Carmichael numbers

This paper begins with a short description of Carmichael numbers, and the characterization of Carmichael numbers due to Chernick and the proof of this characterization. This along with Carmichael's original work leads naturally into a discussion of some bounds on Carmichael numbers in terms of the primes in their decomposition. Some bounds are presented and some examples given that show Carmichael numbers that attain these bounds. Next, Chernick's universal forms are examined, and a general universal form with an arbitrary number of linear factors is established. Some heuristic evidence is presented that supports the conjecture of the existence of infinitely many Carmichael numbers.