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arXiv · 2505.14516

Prime Factorization in Models of PV$_1$

Ondřej Ježil
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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.

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Ondřej Ježil. 2025-05-20. Prime Factorization in Models of PV$_1$. https://doi.org/10.46298/lmcs-22(2%3A1)2026

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