Fermat numbers

Fermat numbers are these:

F_n=2^{2^n}+1

Are they all prime? If so, we’d have a truly incredible proof. Unfortunately, they are not. While F_0,F_1,F_2,F_3,F_4 are prime, F_5 is not (it’s divisible by 641). Up through around F_{32} 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 F_{11} is 319,489. The smallest prime dividing F_{10} is 45,592,577. Anyway, this isn’t relevant to us.

Here’s a fun fact I won’t prove. You can prove it via induction.

F_n=F_0\cdot F_1\cdot\dots\cdot F_{n-1}+2.

So suppose p is some prime dividing F_j, and n>j. Obviously p is odd, and since F_j shows up in the product, F_n cannot be divisible by p. 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 k Fermat numbers, I also have a collection of at least k primes. Since there are infinitely many Fermat numbers, there must be infinitely many primes.

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One Response to Fermat numbers

  1. Pingback: No proof today « Andy Soffer

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