Variants of Korselt's Criterion

TL;DR

This paper generalizes Korselt's criterion by proving the existence of infinitely many integers with prime factors satisfying specific divisibility conditions related to an arithmetic progression, extending properties of Carmichael numbers.

Contribution

It introduces a new generalization of Korselt's criterion and proves the existence of infinitely many such integers under certain conditions.

Findings

Infinitely many integers n exist with prime factors p satisfying p - a | n - a.

Generalization of Carmichael numbers to broader divisibility conditions.

Establishes conditions under which these integers occur infinitely often.

Abstract

Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer $a$, there are infinitely many $n\in \mathbb N$ such that for each prime factor $p|n$, we have $p-a|n-a$. This can be seen as a generalization of Carmichael numbers, which are integers $n$ such that $p-1|n-1$ for every $p|n$.