Year: 2026
Type: article
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 $BB(Σ^b_0)$, 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.
Author: Ondřej Ježil
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Awards: G5362880987, G6203253224, G7595993604
OpenAlex ID: W4415024037
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