mersenneforum.org Suggestion for new sieving software
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 2012-03-30, 10:34 #1 ATH Einyen     Dec 2003 Denmark 56278 Posts Suggestion for new sieving software Sascha77 came up with this problem/puzzle: http://www.mersenneforum.org/showpos...7&postcount=74 Which corresponds to: Does any factor of a mersenne number 2np+1 have n=k*p. I checked all of GIMPS 33.5 million known factors and found 1 solution only: http://www.mersenneforum.org/showpos...0&postcount=81 http://www.mersenneforum.org/showpos...6&postcount=83 so this like a new kind of Wieferich prime, just as rare. So I wrote some programs and have been looking for other examples, but my programming skill is limited to C with GNU MP, so they are not super fast. It would be very nice with an efficient siever like newpgen/srsieve for numbers of the form 2*k*p2+1, p prime (or k*p2+1 whichever is easier) over k and p intervals. Maybe if people like Prime95/geoff/rogue/Jean Penné who has the skills finds this interesting and have the time :) Last fiddled with by ATH on 2012-03-30 at 10:40
 2012-03-30, 11:19 #2 wreck     "Bo Chen" Oct 2005 Wuhan,China 7·23 Posts I think there should be infinite solutions to this problem, though it is rare. For any given p, Consider the minimum number of A (A>1) that satisfy A^p = 1 (mod p^2) maybe help.
 2012-03-30, 14:58 #3 LaurV Romulan Interpreter     Jun 2011 Thailand 22·3·739 Posts There are certainly an infinite amount of solutions for composite p, and composite factors 2*k*p^2+1, as for example p=28, 56, 100, 126, 140, 240, 256, 280, 312, 336, 364, 512, 624, 748 etc. The factors are prime for the trivial p=32, 64, 128, but also for other which are not powers of 2, like for example p=678: we have 2^678-1 is divisible by the prime 10113049, which is 2*11*678^2+1. Contrary, there are not many solutions for odd composite p either. The only one I found 339: 2^339-1 is divisible by the same prime, 10113049, which this time is 2*(11*4)*339^2+1, so k=44, considering that 678=2*339 (=2*3*113). But 2^113-1 does not have any factor f whose f-1 contain some square of 113.
2012-04-04, 13:03   #4
science_man_88

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Jul 2009
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Quote:
 Originally Posted by ATH Sascha77 came up with this problem/puzzle: http://www.mersenneforum.org/showpos...7&postcount=74 Which corresponds to: Does any factor of a mersenne number 2np+1 have n=k*p. I checked all of GIMPS 33.5 million known factors and found 1 solution only: http://www.mersenneforum.org/showpos...0&postcount=81 http://www.mersenneforum.org/showpos...6&postcount=83 so this like a new kind of Wieferich prime, just as rare. So I wrote some programs and have been looking for other examples, but my programming skill is limited to C with GNU MP, so they are not super fast. It would be very nice with an efficient siever like newpgen/srsieve for numbers of the form 2*k*p2+1, p prime (or k*p2+1 whichever is easier) over k and p intervals. Maybe if people like Prime95/geoff/rogue/Jean Penné who has the skills finds this interesting and have the time :)
here's an inefficient code to find them:

Code:
for(x=1,10000,for(k=1,x,if((2^x-1)%(2*k*x^2-1)==0,print(k","x))))
thought part of the slowness is because I don't break once I find one.

Last fiddled with by science_man_88 on 2012-04-04 at 13:08

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