Connections on the Rational Korselt Set of pq

TL;DR

This paper investigates the structure of Korselt bases related to the product of two distinct primes, establishing conditions under which certain bases generate others and characterizing the set of integer Korselt bases.

Contribution

It proves that for the product of two primes, all non-integer Korselt bases generate others, and characterizes the set of integer Korselt bases, specifically identifying when it contains only one element.

Findings

If the rational Korselt set excluding integers is empty, then the integer Korselt set contains only q+p-1.

Each pq-Korselt base outside q+p-1 generates additional bases in the rational set.

The paper characterizes the structure of Korselt bases for the product of two primes.

Abstract

For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A}\setminus \{0,N\}$ verifying $\alpha_{2}r-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $r$ of $N$. The set $\mathbb{A}$-$\mathcal{KS}(N)$ is called the set of $N$-Korselt bases in $\mathbb{A}$. Let $p, q$ be two distinct prime numbers. In this paper, we prove that each $pq$-Korselt base in $\mathbb{Z}\setminus\{ q+p-1\}$ generates other(s) in $\mathbb{Q}$-$\mathcal{KS}(pq)$. More precisely, we will prove that if $(\mathbb{Q}\setminus\mathbb{Z})$-$\mathcal{KS}(pq)=\emptyset$ then $\mathbb{Z}$-$\mathcal{KS}(pq)=\{ q+p-1\}$.