You might ask, after thousands of years, haven’t those clever mathematicians come up with better and more sophisticated methods of testing for primality? The answer is yes. In 2002, a relatively quick way was discovered to test if a given number is prime. The so-called‘AKS primality test’does not, however,provide the factorization of the given number if it happens to be composite. The problem ofnding the prime factors of a given number, although in principle solvable by trial, still seems practically intractable for extremely large integers, and for that reason it forms the basis of much ordinary encryption on the Internet, a subject to which we will return in Chapter 4. Before that we shall, in the next two chapters, look a little more closely at primes and factorization.