Speaking of Fermat number factorization - this page:
http://www.prothsearch.net/fermat.html
lists the current state-of-play.
Note that only F5-F11 have been completely factored. The smallest Fermat number with no known factors is F14, and the smallest Fermat number whose compositeness (ie factorizability) has not been proven is F33.
[for additional/preparatory reference - Wikipedia article on Fermat numbers: http://en.wikipedia.org/wiki/Fermat_number]
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The factorisation of Fermat numbers has fascinated me ever since I began studying number theory on the web.
I have read a good deal about these numbers, and believe that many people are unaware we are fortunate to have got as far as we have factoring Fermat numbers. The paper on the factorisation of the tenth and eleventh Fermat numbers, published in 1995, shows that we are exceedingly unlikely to completely factor larger Fermat numbers. The eleventh Fermat number could not ordinarily be completely factored by any existing algorithm.
It was completely factored because its second-largest factor is unusually small. The probability that a random 617-digit composite integer has so small a second-largest factor is only six percent.
If we turn to generalised Fermat numbers [b^(2^n)+1 where b ≠ 2^k] my prediction is born out. For example, though smaller than the eleventh "ordinary" Fermat number, the ninth generalised Fermat number in base 10 (10^512+1) is still not fully factored, nor is the eighth GFN in base 12.
Logically, if we cannot factor these, we are not going to successfully factor numbers with twice as many decimal digits.
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