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Old 2020-11-05, 13:05   #7
Dr Sardonicus
 
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Feb 2017
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Things any do-it-yourselfers should keep in mind

Ask yourself, "Do I really want to do this myself?" If you're sure...

1) Understand the job you are trying to do
2) Know the techniques for doing it
3) Get (borrow, rent, buy) good tools and equipment
4) Make sure you know how to use them

Now, I'm unlikely to go searching for factors of Fermat numbers myself. Even so, I believe I can understand (1) and, to some degree, (2) for this task. So when I see a new announcement, I can have some appreciation of the hard work that went into making it happen.

The Fermat numbers being looked at are so large, about the only plausible way to find factors is to search through "small" candidates and hope you get lucky.

About all I know in this regard is that, for n > 1, all factors of Fn are congruent to 1 (mod 2n+2). So you look at N = k*2n+2 + 1 and see whether N divides Fn. This would involve repeated squaring starting with Mod(2, N).

At this point, I ask myself, "Is it worthwhile to weed out N-values with small prime factors to avoid doing this test on them?" My guess is, "Seems pretty likely."

So, I'm guessing you need something that can do some fast sieving, and something that can handle repeated squarings (mod N) for largish-to-large N quickly.

If I were seriously interested in trying it myself, I would inquire further into what the right tools for the job might be, and how I can get the use of them within my budget.

The regular contributors to the various forums and threads devoted to specific factoring efforts know this stuff as well as anyone on the planet. I'm sure they would be more than happy to help someone who asks respectfully. Part of that respect is IMO putting some effort of your own into learning (1) and (2) above before seeking help.

Just my
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