The automaticity of the set of primes

TL;DR

This paper investigates the automaticity of the set of prime numbers, establishing a lower bound that approaches the maximum possible automaticity, thereby revealing the complexity of recognizing primes with finite automata.

Contribution

It provides a new lower bound on the automaticity of primes, advancing understanding of their computational recognition complexity.

Findings

Lower bound on prime set automaticity is approximately x times an exponential decay factor.

Automaticity of primes is shown to be close to the maximal possible automaticity.

The result connects number theory with automata theory, highlighting the complexity of prime recognition.

Abstract

The automaticity $A(x)$ of a set $\mathcal{X}$ is the size of the smallest automaton that recognizes $\mathcal{X}$ on all words of length $\leq x$. We show that the automaticity of the set of primes is at least $x\exp\left(-c(\log\log x)^2\log\log\log x\right)$, which is fairly close to the maximal automaticity.