Recursive Prime Factorizations: Dyck Words as 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.