On the Rigidity of Prime-Preserving Bijections

Abstract

We consider multiplicative prime-preserving bijections
    \( f: \mathbb{N} \to \mathbb{N} \).
    We prove that if such an \(f\) satisfies a uniform gap condition
    on primes \(p\) for which \(f(p)\neq p\), and if the set of these
    moved primes has a divergent reciprocal sum, then \(f\) can permute
    only finitely many primes.
    Contrary to earlier claims, such bijections may nontrivially permute
    infinitely many composites associated with the finitely many moved primes.
    Our result refines previous arguments, thereby addressing a nuanced
    rigidity question in arithmetic systems.