## Re: [PrimeNumbers] Examination of Feitsma's ideas for finding all base-2 Fermat pseudoprimes <2^64

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• ... That indeed was one of my sources of unhappiness. ... Even after having an aha! moment reading your wonderfully clear explanation, I went back to the
Message 1 of 37 , Mar 14, 2013
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--- On Thu, 3/14/13, WarrenS <warren.wds@...> wrote:
> Feitsma then gets more time-savings by claiming
> "we need only consider q<=x^(2/3)"
> unfortunately without saying what the hell "x" is, so I have
> no idea what he is talking about?

That indeed was one of my sources of unhappiness.

> Hence we can rename f*s as the new q and q/f as the new s,

Even after having an aha! moment reading your wonderfully clear explanation, I went back to the Feitsma page, and my first reaction was that I still couldn't map what he was writing onto what you've just said. On the 3rd take, with pen and paper in hand, I convinced myself he did mean what you wrote.

Thanks for going through this with a fine-toothed comb, Warren, it's much appreciated.

Phil
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• ... Apologies, my Pari/GP script-fu is definitely not on par with DJB s. I just wanted to provide a script so that people could plug and chug an r,p pair to
Message 37 of 37 , Mar 14, 2013
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On 3/13/2013 5:47 AM, Phil Carmody wrote:
> No I couldn't. That's so overly verbose and redundant it makes me twitch, I can
> barely bring myself to repeat it!
> "lift(Mod(p*s, lift(znorder(Mod(2,s))))) == 1" is just
> "p*s % znorder(Mod(2,s)) == 1"
> Having 3 exit conditions to the loop is overkill too.

Apologies, my Pari/GP script-fu is definitely not on par with DJB's. I just
wanted to provide a script so that people could plug and chug an r,p pair to see
what psp's would be generated. Also, I didn't know that the % (mod) operator
still worked in Pari. I thought everything had to be done with Mod(). Thanks
for that insight.

> The latter worries me a bit, as it might imply wasted effort. I'm trying to
> picture how these duplicates arise. Given a n, the maximal prime factor p|s is
> uniquely defined, and r as order_2(p) is uniquely defined. Therefore n can only
> appear with pair (r,p)?

And apologies here too, I mis-remembered a statement from his Category S page
and mis-spoke by applying it to the Category E psp's.

On 3/14/2013 11:02 AM, WarrenS wrote:
> 2. Consulting the Cunningham project pages,
> http://homes.cerias.purdue.edu/~ssw/cun/index.html
> every Mersenne-form number 2^r - 1 now is fully factored if
> r<929. Apparently the first two open cases are r=929 and 947
> yielding 214 and 217 digit numbers to factor.

An update here: M929 has been factored, and a group of people have already
started factoring M947. You can find the factor for M929 here:
http://homes.cerias.purdue.edu/~ssw/cun/page125
And, you can see who is factoring which Cunningham number here:
http://homes.cerias.purdue.edu/~ssw/cun/who
And more importantly, you can find all known*1 factors for all important*2
numbers in the online factor database here:
factordb.com
Once there, you can type in 2^929-1, and it will show you all the factors of
that number and that it is Fully Factored (FF). Currently, you can type in
2^947-1 and see that it is a CF, composite number, with factors known, but not
yet fully factored.

*1 = All known factors that have been stored into the factordb.
*2 = All numbers that people are interested in and store in the factordb.

Also, the factordb stores prime numbers too. Below 300 digits it will just
prove the number prime, and above that it will accept Primo certificates and
verify them locally.

-David C.
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