User:Sirac ascade

 Fermat primes and the Pellian  At the time Fermat mentioned these numbers they had no mathematical significance ; he was looking for a function that yielded primes all the time. It is also well known that he initiated the study of the Pellian.If we combine these two facts we will see why he made his mistaken statement about fermat primes and how close he came to finding the function he was looking for. Under ordinal number N the table below yields the fractional approximations YN/XN for the square root of 2 obtained by solving the Pellian for this number.

Statement: Whenever N is an n-th power of 2 the numerator YN is a) a fermat prime or b) a prime of the form [(2^2n)R^2]+1 or c) if composite one of its divisors must be a fermat prime (easily demonstrable)

Besides the trivial 3 and 17 above, when N=32 and n=5 we have YN=886,731,088,897 = 257x3,450,315,521 When N = 2048 and n = 11 we have YN = 65537x ....(763 digit number)

If there are no more fermat primes, we will have to conclude that all Yn beyond n = 11 are all prime (the function Fermat was looking for). --20:29, 3 March 2011 (UTC)Sirac ascade (talk)