On the finiteness of Carmichael numbers with Fermat factors and $L=2^{\alpha}P^2$

TL;DR

This paper classifies all Carmichael numbers with a Fermat prime factor where the least common multiple of p-1 is of the form 2^α P^2, identifying exactly eleven such numbers.

Contribution

The paper explicitly determines all Carmichael numbers with a Fermat prime factor satisfying the specific L=2^α P^2 condition, providing a complete classification.

Findings

Identified eleven Carmichael numbers with the specified property.

Established the structure of L for these Carmichael numbers.

Contributed to the understanding of the distribution of Carmichael numbers.

Abstract

Let $m$ be a Carmichael number and let $L$ be the least common multiple of $p-1$, where $p$ runs over the prime factors of $m$. We determine all the Carmichael numbers $m$ with a Fermat prime factor such that $L=2^{\alpha}P^2$, where $k\in \mathbb{N}$ and $P$ is an odd prime number. There are eleven such Carmichael numbers.