Quote:
Originally Posted by Raman
I don't think that it will be possible to factor completely any more Fermat numbers with the current technology.
Either computational power needs to be increased either by using multiple computers or by using micro processor speed or improvements with in factoring algorithms need to be made or quantum computers  Shor's algorithm should need to be built out effectively and then efficiently up (Integer Factorization Problem Improvements ⇔ Discrete Logarithm Problem Improvements)! (Google Search Engine features much more symbols than character map).
If 2,1024+ did not drop off a small 40digit factor, it would be only factored recently, right now. 2,1039, first kilo bit SNFS factorization had been done how ever as early as Monday 21 May 2007.
It had been very lucky enough that the 564 digit cofactor of 2,2048+ is being prime number candidate, or it would also have been infeasible  computationally out of reach too, right now! Given that fixed penultimate prime factor candidate of 2,2048+ which is being a smaller number candidate  not  not  larger number candidate!

In reality it is just a matter of luck. We find a few fermat factors each year. Any one of these could leave a prp cofactor. This happens for Mersenne numbers. The issue is that the fermat numbers get bigger much quicker and less factors are found.
I don't know whether the cofactor has to be proven prime here. That would limit us to rather a small number of candidates.