On Williams Numbers with Three Prime Factors

Abstract

Let $a\in \mathbb{Z}\setminus \{0\}$. A positive squarefree integer $N$ is said to be an $a$-Korselt number ($K_{a}$-number, for short) if $N\neq a$ and $p-a$ divides $N-a$ for each prime divisor $p$ of $N$. By an $a$-Williams number ($W_{a}$-number, for short) we mean a positive integer which is both an $a$-Korselt number and $(-a)$-Korselt number. This paper proves that for each $a$ there are only finitely many $W_{a}$-numbers with exactly three prime factors, as conjectured in 2010 by Bouallegue-Echi-Pinch.