Recursive Prime Factorizations: Dyck Words as Numbers

TL;DR

This paper introduces a novel non-positional numeral system using Dyck words to represent natural and rational numbers through recursive prime factorization, exploring its structure and potential research avenues.

Contribution

It presents a new class of numeral systems based on Dyck words, extending prime factorization concepts to non-positional representations and analyzing their properties.

Findings

Two subsets of Dyck language can uniquely represent all natural numbers.

A superset of rational numbers can be represented within this system.

Discussion of Dyck-complete languages where all Dyck words represent numbers.

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.