Recursive Prime Factorizations: Dyck Words as Numbers
Formal Languages and Automata Theory
2026-02-18 v4 Number Theory
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.
Keywords
Cite
@article{arxiv.2102.02777,
title = {Recursive Prime Factorizations: Dyck Words as Numbers},
author = {Ralph L. Childress},
journal= {arXiv preprint arXiv:2102.02777},
year = {2026}
}
Comments
Minor corrections and clarifications after the previous major revision. No changes to results or references. Ancillary files added