Quote:
Originally Posted by siegert81
I'm assuming that various researchers have attempted to factor them and that their attempts have been considerably deep. Given they have effectively ruled out many small potential factors, what are the "probabilities" that these numbers are prime?

It might have been "deep" factoring considering the amount of work it took, but comparing to the size of F33, F34 and F35 it is close to nothing:
http://www.prothsearch.com/fermat.html#Search
For F33: k*2
^{35} searched to k=4.8*10
^{17} and k*2
^{36} searched to k=7*10
^{17} and lower for k*2
^{37}, k*2
^{38} etc.
Now: 7*10
^{17} * 2
^{36} is about 2
^{95} so 95 bits. Compare that to F33 = 2
^{8,589,934,592} + 1. So it has been factored (almost) up to the 90,000,000th root of F33.
For a Mersenne number in the 90M range the 90,000,000th root is 2, so that would correspond to no factoring at all on a 90M exponent.
For F34 and F35 it is even worse: ~2
^{95} vs 2
^{17,179,869,184} and 2
^{34,359,738,368}