Pseudoprimes

Abstract

This chapter returns to the question of deciding whether a given odd number m is prime. The a-pseudoprime test of Chapter 10D will not work on Carmichael numbers. We first describe a recent idea of Alford which shows that there are many Carmichael numbers. Then we develop the strong a-pseudoprime test and present a theorem of Rabin that every composite number m fails the strong a-pseudoprime test for most a < m. We conclude this chapter with a proof of a weak version of Rabin’s theorem; the next chapter gives a proof of the strong version of Rabin’s theorem.