  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.
