Prime Factorization in Models of PV$_1$

TL;DR

The paper explores the limitations of the bounded arithmetic theory PV$_1$ in proving prime factorization properties, assuming certain complexity-theoretic conjectures about polynomial-size circuits.

Contribution

It demonstrates that under a complexity assumption, PV$_1$ cannot prove every number has a prime divisor, implying the existence of models with nonstandard numbers lacking prime factors.

Findings

Assuming no polynomial-size circuit family can factor many products of two primes.

PV$_1$ cannot prove that every number has a prime divisor under this assumption.

Existence of a model of PV$_1$ with a nonstandard number without prime factors.

Abstract

Assuming that no family of polynomial-size Boolean circuits can factorize a constant fraction of all products of two $n$-bit primes, we show that the bounded arithmetic theory $\text{PV}_1$, even when augmented by the sharply bounded choice scheme $BB(\Sigma^b_0)$, cannot prove that every number has some prime divisor. By the completeness theorem, it follows that under this assumption there is a model $M$ of $\text{PV}_1$ that contains a nonstandard number $m$ which has no prime factorization.