h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

First things first, I like this entry, nice and neat. It's good the way it introduces the conjecture that every Fermat number is prime before telling us it's wrong. Of course, I already knew that.

However, I have to point out that none of the four skins renders "22n" as we want. Instead it looks like 2^(2n). I don't know what can be done, but it's misleading as it is.

Also, Polya's result isn't thatinteresting given that Euclid knew there were infinitely many primes. With a simpler proof too. Didn't Polya actually show there are infinitely many Fermat primes? Or was that someone else?

Hi Bagpuss. Sorry the superscript on the superscript didn't work for you. It looks OK on my browser.

I think Polya's proof is interesting. Of course Euclid had already proved it, but Polya's one is neat even if a little long winded, and it doesn't use 'proof by contradiction'. Some people are not 100% happy with that method of proof.

Well I'm in Internet Explorer here. Mozilla handles it fine. Sorry, I don't understand computers so it didn't occur to me that it could be the browser.

I guess you're right about Polya. There's an easier way without using contradiction, though. You can prove that any natural number has a prime larger than it by basically Euclid's argument, but without reductio ad absurdam.

You know, it'd be good if some of the undergrads here read this article. It's a good demonstration of why proof by example, even several examples, is wrong.