The examples of complexity that I have shown so far in this book are almost all completely new. But the first few hundred primes were no doubt known even in antiquity, and it must have been evident that there was at least some complexity in their distribution.
However, without the whole intellectual structure that I have developed in this book, the implications of this observation—and its potential connection, for example, to phenomena in nature—were not recognized. And even though there has been a vast amount of mathematical work done on the sequence of primes over the course of many centuries, almost without exception it has been concerned not with basic issues of complexity but instead with trying to find specific kinds of regularities.
Yet as it turns out, few regularities have in fact been found, and often the results that have been established tend only to support the idea that the sequence has many features of randomness. And so, as one example, it might appear from the pictures on the previous page that (c), (d) and (e) always stay systematically above the axis. But in fact with considerable effort it has been proved that all of them are in a sense more random—and eventually cross the axis an infinite number of times, and indeed go any distance up or down.
So is the complexity that we have seen in the sequence of primes somehow unusual among sequences based on numbers? The pictures on the facing page show a few other examples of sequences generated according to simple rules based on properties of numbers.
And in each case we again see a remarkable level of complexity.
Some of this complexity can be understood if we look at each number not in terms of its overall size, but rather in terms of its digit sequence or set of possible divisors. But in most cases—often despite centuries of work in number theory—considerable complexity remains.
And indeed the only reasonable conclusion seems to be that just as in so many other systems in this book, such sequences of numbers exhibit complexity that somehow arises as a fundamental consequence of the rules by which the sequences are generated.