The Distribution of Primes
Although prime numbers of the Mersenne form are few and far between, we must remember that they are not representative of primes in general and have achieved special interest only because efficient methods for testing them for primeness have been devised. In this chapter we wish to consider all the known prime numbers in increasing order 2, 3, 5, 7, 11, 13, … onward and ever upward. Is there any pattern or regularity, for example, in their appearance? We must first recognize that calculating all the primes, without missing one, is a far more arduous task that just looking for a particular large but specific prime number. As a result we shall not be concerned here with prime number data out to thousands of digits, as we were with the Mersennes. Nevertheless, modern computer facilities have made it possible to generate every prime up to values of more than a trillion (that is 1012) so there are certainly plenty of raw data to maintain our interest. Two seemingly contradictory facts can be established from this data. Firstly, the primes appear to obey no other law than that of chance, meaning that it is impossible to predict in advance where the next one will be except by testing all the subsequent numbers in order. That much is rather discouraging and is, perhaps, what we have come to expect of prime numbers. Secondly, however, in complete contradiction, they exhibit a stunning regularity when looked at ‘from a distance’, and in this respect the laws governing their general behavior are obeyed with great precision.