\xb7b+1$.

- for variety S numbers, ${2}^{2k+1}\xb7b\xb7{\left(\right)}^{\frac{3}{4}}k$, which proves
**Rule Four**and - for variety V numbers, $4\xb7{4}^{k}\xb7b\xb7{\left(\right)}^{\frac{3}{4}}k$ which proves
**Rule Three**because**Rule One**already provides that $V(b\xb7{3}^{k})\equiv b\xb7{3}^{k}$.

#### 5.4. Proving Rule Five

**Rule Two**, we showed that any number of rank $n>1$ is finitely mapped by the Collatz dynamics to $G({3}^{n-1}\xb7{G}^{-1}\left({S}^{-(n-1)}\left(a\right)\right))$, which combined with

**Rule Two**itself gives

**Rule Five**.

**Rules Three, Four, and Five**plotted in gold.

**Rules One and Two**are plotted in black. Whenever a number is connected to 1 by a finite path of black and/or gold edges, it is proven to converge to 1.

## 6. The Golden Automaton

**Definition**

**7.**

**Rules One, Two, Three, Four, and Five**from number 1 onward, in the natural order on $\mathbb{N}$ is called the

**“Golden Automaton”**.

#### 6.1. “Golden Arithmetic”

**“Golden arithmetic”**involves words taken in the alphabet $\{G;S;V;3\}$, which we call in their order of application, such as in turtle graphics. For example, VGS3 means $3\xb7S\circ G\circ V$

**Rules One and Two**are still assumed:

**Rule Three:**Let b be of type B; then, $b\equiv VGS{3}^{-1}$ from b. We will all this action ${R}_{b}\left(x\right)=16\frac{x}{3}+1$ and it is defined in $6{\mathbb{N}}^{*}+3$.**Rule Four:**Let c be of type C; then, $c\equiv GS{3}^{-1}$ from c. We call this action ${R}_{c}\left(x\right)=\frac{4x-1}{3}$ and it is defined in $6{\mathbb{N}}^{*}+1$.**Rule Five:**Let a be of type A; then, $a\equiv G{3}^{-1}$ from a. We call this action ${R}_{a}\left(x\right)=\frac{2x-1}{3}$ and it is defined on $6{\mathbb{N}}^{*}+5$.

**Rules One and Two**ensure that the quiver generated by the Golden Automaton branches, with each type B number that is vertical even providing both a new A type and a new B type number to keep applying, respectively, Rules 5 and 3, we may follow only the pathway of type A numbers to define a single non-branching series of arrows, forming a single infinite branch of the quiver. The latter, if computed from number 15, leads straight to 31 and 27, solving a great deal of other numbers on the way:

**Conjecture**

**1.**

**Rule One**; then, 3 proves all numbers from 5 to 15, which in turn prove all numbers from 33 to 127. In the next subsection, we render larger quivers generated by the Golden Automaton to provide a better understanding of their geometry and fundamental properties and to demonstrate why it is so and, more generally, how, granted Goodstein sequences converge (meaning this requires the axiom of choice), it can be proven that they can reach any number in $2{\mathbb{N}}^{*}+1$.

## 7. The Golden Automaton Well-Behaves as a Collatz Convergence on the Binary Tree

**Black**, meaning the odd number is not (yet) proven to converge under the iterated Collatz transformation or, equivalently, that it is only equivalent to another black number;**Gold**, meaning the odd number is proven to converge and the consequences of its convergence have not yet been computed, i.e., it can have an offspring; and**Blue**, meaning the number is proven to converge and the consequences of its convergence have been computed i.e., its offspring has already been turned gold.

**Rule 1**: if a cell on x is gold, color the cell on $V\left(x\right)$ in gold;**Rule 2**: if a cell on x is gold, color the cell on $S\left(x\right)$ in gold depending on the precise conditions of rule 2;- If a cell on x of type A is gold, then color the cell on ${R}_{a}\left(x\right)$ in gold;
- If a cell on x of type C is gold, then color the cell on ${R}_{c}\left(x\right)$ in gold; and
- After applying the previous rules for a gold cell, turn it blue.

**Rule 1**, then for ${R}_{c}$, the algorithm needs not implement a defined ${R}_{b}$ and we can in fact compress it to only four rules.

## 8. Cost and Complexity of the Algorithms for Linearizing the Collatz Convergence

#### 8.1. Golden Automaton I (Implemented in Python)

#### 8.2. Golden Automaton II (Implemented in C++) and Its Output

## 9. The Golden Automaton as a Hydra Game

- If a number is written $x\underset{\mathrm{n}}{\underbrace{1\dots 1}}$ in base 2, then it is finitely mapped to the result of operation G on the number written $y\underset{\mathrm{n}}{\underbrace{1\dots 1}}$ in base 3 with $y=(x+1)/2$. Note that this is the one and only way an orbit can grow in the Collatz dynamics.
- If a number is written $z\underset{\mathrm{n}}{\underbrace{2\dots 2}}1$ in base 4, then it is immediately mapped to a number written $x\underset{2\mathrm{n}+1}{\underbrace{1\dots 1}}$ in base 2.
- If a number is written $s\underset{2\mathrm{n}+1}{\underbrace{0\dots 0}}1$ in base 2, then it is equivalent to the result of operation S on $r\underset{\mathrm{n}}{\underbrace{0\dots 0}}$ in base 3 with r as the base 3 representation of s.
- If a number is written $v\underset{2\mathrm{n}}{\underbrace{0\dots 0}}1$ in base 2, then it is equivalent to $w\underset{\mathrm{n}}{\underbrace{0\dots 0}}$ in base 3 with w as the base 3 representation of v.

**but this time studying not the vertices but the edges of the graph**. To streamline its algorithmic scaling, we use the simplified rules we defined in the previous subsection, again, without loss of generality. Our precise purpose is to pave the way for a formal demonstration that proving the convergence of odd numbers up to n is always isomorphic to a Hydra game, which justifies that we now study edges and not vertices. In Figure 18, Figure 19, Figure 20 and Figure 21, we color all of the elements of $24{\mathbb{N}}^{*}+17$, for example $\{17,41,65,\dots \}$, in red; as we demonstrate in the next section, they are precisely from the “heads” of the Hydra Game.

**Theorem**

**9.**

**Definition**

**8.**

**Hydra**is a rooted tree with arbitrarily many and arbitrarily long finite branches. Leaf nodes are called

**heads**. A head is

**short**if the immediate parent of the head is the root and

**long**if it is neither short nor the root. The object of the

**Hydra game**is to cut down the Hydra to its root. At each step, one can cut off one of the heads, after which the Hydra grows new heads according to the following rules:

- If the head was long, grow n copies of the subtree of its parent node minus the cut head, rooted in the grandparent node.
- If the head was short, grow nothing.

**Lemma**

**10.**

**at worst**a Hydra game over a finite subtree of the complete binary tree over $24{\mathbb{N}}^{*}+17$.

**Proof.**

**at worst**a Hydra game are as follows:

- What are the Hydra’s heads?
- How do they grow?
- Does the Golden Automaton cut them according to the rules (at worst)?

**Definition**

**9.**

**vertical even**is called an ${A}_{g}$. The set of ${A}_{g}$ numbers is $24{\mathbb{N}}^{*}+17$. Type B numbers that verify $b\equiv S\left(b\right)$ and type C numbers that verify $c\equiv S\left(c\right)$ under

**Rule Two**are called

**Bups**and

**Cups**, respectively.

#### 9.1. What Are the Hydra’s Heads?

**Rule Five**is smaller than them since action ${R}_{a}$ strictly decreases. Thus, up to the nth ${A}_{g}$, there are $2n$ (Bups + Cups) of rank 2 or more and half of them are equivalent to those ${A}_{g}$ (e.g., between 17 and 41, Bup 27 is equivalent to ${A}_{g}$ 41, which is equivalent to Cup 31 by

**Rule Four**).

#### 9.2. How Do They Grow?

- Eight non-A numbers;
- One at most mapped to the second ${A}_{g}$;
- Three at most “ups” (Bup or Cup) of rank 2 or more.

- Let b be of type B; there are $\frac{2b}{3}$ numbers of type ${A}_{g}$ that are smaller than ${V}^{2}\left(b\right)$;
- Let c be of type C; there are $\frac{S\left(c\right)}{3}$ numbers of type ${A}_{g}$ that are smaller than ${V}^{2}\left(c\right)$;
- Let 3c be of type B, where c is of type C; there are $\frac{S\left(c\right)}{3}$ numbers of type ${A}_{g}$ up to ${R}_{b}\left(3c\right)$ included; and
- Let 3a be of type B, where a is of type A; there are $\frac{G\left(a\right)}{3}$ numbers of type ${A}_{g}$ smaller than ${R}_{b}\left(3a\right)$,

#### 9.3. Does the Golden Automaton Play a Hydra Game?

**Rule 5**specifically so that the reader can now report it more easily because, each time this rule is used, a head (that is, an ${A}_{g}$) has just been cut.

## 10. The Golden Automaton as a Winning Cellular Game Represented as a 3D L-System, with Some Important Applications in Industrial Cryptography

**its basin of attraction inflates along with its orbit.**

## 11. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Algorithm of the Golden Automaton II (Implemented in C++ by Baptiste Rostalski)

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**Figure 2.**Quiver connecting all odd numbers from 1 to 31 with the arrows of actions S, V, and G. The set $2{\mathbb{N}}^{*}+1$ is thus endowed with three unary operations without a general inverse that are noncommutative with $G\circ S=V$. Whenever we mention the inverse of these operations, we assume that they exist on $\mathbb{N}$. Type A numbers are circled in teal, type B is in gold, and type C is in purple.

**Figure 3.**Collatz feather rendered in Blender, this time with the same ternary typology as defined in Definition 4 [42].

**Figure 5.**The five rules completing the binary tree row by row in our first Python implementation of the Golden Automaton (“GAI”) [42].

**Figure 11.**At the same instant, state of row 11 (going from 1025 to 2047: each line has about 100 dots).

**Figure 12.**Amount of extra numbers proven to converge above row n when it has just been finished by the Golden Automaton in either linear or log scales [42].

**Figure 13.**Amount of extra numbers proven to converge above row n, this time in a ternary tree, when it has just been finished by the Golden Automaton, in either linear or log scales [42].

**Figure 14.**Amount of extra numbers proven by Golden Automaton II (log scale) after finishing any row n of the binary tree (x-axis), fit to the ${3}^{n+1}$ function (red).

**Figure 15.**Log scale of the proportion (in %) of presumed unproven numbers in each row $n+1$ when row n is finished by Golden Automaton II. The line of ${1.7}^{(2-n)}$ is shown in red for comparison.

**Figure 16.**Average difference between row n and $n+1$ in absolute values (not proportion) and not logarithmic ones.

**Figure 17.**Cumulative time to compute any row n, with ${3}^{n}-10$ in red for comparison (log scale).

**Figure 18.**Golden Automaton confined to numbers smaller than 32 [42].

**Figure 19.**Golden Automaton confined to numbers smaller than 64 [42].

**Figure 20.**Golden Automaton confined to numbers smaller than 128 [42].

**Figure 21.**Golden Automaton confined to numbers smaller than 256 [42].

**Figure 22.**Orthogonal view of the Golden Automaton starting from 1 (in blue) obtained from the code in [42]. All its intersections with the automaton starting independently from 1457 (the first ${A}_{g}$ in the forward Collatz trajectory of 127) are shown in green. As expected from our 2D works in Section 7, the Golden automaton starting from 1 covers all numbers. This figure also provides the first trigonometric representation of the inflation propensity of Collatz orbits, which Bocart [19] has proven constitutes a reliable proof-of-work for blockchain applications: the number of green lines (overlapping the inverse orbit of 1457 and that of 1) is directly tied to the inflation propensity of a given orbit; simply put, the more an orbit inflates, the more green lines are shown on this disc, but their distribution cannot be faked and thus forms a functional authentication fingerprint. As green lines also represent particular trajectories, this figure also suggests that other promising proofs-of-work, comparable with that of Bocart, could be obtained from the study of non-ergodic billiards.

**Figure 23.**Isometric projection of Figure 22 (code available at [42]). The green lines visible on the sides represent a binary transformation (e.g., operations S, V, or G) and those visible on the top of the cone represent ternary operations (e.g., $\times 3$), thus decomposing the correlates (one in base 2 and one in base 3) of the inflation propensity of the number’s orbit in two dimensions. From this figure, it could be possible to implement faster proof-of-work verification of the Bocart protocol by just scanning the side lines, though this would admit a certain margin of error.

**Figure 24.**Orthogonal view of the Golden Automaton starting from 1 (blue), which overlaps the one starting from 161 (green). We first observe that the basin of 161 (the first ${A}_{g}$ of 31) now occupies a much larger proportion of the basin of 1 than did the basin of 1457 (the first ${A}_{g}$ of 127). Simply put, the more a number diverges, the longer the trail of type A numbers it leaves and the more its basin of attraction inflates, ultimately making a collision with the basin starting from 1 inevitable. (Another important property of Mersenne 31 is that, as defined by OEIS [43], it is “self-contained”, meaning its orbits contains multiples of itself (i.e., the number 155).) This representation of base 3 correlates with the inflation propensity of Collatz orbits is in fact directly scannable, similar to a QR code, with the central truncated caustic forming the standard reference point of the scan and the pseudorandom distribution of the green lines using a direct verification protocol of the Bocart proof-of-work.

**Figure 25.**The Golden Automaton starting from 161 with all its collision edges, with the one starting from 1 shown in green. Although they are related, both the side and top distributions of the green edges can be used for cryptographic applications. The angle of the display remains important for scannable applications, as only the central convexoid of the figure may be used as a standard reference for the scan and must therefore be visible or otherwise indicated even if only the side lines are scanned.

**Figure 26.**The ternary tree over the binary tree embedded on the unit circle. The shape of this figure is essential in generating the pseudorandomness of the inflation propensity of Collatz orbits and, thus, of the Bocart proof-of-work. More generally, it forms the base of the pseudorandomness of conversions between bases 2 and 3, which led Furstenberg to later state his eponymous $\times 2\times 3$ conjecture.

**Figure 27.**The truncated caustic generated by the $\xb73$ map on the binary tree, this time with gradient-colored lines from the domain (red) to codomain (yellow), underlining the non-ergodicity of the $\times 3$ map on the binary tree and why other number-theoretical proof-of-work comparable with that of Bocart, in particular, independent of large prime numbers, may be obtained from the study of non-ergodic number billiards. The code repository for this figure is also available at [42].

**Figure 28.**Clockwise, from the upper left: number of points on each side of the unit circle (side 7 is “top”, and side 5 is “bottom") after iterations of the multiplication by 3 on number 1. The top/bottom ratio, on the next figure, converges to approximately 0.7. The next two figures display the cosine and sine of the multiplication by 3 of each point of the unit circle: on one third of the domain, this operation multiplies the angle of the starting point by 1.5, and on the other two thirds, the operation multiplies the angle by 0.75, thus explaining the asymmetry of the truncated caustic, in turn explaining the non-ergodicity of the $\times 3$ map on the binary tree embedded in the unit circle.

**Figure 29.**The amount N of numbers proven by applying the five rules from points x to ${2}^{n}x$ is plotted here against n. Function ${2}^{n}$ is shown in orange as a reference. Clockwise from the upper left, the starting points are 31, 161, 13121, and 511. [42].

**Figure 30.**The observed growth rates of the basin of attraction of different Mersenne numbers, with 2 in orange for reference.

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**MDPI and ACS Style**

Rahn, A.; Sultanow, E.; Henkel, M.; Ghosh, S.; Aberkane, I.J.
An Algorithm for Linearizing the Collatz Convergence. *Mathematics* **2021**, *9*, 1898.
https://doi.org/10.3390/math9161898

**AMA Style**

Rahn A, Sultanow E, Henkel M, Ghosh S, Aberkane IJ.
An Algorithm for Linearizing the Collatz Convergence. *Mathematics*. 2021; 9(16):1898.
https://doi.org/10.3390/math9161898

**Chicago/Turabian Style**

Rahn, Alexander, Eldar Sultanow, Max Henkel, Sourangshu Ghosh, and Idriss J. Aberkane.
2021. "An Algorithm for Linearizing the Collatz Convergence" *Mathematics* 9, no. 16: 1898.
https://doi.org/10.3390/math9161898