💻 computer science

Recursive Prime Factorizations: Dyck Words as Numbers

This paper proposes a class of non-positional numeral systems based on recursive prime factorizations that utilize Dyck words to uniquely represent natural numbers and rational numbers, while also exploring "Dyck-complete" languages where every Dyck word corresponds to a number.

Original authors: Ralph L. Childress

Published 2026-02-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ralph L. Childress

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

Imagine you are trying to describe a number, like 520.

In the school system we all know, we write 520. This is a "positional" system. It's like a hotel where the room number tells you the value based on where the digits sit: the '5' is in the "hundreds" room, the '2' is in the "tens" room, and the '0' is in the "ones" room. But here's the catch: Why 10? Why not 12? Why not 7? The number 520 is just a collection of building blocks (primes: $2 \times 2 \times 2 \times 5 \times 13$ ), but our decimal system hides those blocks behind the arbitrary choice of "10."

Ralph Childress, in this paper, asks: What if we built a number system that doesn't use a "base" at all? What if the number was its own recipe?

He proposes a system called Recursive Prime Factorization (RPF). Instead of using digits 0–9, this system uses only two symbols: an open parenthesis ( and a closed parenthesis ).

Think of it like a Russian Nesting Doll made entirely of parentheses.

The Core Idea: The Parenthesis Recipe

In this system, every number is a unique string of balanced parentheses, like (()(())).

Here is how the magic works:

The Ingredients (Primes): Imagine the prime numbers (2, 3, 5, 7, 11...) are the "flavors."
- The 1st flavor is 2.
- The 2nd flavor is 3.
- The 3rd flavor is 5.
- And so on.
The Container (Parentheses): A pair of parentheses () acts as a container.
- () by itself represents the number 1.
- ( ) (an empty container) represents 0.
The Recipe (Nesting):
- If you have a container ( ... ), the stuff inside tells you the exponent (how many times to multiply).
- The position of the container in the line tells you which prime it is.
- The first container in the line is about the prime 2.
- The second container is about the prime 3.
- The third is about 5.

Let's decode a simple example:
Imagine the string: (())(())

First part: (()). This is the 1st container. It contains ().
- () inside means the exponent is 1.
- So, this is $2^1 = 2$ .
Second part: (()). This is the 2nd container. It also contains ().
- So, this is $3^1 = 3$ .
The Math: Multiply them together: $2 \times 3 = 6$ .
Result: The string (())(()) is the number 6.

Now, let's look at a complex one (The "Recursive" part):
What if the exponent itself is a big number?
In normal math, we write $2^5$ . In this system, we can't just write "5". We have to write the recipe for 5 inside the exponent slot.

The recipe for 5 is ()()(()) (which means $2^1 \times 3^1 \times 5^1$ ).
So, to write $2^5$ , we put that whole recipe inside the first container:
(()()(())
This looks like a nesting doll: The outer shell is the prime 2. Inside, it holds the entire structure of the number 5.

The "Dyck Word" Connection

The paper calls these strings Dyck Words. Think of a Dyck Word as a perfectly balanced sentence of parentheses.

(() is broken (unbalanced).
()) is broken.
(()()) is perfect.

Childress proves that there is a special subset of these perfect sentences that corresponds exactly to every natural number (0, 1, 2, 3...). No two numbers share the same sentence, and every number has a sentence. It's a perfect 1-to-1 map.

The Super-Power: Going Beyond Whole Numbers

The most exciting part of the paper is the "Superrational" extension.

In standard math, if you want to write a fraction like $1/2$ , you need a slash or a decimal point. If you want to write a negative number, you need a minus sign.
Childress's system is so clever that it doesn't need new symbols.

Negative Numbers: He uses a "trick" at the very end of the string. If a string ends with an extra empty pair (), it flips the sign.
- ... = Positive 520.
- ...() = Negative 520.
Fractions and Roots: Because the system is recursive, you can put a "negative" inside an exponent.
- $2^{-1}$ (which is $1/2$ ) is written by putting a "negative" marker inside the exponent slot.
- $\sqrt{2}$ (which is $2^{0.5}$ ) is written by nesting the recipe for a fraction inside the exponent.

This means you can write irrational numbers (like $\sqrt{2}$ ) and even transcendental numbers (like $2^{\sqrt{2}}$ ) using only parentheses. No decimals, no minus signs, no square root symbols. Just ( and ).

Why Does This Matter? (The "So What?")

You might ask, "This is cool, but can I use it to balance my checkbook?"
No. It's terrible for that. It's way too long and hard to read for humans.

But, it's fantastic for computers and pattern recognition.

No Arbitrary Bases: It reveals the "true DNA" of numbers. The patterns you see in the parentheses are patterns of the numbers themselves, not artifacts of our base-10 habit.
Computer Science: Since these are just strings of parentheses, computer scientists can use "Grammar" tools (like those used to parse code) to analyze numbers.
AI Learning: The paper suggests that if we teach an AI (like a Large Language Model) to read these parenthesis strings, it might learn the structure of math better than if we just fed it decimal numbers. It forces the AI to understand the "recipe" of the number rather than just memorizing digits.
Compression: Because the system is recursive, it might be a very efficient way to store complex mathematical structures in a computer.

The Analogy Summary

Decimal System: Like describing a house by its address (Street, House Number, Zip Code). It works, but the "Street" (Base 10) is arbitrary.
RPF System: Like describing a house by its blueprint. You don't say "House #520." You say, "It's made of 2 bricks, 3 beams, and 5 windows."
- If the house has a basement, the blueprint nests inside the main room.
- If the house is a "negative" house (a ghost?), the blueprint has a special tag at the end.
- If the house is a fraction, the blueprint has a tiny, nested blueprint inside it.

Childress has built a universal language of parentheses where every number, fraction, and root is a unique, perfectly balanced sentence. It's a way of looking at math that strips away the "human convention" of base-10 and leaves only the pure, recursive structure of numbers themselves.

Technical Summary: Recursive Prime Factorizations: Dyck Words as Representations of Numbers

1. Problem Statement

The paper addresses the inherent arbitrariness of positional numeral systems (e.g., base-10), which rely on an arbitrarily selected base and require exponentiation, multiplication, and addition for evaluation. These systems obscure intrinsic patterns in numbers and introduce redundancy (e.g., multiple representations for the same number like $1, 01, 001$).

The author, Ralph "Tim" Childress, Jr., seeks to develop a non-positional numeral system that:

Is based directly on prime factorization, eliminating the need for an arbitrary base.
Requires no addition for evaluation (relying solely on multiplication and exponentiation).
Uses the smallest possible alphabet (two symbols).
Uniquely represents natural numbers and extends to a superset of rational numbers, including certain algebraic and transcendental numbers.

2. Methodology

The core methodology involves mapping numbers to Dyck words (strings of balanced parentheses) through a recursive extension of prime factorization.

2.1. The Standard Minimal RPF Natural Interpretation ( $RPFN_{rmin}$ )

Alphabet: $\{(, )\}$ .
Base Cases:
- $0 \to \epsilon$ (empty string).
- $1 \to ()$ .
Recursive Step: For any natural number $n > 1$ $n > 1$ , let $n = p_1^{a_1} p_2^{a_2} \dots p_m^{a_m}$ $n = p_{1}^{a_{1}} p_{2}^{a_{2}} \dots p_{m}^{a_{m}}$ be its prime factorization, where $p_k$ $p_{k}$ is the $k$ $k$ -th prime.
- The representation is constructed by concatenating the representations of the exponents $a_i$ , each enclosed in parentheses:
  $\gamma_{Nr}(n) = \bigcup_{i=1}^m ( \gamma_{Nr}(a_i) )$
- Crucially, the system uses a "padded" factorization where all primes up to the greatest prime factor $p_m$ are included, even if their exponent is 0. The representation of $0$ (empty string) inside parentheses () represents the exponent 0.
Evaluation: The inverse function $\alpha_{Nrmin}$ evaluates a Dyck word by treating the $i$ -th "chunk" (a substring enclosed in the outermost matching parentheses) as the exponent for the $i$ -th prime $p_i$ .
$\alpha_{Nr}(w) = \prod_{i=1}^{dim(w)} p_i^{\alpha_{Nr}(content_i)}$

2.2. The Standard Minimal RPF Superrational Interpretation ( $RPFS_{rmin}$ )

To extend the system beyond natural numbers ( $\mathbb{N}$ ) to a set $S$ containing rationals, algebraic irrationals, and some transcendental numbers, the author introduces negation via "inflation."

Negation Mechanism: A negative sign is encoded by appending a single empty pair of parentheses () to the end of a word.
- $s \to \gamma_{Sr}(s) \cdot ()$ represents $-s$ .
Recursive Negation: Because the system is recursive, negative signs propagate to exponents. This allows for negative exponents (rational numbers) and nested negative exponents (irrational numbers).
- Example: $\sqrt{2} = 2^{1/2} = 2^{2^{-1}}$ . The representation involves nested structures encoding the negative exponent.
Set $S$ (Superrational Numbers): Defined as the smallest subset of $\mathbb{R}$ containing $0$, closed under prime-base exponentiation, multiplication, and negation. This set includes $\mathbb{N}$ , $\mathbb{Z}$ , $\mathbb{Q}$ , $\sqrt{2}$ , and the Gelfond-Schneider constant $2^{\sqrt{2}}$ .

2.3. Formal Language Properties

Dyck Language ( $D$ ): The set of all well-balanced parenthesis strings.
Minimal Languages: The author defines specific subsets of $D$ $D$ ( $D_{rmin}$ $D_{r min}$ for naturals, $D_{rqmin}$ $D_{r q min}$ for superrationals) that serve as minimal representations.
- $D_{rmin}$ is a proper subset of $D$ characterized by avoiding the substring )()) and the suffix )().
- $D_{rqmin}$ allows for exactly one "inflationary" empty pair at the end (to denote sign) but prohibits multiple consecutive terminal empty pairs.
Inflation/Deflation: The paper defines "inflation" as adding non-essential empty pairs to a word without changing its numerical value. The minimal languages contain only the "deflated" (canonical) forms.

3. Key Contributions

3.1. Novel Numeral Systems

The paper introduces two bijective numeral systems:

Natural RPF: A bijection between $\mathbb{N}$ and a proper subset of the Dyck language $D_{rmin}$ .
Superrational RPF: A bijection between the set $S$ $S$ (superrationals) and a subset $D_{rqmin}$ $D_{r q min}$ .
- Significance: Unlike positional systems, these systems use the same two-symbol alphabet for integers, rationals, and specific irrationals/transcendentals. No new symbols (like minus signs or decimal points) are required.

3.2. Theoretical Proofs

Bijectivity: Proves that the spelling function $\gamma$ and evaluation function $\alpha$ are mutual inverses for both natural and superrational interpretations.
Context-Free Nature: Demonstrates that the minimal natural language $D_{rmin}$ is a context-free language (specifically, the intersection of the Dyck language and a regular language defined by a DFA).
Non-Minimality of Full Dyck Language: Shows that the full Dyck language $D$ allows for multiple representations of the same number (via inflation), making it a non-minimal interpretation.

3.3. Algorithmic Tools

Prime Factorization Algorithm: Provides an efficient algorithm to extract the prime factorization of a Dyck natural number directly from its string structure.
Inflation/Deflation Logic: Formalizes the concept of "inflating" a number word to create non-minimal representations, useful for understanding the structure of the Dyck language.

4. Results

Representation Examples:
- $520 \to (()(()))()(())()()(()))$
- $-0.2 \to (())((())())()$
- $\sqrt{2} \to ((()()))$
- $2^{\sqrt{2}} \to (((()())))$
Pattern Discovery: The paper identifies "stripes" in the sequence of natural numbers based on the length of their Dyck representations. It notes that the first term of each stripe corresponds to a prime number, and the last terms follow a base-2 tetration pattern ( $2, 2^2, 2^{2^2}, \dots$ ).
Learnability Implications: The author suggests that these representations could serve as structured corpora for training AI models (Transformers) to learn arithmetic structures, as the Dyck words encode the hierarchical tree structure of prime factorization losslessly.

5. Significance and Future Directions

Formal Language Theory in Number Theory: The paper bridges formal language theory (Dyck languages, context-free grammars) with number theory, offering a "grammatical number theory" perspective.
Pattern Recognition: By removing the "noise" of arbitrary bases, the system reveals intrinsic patterns in number sequences (e.g., the relationship between prime indices and nesting depth).
AI and Machine Learning: The author proposes using these Dyck-word corpora to test the learnability of arithmetic rules in neural networks. The "lossless" nature of the superrational interpretation preserves positional information of primes that is lost in standard factorization trees.
Compression and Detection: The paper suggests using grammar-based compression algorithms to detect patterns in sequences of these numbers; higher compression ratios would indicate underlying structural regularities.
Cryptography: The ability to characterize subsets of these languages (e.g., square-free semiprimes) using regular expressions offers new ways to analyze cryptographic moduli.

In conclusion, Childress presents a mathematically rigorous framework where numbers are not just values but syntactic structures. This system eliminates the arbitrariness of bases, unifies the representation of diverse number types under a single binary alphabet, and opens new avenues for studying arithmetic through the lens of formal languages.