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.

翻译：假设不存在多项式规模布尔电路族能够分解常数比例的所有两个 $n$ 比特素数的乘积，我们证明有界算术理论 $\text{PV}_1$，即使增强以锐利有界选择方案 $BB(Σ^b_0)$，也无法证明每个数都具有某个素因子。根据完备性定理，在此假设下，存在一个 $\text{PV}_1$ 的模型 $M$，其中包含一个没有素数分解的非标准数 $m$。