Fermat Primes are primes in the form of
Fermat (http://en.wikipedia.org/wiki/Fermat) conjectured that this is always prime for all n.

Then Euler proved that for n=5, it is divisible by 641. Now, all Fermat Numbers have been proven up to (i.e. n = 2167797 in the equation ) and all are in the form .

Now the question is, "is composite for all ?"

My idea is to find a correlation between the prime factor n and the exponent of the Fermat Prime? And if so, then I can prove using Sierpinski Numbers (http://en.wikipedia.org/wiki/Sierpinski_number) which are numbers in the form for all odd ks. So if you cannot get a prime factor from that form, then there is no prime factor for that Fermat Number, and if I can prove the opposite, then I can prove that there in no Fermat Prime for n>4!

Best of Luck on this Problem!
Zach