Generalized Schatunowsky theorem in a weak arithmetic

TL;DR

This paper extends a classical number theory theorem to weak arithmetic systems, demonstrating the finiteness of certain prime-related sets under generalized conditions.

Contribution

It proves that the generalized Schatunowsky theorem holds within weak arithmetic frameworks, broadening its applicability.

Findings

The theorem's generalization is valid in weak arithmetic.

Finiteness of specific prime-related sets is established.

Supports the robustness of number theoretic results in weaker logical systems.

Abstract

Schatunowsky's 1893 theorem, that 30 is the largest number all of whose totatives are primes, has been recently generalized by Kaneko and Nakai. In its generalized form, it states the finiteness of the set of all positive numbers $n$, which, for a fixed prime $p$, have the property that all of $n$'s totatives that are not divisible by any prime less than or equal to $p$ are prime numbers. It is this generalized form that we show holds in a weak arithmetic