Prime time

I remember that, when I first came across Euclid's proof that there is no highest prime number, it just struck me as the goods - the simplicity and economy of it.

Stating the proof in a 'literary' way so as not to put anyone off, it goes roughly like this (and I will doubtless make some technical error here, but then you can pick me up on it):

(Prime numbers being numbers divisible only by themselves and 1…)
> Assume there is a highest prime number, p.
> Then, multiply all the primes up to and including p together and add 1 to the product, obtaining the number n.
> This number, n, is now either itself a prime, or it isn't.
> If n is a prime, then there is a prime higher than p.
> And if n isn't a prime, then as n cannot be divisible by any of the primes up to and including p (because dividing any of these into n will leave a remainder of 1), n must be divisible by a prime lying between p and n; so, again, there is a prime higher than p.
> By the same method, there must also be a prime higher than n, and so on.

Or: see, more briefly, here.

It also intrigued me when I first read of Goldbach's Conjecture - that every even number is the sum of two primes - and read that though no counter-instance to this conjecture had yet been discovered, neither was there a proof for it. How odd! Whether this is still the state of affairs obtaining I couldn't say.

In my thirties I took a course in the evenings to get me to the equivalent of A-level or university entrance maths. One of the new things I learned then was some elementary computer programming, and I devised an algorithm for generating primes. I still have it somewhere.

Why am I telling you all this? Why not? It's Saturday. I read a piece in the Graun a few days ago by Marcus du Sautoy, about maths education. He said:

One certainly can't shy away from teaching the technical side. Learning a musical instrument or a language requires a certain amount of tedious hard graft, too. But why can't this be balanced by learning about the great ideas, history and people that make up the true story of mathematics.

In making this argument du Sautoy focused, among other things, on the fascination of prime numbers. You better believe it.