May 14, 2012 1 Comment
Fermat numbers are these:
Are they all prime? If so, we’d have a truly incredible proof. Unfortunately, they are not. While are prime, is not (it’s divisible by ). Up through around there are no more Fermat numbers which are prime. It’s weird though, because the primes which do divide Fermat numbers seem to get big fast. The smallest prime dividing is . The smallest prime dividing is . Anyway, this isn’t relevant to us.
Here’s a fun fact I won’t prove. You can prove it via induction.
So suppose is some prime dividing , and . Obviously is odd, and since shows up in the product, cannot be divisible by . This means that once a Fermat number is divisible by a prime, no other Fermat number will be ever again (in some sense, this is why the primes dividing Fermat numbers grow so quickly). Anyway, this means that if I have a collection of Fermat numbers, I also have a collection of at least primes. Since there are infinitely many Fermat numbers, there must be infinitely many primes.