Kleene Star of the Primes is not Regular in Any Base

TL;DR

This paper proves that the set of prime numbers expressed in any base, when repeated any number of times, does not form a regular language, extending a classical result about primes not forming regular languages.

Contribution

It establishes that the Kleene star of the primes in any base is not regular, strengthening the known result that primes themselves do not form regular languages.

Findings

$P_b^*$ is not regular for any base $b$

Extends classical non-regularity of primes to their Kleene star

Supports the complexity of prime representations in formal language theory

Abstract

Let $P_b$ denote the primes expressed in base-$b$. In this note, we prove that $P_b^*$ is not regular. This strengthens a classical result that $P_b$ is not regular, due to Minsky and Papert in 1966.