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Recursive Prime Factorizations: Dyck Words as Numbers

Ralph L. Childress

TL;DR

This work introduces Recursive Prime Factorizations (RPF), a framework mapping numbers to Dyck words via recursive prime-factor encodings in a nonpositional numeral system. It develops the Standard Minimal RPF Natural Interpretation $\text{RPF}_{\mathbb{N}_{r_{\text{min}}}}$ with the Dyck-language-based domain $\mathcal{D}_{r_{\text{min}}}$, and proves a precise bijection with $\mathbb{N}$ through the spelling $\upgamma_{\mathbb{N}_r}$ and evaluation $\upalpha_{\mathbb{N}_{r_{\text{min}}}}$; it also shows $\mathcal{D}_{r_{\text{min}}}$ is a proper subset of the Dyck language $\mathcal{D}$. The paper then extends to a Standard Minimal RPF Superrational Interpretation, introducing the superrational numbers $\mathbb{S}$ and a corresponding two-symbol spelling/evaluation system that yields a surjective map from a Dyck-based language $\mathcal{D}_{r_{\text{qmin}}}$ to $\mathbb{S}$, and develops the mutual inverses $\upgamma_{\mathbb{S}_r}$ and $\upalpha_{\mathbb{S}_{r\text{min}}}$. Further, it surveys Dyck-complete interpretations where the entire Dyck language underlies the numeral system, and outlines rich directions for future work, including stripe analysis, learnability with lossless factorization trees, grammar-based pattern detection, and automata-theoretic characterizations of recursive prime factorizations. The results illuminate a grammar-centric perspective on number representations with potential crossovers to formal languages, cryptography, and computational learnability.

Abstract

I propose a class of non-positional numeral systems where numbers are represented by Dyck words, with the systems arising from a recursive extension of prime factorization. After describing two proper subsets of the Dyck language capable of uniquely representing all natural numbers and a superset of the rational numbers respectively, I consider "Dyck-complete" languages, in which every member of the Dyck language represents a number. I conclude by suggesting possible research directions.

Recursive Prime Factorizations: Dyck Words as Numbers

TL;DR

This work introduces Recursive Prime Factorizations (RPF), a framework mapping numbers to Dyck words via recursive prime-factor encodings in a nonpositional numeral system. It develops the Standard Minimal RPF Natural Interpretation with the Dyck-language-based domain , and proves a precise bijection with through the spelling and evaluation ; it also shows is a proper subset of the Dyck language . The paper then extends to a Standard Minimal RPF Superrational Interpretation, introducing the superrational numbers and a corresponding two-symbol spelling/evaluation system that yields a surjective map from a Dyck-based language to , and develops the mutual inverses and . Further, it surveys Dyck-complete interpretations where the entire Dyck language underlies the numeral system, and outlines rich directions for future work, including stripe analysis, learnability with lossless factorization trees, grammar-based pattern detection, and automata-theoretic characterizations of recursive prime factorizations. The results illuminate a grammar-centric perspective on number representations with potential crossovers to formal languages, cryptography, and computational learnability.

Abstract

I propose a class of non-positional numeral systems where numbers are represented by Dyck words, with the systems arising from a recursive extension of prime factorization. After describing two proper subsets of the Dyck language capable of uniquely representing all natural numbers and a superset of the rational numbers respectively, I consider "Dyck-complete" languages, in which every member of the Dyck language represents a number. I conclude by suggesting possible research directions.

Paper Structure

This paper contains 27 sections, 13 theorems, 84 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Standard Minimal RPF Natural Interpretation $\text{RPF}_{\mathbb{N}_{r_{\text{min}}}}$
  3. Informal Treatment by Example
  4. The Minimal Parenthesized Padded Prime Factorization
  5. Finding the RPF$_{\text{min}}$ Equivalent of Decimal 520
  6. Finding the Decimal Equivalent of (()(()))()(())()()(())
  7. A Preview of the Superrational Extension
  8. Interpretations and Representations
  9. Standard and Other Prime-permuted RPF Interpretations
  10. $\boldsymbol{\upgamma_{\mathbb{N}_{r}}}$ and $\text{RPF}_{\mathbb{N}_{r_{\text{min}}}}$
  11. The Language $\mathcal{D}$ and the Dyck Natural Numbers
  12. The Standard RPF Natural Evaluation Functions $\boldsymbol{\upalpha_{\mathbb{N}_{r}}}$ and $\boldsymbol{\upalpha_{\mathbb{N}_{r_\text{min}}}}$
  13. Prime Factorization of Dyck Naturals
  14. Generalization to All Minimal Natural Interpretations
  15. The Standard Minimal RPF Language Reconsidered
  16. ...and 12 more sections

Key Result

Theorem 2.1

The language $\mathcal{D}_{r_{\text{min}}}$ is a proper subset of the Dyck language $\mathcal{D}$.

Figures (1)

  • Figure 1: Euler diagram illustrating the hierarchy of standard RPF languages.

Theorems & Definitions (128)

  • Remark
  • Remark
  • Remark
  • Remark
  • Remark
  • Remark
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 118 more