Prime Factorization in Models of PV$_1$
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910129297817600 |
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| author | Ježil, Ondřej |
| author_facet | Ježil, Ondřej |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime Factorization in Models of PV$_1$ Ježil, Ondřej Logic Logic in Computer Science 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. |
| title | Prime Factorization in Models of PV$_1$ |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2505.14516 |