# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1702.08066