The Cosmic Web

Yesterday I asked the question “how many prime numbers are there?” (a prime number being one which is a multiple of exactly two numbers: itself and 1). I promised that I’d prove it using ‘proof by contradiction’, the idea that if we start from some assumption and use it to reach a conclusion we know to be false, then our assumption must have been false too. Here’s the proof.

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At the genesis of this blog I threatened to write some posts about maths: here’s one of them. There’s nothing very complicated in it - certainly no esoteric mathematical notation or necessary knowledge - so I hope you can follow me.

I’m going to begin today by asking a question and discussing the method I’ll use to answer it. Then tomorrow I’ll show you the full proof.

How many primes are there?

A prime number is one which is a multiple of exactly two numbers: itself and 1. So 2, 3 and 5 are all prime, but 4 and 6 aren’t since they’re both multiples of 2.

As you climb through the integers, primes become further and further apart. There are four primes between 1 and 10 (2,3,5 and 7) but only one between 1001 and 1010 (1009).

Does this thinning of the primes continue until we reach a last prime, with no more after that point? If so, how many primes are there and what is the biggest? Or do the primes continue for ever, an infinite number of them? That’s not impossible just because they thin out: the square numbers also get more spread out as they go, but there is no biggest square number.

This seems like an interesting question to ask (at least to me it does - I hear tell of some strange people who do not find numbers fascinating). But how can we answer it? If there are finitely many primes and we manage to write down every single one, how will we know there isn’t another one hiding somewhere? And if there are infinitely many, how can we possibly prove that?

We can prove it, and I’m going to do so using something called proof by contradiction.

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