The Sequence of Primes

In the sequence of all possible numbers 1, 2, 3, 4, 5, 6, 7, 8, ... most are divisible by others—so that for example 6 is divisible by 2 and 3. But this is not true of every number. And so for example 5 and 7 are not divisible by any other numbers (except trivially by 1). And in fact it has been known for more than two thousand years that there are an infinite sequence of so-called prime numbers which are not divisible by other numbers, the first few being 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

The picture below shows a simple rule by which such primes can be obtained. The idea is to start out on the top line with all possible numbers. Then on the second line, one removes all numbers larger than 2 that are divisible by 2. On the third line one removes numbers divisible by 3, and so on. As one goes on, fewer and fewer numbers remain. But some numbers always remain, and these numbers are exactly the primes.

Given the simplicity of this rule, one might imagine that the sequence of primes it generates would also be correspondingly simple. But just as in so many other examples in this book, in fact it is not. And indeed the plots on the facing page show various features of this sequence which indicate that it is in many respects quite random.

A filtering process that yields the prime numbers. One starts on the top line with all numbers between 1 and 100. Then on the second line, one removes numbers larger than 2 that are divisible by 2—as indicated by the gray dots. On the third line, one removes numbers larger than 3 that are divisible by 3. If one then continues forever, there are some numbers that always remain, and these are exactly the primes. The process shown is essentially the sieve of Eratosthenes, already known in 200 BC.

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From Stephen Wolfram: A New Kind of Science [citation]