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Prime Factorization in Models of PV$_1$

Ondřej Ježil

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.

Prime Factorization in Models of PV$_1$

Abstract

Assuming that no family of polynomial-size Boolean circuits can factorize a constant fraction of all products of two -bit primes, we show that the bounded arithmetic theory , even when augmented by the sharply bounded choice scheme , cannot prove that every number has some prime divisor. By the completeness theorem, it follows that under this assumption there is a model of that contains a nonstandard number which has no prime factorization.

Paper Structure

This paper contains 6 sections, 12 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Student-teacher protocols
  4. The unprovability in $\text{PV}_1$
  5. The Unprovability with the Sharply Bounded Choice Scheme
  6. Concluding remarks

Key Result

Theorem 2.4

Let $\varphi(x,y,z)$ be an open formula. If then there is a number $c\in\mathbb{N}$, and $\text{PV}$-symbols $f_1,\dots,f_c$ such that Moreover, there is a polynomial-time function $s$, such that for any input $x$ and any teacher $t$ which is $\varphi$-correcting on the input $x$, we have that the computation of the protocol $(s,t,c)$ on an input $x$ contains some $y_i$ which satisfies $\mathbb{

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: The KPT theorem krajicek1991bounded
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • ...and 21 more