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Old 2008-09-19, 00:43   #1
Syd
 
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Sep 2008
Krefeld, Germany

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Default Mersenne-version of the quadratic sieve

Hi everyone,

i tried to modify the quadratic sieve for mersenne primes, can anyone please comment on this one?

All factors of mersenne numbers have the form (k*x+1), for x being the exponent.

So:
2^x-1 = ((k+t)*x+1)((k-t)*x+1)

2^x-1 = k^2*x^2 - t^2*x^2 + 2kx + 1

k^2*x + 2kx - (2^x-2)/x = t^2*x

now we need a k to make the left side positive and start with

k=(sqrt(2^x-1)-1)/x

and go up from that until the left side is divisible by x (any way to improve that?)
When found, we increase k by x in the next steps and get a divisable one every step.

Divise all them by x and continue:

Factorize all numbers found that way and cut out double prime factors, multiply all remaining factors and put that number in a big list.

Check every new number coming in against all the others on the list in that way:
is new*old/ggt(new,old)^2 < new? Then add new to the list. x*y/ggt(x,y)^2 wipes out all common prime factors.

If we finally get a zero we have a factor.

Will this algorithm work fast? Will it even work?

Thanks for reading
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