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 $BB(Σ^b_0)$, 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.

@article{paperbot4007,
  title = {Prime Factorization in Models of PV$_1$},
  author = {Ondřej Ježil},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 22, Issue 2},
  year = {2026},
  doi = {10.46298/lmcs-22(2:1)2026}
}