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Collatz Representations With Bounded Partial Quotients

The paper defines Collatz representations for a subset of rational numbers and proves that any real number outside the interval $(-1, 1)$ can be arbitrarily well approximated by rational numbers whose Collatz representations consist exclusively of the digits 1 and 2.

Original authors: Franciszek Kobus

Published 2026-07-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Franciszek Kobus

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Infinite Labyrinth of Numbers

Imagine you are standing in a vast, infinite library where every book is a number. Most of us are familiar with the standard way of writing numbers: decimals like 3.14 or fractions like 1/2. But mathematicians have a habit of inventing new ways to look at old things, much like how a sculptor might see a statue in a block of stone that others just see as rock. One of the most famous puzzles in this library is the "Collatz Conjecture." It's a simple game: take any whole number, and if it's even, cut it in half; if it's odd, triple it and add one. Repeat this forever. The big question is: does every starting number eventually get stuck in a tiny loop of 1, 4, 2, 1, 4, 2...? Nobody knows for sure, but it's a riddle that has stumped the world's best minds for decades.

To solve riddles like this, mathematicians often use "maps." A common map is called a "continued fraction," which breaks a number down into a chain of smaller integers, like a recipe. This new paper, written by Franciszek Kobus, invents a brand-new kind of map called a "Collatz representation." Instead of just breaking numbers down, this map records the specific steps a number takes when it plays the Collatz game. It turns out that by looking at these steps, we can describe numbers in a way that reveals hidden patterns, almost like finding a secret code in the DNA of mathematics. The paper asks a fascinating question: If we only allow the map to use the numbers 1 and 2 in its recipe, how many numbers can we actually reach?

The Magic Recipe of 1s and 2s

In this paper, the author introduces a special way to write down rational numbers (fractions with odd numbers on top and bottom) using the Collatz game. Think of the Collatz algorithm as a machine. You feed it a number, and it spits out a sequence of instructions. If the number is odd, the machine multiplies it by 3 and adds 1. Then, it keeps dividing by 2 until the result is odd again. The number of times it had to divide by 2 is recorded as a "partial quotient."

For example, if you start with the number 1, the machine does: $3(1)+1 = 4$ , then divides by 2 twice to get back to 1. So, the instruction is "divide by 2 twice," which we write as the number 2. The "Collatz representation" of 1 is just the repeating sequence of 2s. If you start with -1, the machine does: $3(-1)+1 = -2$ , then divides by 2 once to get back to -1. So, the representation of -1 is a repeating sequence of 1s.

The paper proves a very cool fact: every rational number that eventually loops back to itself (like 1 or -1) has a unique, repeating recipe of these instructions. It's like a fingerprint; no two different numbers have the exact same repeating sequence of steps.

The Fractal Forest of 1s and 2s

The real magic happens when the author asks: "What if we only use the numbers 1 and 2 in our recipe?"

Imagine you are building a tree. You start with a single point. Then, you branch out. If you add a "1" to your recipe, you go one way; if you add a "2," you go another. The paper shows that if you keep doing this, creating a map of all the numbers you can make using only 1s and 2s, you don't just get a random scatter of dots. You get a fractal.

A fractal is a shape that looks the same no matter how much you zoom in, like a fern leaf or a snowflake. The author draws these points on a graph and connects them with lines. The result is a beautiful, self-repeating pattern. The paper proves that these shapes are "similar," meaning they are the same shape but just scaled up or down. Specifically, if you have two shapes that share a corner, one is exactly 2/3 the size of the other if they are side-by-side, or 4/3 the size if one is stacked on top of the other. It's like a set of Russian nesting dolls where the dolls are made of math.

Filling the Gaps

The most surprising discovery is about how "full" this fractal is. The author proves that if you look at the number line from negative infinity up to -1, and from 1 up to positive infinity, you can find a number made of only 1s and 2s that is arbitrarily close to any number in those ranges.

Think of it like trying to hit a target with a dart. If the target is a number like -5.738, you might not be able to hit it exactly with a dart made of 1s and 2s. But the paper proves that you can get as close as you want. You can throw a dart that lands at -5.7380001, or -5.7380000001. No matter how small a gap you draw around your target, there is a "1-and-2" number inside that gap.

The author provides a step-by-step recipe (a constructive proof) to find these numbers. It's like a game of "hot and cold." You start with a number, and if you're too high, you add a "2" to your recipe to drop the value. If you're too low, you add a "1" to nudge it up. By following this logic, you can zero in on any number in the range $(-\infty, -1] \cup [1, \infty)$ with infinite precision.

What This Means (and What It Doesn't)

The paper is very careful about what it claims. It proves that for any repeating sequence of 1s and 2s, there is exactly one rational number that matches it. However, it also points out that not every infinite sequence of 1s and 2s corresponds to a number in the set of rational numbers with odd numerators and denominators. Some sequences are just too wild to land on a specific rational spot.

The author also touches on the famous Collatz Conjecture. In this new language, the conjecture says that the only positive whole number that gets stuck in a loop is the number 1. The paper lists other loops that exist for negative numbers (like -1, -5, -7, etc.), but for positive integers, 1 is the only known "absolutely periodic" number.

So, what have we learned? We've learned that the chaotic dance of the Collatz game can be translated into a structured, beautiful language of 1s and 2s. This language creates a fractal map that covers the number line in a very specific way, allowing us to approximate almost any number in the outer regions of the number line with incredible precision. It's a reminder that even in the most stubborn mathematical puzzles, there are hidden patterns waiting to be drawn, connected, and understood.

Technical Summary: Collatz Representations With Bounded Partial Quotients

Problem Statement
The paper addresses the representation of rational numbers with odd numerators and denominators (denoted as $Q_{odd}$ ) through a novel algorithmic framework inspired by the famous Collatz conjecture. While the standard Collatz conjecture concerns the trajectory of integers under the map $x \mapsto 3x+1$ (with division by 2), this work defines a "Collatz representation" for elements in $Q_{odd}$ analogous to continued fractions. The central problem investigated is the structure of the set of rational numbers whose Collatz representations consist exclusively of the partial quotients 1 and 2. Specifically, the author seeks to determine the density and geometric properties of this set within the real numbers.

Methodology
The author defines the Collatz algorithm for a starting number $a \in Q_{odd}$ :

Initialize $A_1 = 3a + 1$ .
Determine $a_i$ , the 2-adic valuation of the numerator of $A_i$ (in reduced form).
Define $B_i = A_i \cdot 2^{-a_i}$ and iterate with $A_{i+1} = 3B_i + 1$ .

This process generates an infinite sequence of positive integers $C(a) = [a_1, a_2, \dots]$ , termed the Collatz representation of $a$ . The paper establishes a bijection between the set of eventually periodic sequences and the set of rational numbers in $Q_{odd}$ with eventually periodic Collatz representations.

The core methodology involves:

Inverse Mapping: Deriving an explicit formula (Equation 5) to calculate the unique rational number $a = C^{-1}([a_1, \dots, a_n, (b_1, \dots, b_k)])$ for any eventually periodic sequence.
Geometric Construction: Defining a point $P(A) = (C^{-1}[A, (2)], C^{-1}[A, (1)])$ in $\mathbb{R}^2$ for any finite sequence $A$ composed of 1s and 2s. By connecting points derived from sequences of increasing length, the author constructs a sequence of geometric figures $P_n$ .
Analytical Proofs: Using induction and algebraic manipulation of the inverse mapping formula to prove scaling properties and convergence limits of these geometric structures.

Key Contributions and Results

Existence and Uniqueness: The paper proves that for every eventually periodic sequence of positive integers, there exists precisely one rational number in $Q_{odd}$ that generates it. This establishes a well-defined bijection between eventually periodic sequences and a specific subset of $Q_{odd}$ .
Fractal Structure of Bounded Quotients: The set of points generated by sequences containing only 1s and 2s exhibits a self-similar fractal structure. The author proves three specific geometric hypotheses regarding the set $P_n$ :
- Scaling: Triangles formed by adjacent points in the construction are similar with scaling factors of $2/3$ (horizontal adjacency) or $4/3$ (vertical adjacency).
- Boundedness: The set $P_n$ is enclosed by the lines $y - x = 0$ , $x - 1 = 0$ , and $y + 1 = 0$ .
- Limit Behavior: The closure of the intersection of the line $y=x$ and the limit of the sets $P_n$ corresponds to the half-line where $x \leq 1$ .
Density Theorem: The primary result (Theorem 3.8) demonstrates that the set of rational numbers with Collatz representations consisting solely of 1s and 2s is dense in the interval $(-\infty, -1] \cup [1, \infty)$ . The proof is constructive, showing that for any real number $z \leq -1$ and any $\epsilon > 0$ , a sequence of 1s and 2s can be algorithmically constructed to approximate $z$ within $\epsilon$ .
Reformulation of the Weak Collatz Conjecture: The paper notes that in this notation, the Weak Collatz Conjecture can be restated as: "The only positive integer in $Q_{odd}$ with an absolutely periodic Collatz representation is 1." The author lists known absolutely periodic loops for negative integers (e.g., $[-1]$ , $[-5, -7]$ ).
Non-Uniqueness for Non-Periodic Sequences: The paper clarifies that while the mapping is bijective for eventually periodic sequences, it does not hold for all infinite sequences. There exist infinite sequences of 1s and 2s that do not correspond to the Collatz representation of any number in $Q_{odd}$ .

Significance and Claims
The paper claims to provide an "original way of representing rational numbers" that draws a parallel between the Collatz algorithm and continued fractions. Its significance lies in:

Revealing a fractal structure inherent in the subset of rationals with bounded partial quotients (specifically 1 and 2) under the Collatz map.
Proving that this specific subset is dense in the regions $(-\infty, -1]$ and $[1, \infty)$ , implying that real numbers in these domains can be arbitrarily well approximated by rationals with simple Collatz representations.
Offering a geometric perspective (via the $P_n$ construction) that visualizes the distribution of these numbers.

The author remains modest regarding the scope, noting that the results rely on the connection to continued fractions in different settings and that the work focuses on the specific subset of rationals with odd numerators and denominators. The paper does not claim to solve the Collatz conjecture itself but rather uses the conjecture's mechanics to define a new representation system with interesting topological properties.