On the Classification and Algorithmic Analysis of Carmichael Numbers

TL;DR

This paper analyzes Carmichael numbers, classifies them based on their properties, and introduces a new, efficient algorithm combining probabilistic and deterministic tests to distinguish Carmichael numbers from primes.

Contribution

It provides a new classification of Carmichael numbers and develops a novel algorithm that improves detection accuracy and efficiency.

Findings

Classification of Carmichael numbers based on prime factors

Development of a combined probabilistic-deterministic detection algorithm

Enhanced accuracy in distinguishing Carmichael numbers from primes

Abstract

In this paper, we study the properties of Carmichael numbers, false positives to several primality tests. We provide a classification for Carmichael numbers with a proportion of Fermat witnesses of less than 50%, based on if the smallest prime factor is greater than a determined lower bound. In addition, we conduct a Monte Carlo simulation as part of a probabilistic algorithm to detect if a given composite number is Carmichael. We modify this highly accurate algorithm with a deterministic primality test to create a novel, more efficient algorithm that differentiates between Carmichael numbers and prime numbers.