From: Christian Szegedy (szegedy@or.uni-bonn.de)
Date: Wed Jun 19 2002 - 08:09:03 MDT
Lee Corbin wrote:
> But you have just pointed out something VERY WEIRD! If you
> take the 5th Fermat prime, 65537, and make my same mistake
> in its calculation to get 65535, then *that* number factors
> into a product consisting of ALL the previous Fermat primes!
> I wonder what 3 * 5 * 17 * 257 * 65537 plus 2 is. (Perhaps
> similar ratiocinations led Fermat to conjecture that all his
> Fn were prime.) Or was that just a coincidence?
It is neither weird nor coincidence. It is a complete triviality:
2^(2^n)-1 = 3 * 5 * ... (2^(2^(n-1))+1)
as simple induction shows.
However this product does need to be a prime factoriziation as
2^(2^n)+1 does not need to be a prime number.
Fermat came to his (false) conjecture by noticing
the trivial fact that a number of the form 2^n+1 can only be
a prime if n is a power of two. He has simply asked whether the
converse is also true.
Best regards, Christian
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